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Page 1:  · Web viewNature of the Roots of Quadratic Equations . 98 Graphing Quadratic Functions from the Vertex Graphing Quadratic Functions from the Standard Graphing Cubic, Square Root,

A Resource Aligned with New York State Algebra 1 Module 4

Page 2:  · Web viewNature of the Roots of Quadratic Equations . 98 Graphing Quadratic Functions from the Vertex Graphing Quadratic Functions from the Standard Graphing Cubic, Square Root,

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Table of Contents

Multiplying Polynomial Expressions ………………………………………………………………………………………………… 5

Factoring GCF and Difference of Squares ………………………………………………………………………………………… 12

Factoring Trinomials ……………………………………………………………………………………………………………………….. 18

Factoring Quadratic Expressions with a Leading Coefficient ……………………………………………………………. 24

The Zero Product Property …………………………………………………………………………………………………………….. 29

Quadratic Equations ………………………………………………………………………………………………………………………. 35

Creating and Solving Quadratic Equations ……………………………………………………………………………….…….. 41

Exploring the Symmetry in Graphs of Quadratic Functions ……………………………………………………….……. 48

Graphing Quadratic Functions from Factored Form ………………………………………………………………….……. 55

Interpreting Quadratic Functions from Graphs and Tables …………………………………………………………….. 64

Completing the Square ……………………………………………………………………………………………………………….….. 70

Completing the Square with Leading Coefficients …………………………………………………………………………… 76

Solving Quadratics by Completing the Square …………………………………………………………………………….…… 80

Simplifying Radicals ………………………………………………………………………………………………………………………… 87

Quadratic Formula ……………………………………………………………………………………………………………………….…. 92 Nature of the Roots of Quadratic Equations ……………………………………………………………………………………. 98

Graphing Quadratic Functions from the Vertex Form ……………………………………………………………………. 104

Graphing Quadratic Functions from the Standard Form ……………………………………………………………… 109

Graphing Cubic, Square Root, and Cube Root Functions ……………………………………………………………….. 117

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Interpreting Quadratic Functions and Their Graphs ……………………………………………………………………… 122

Key Features of Quadratic Graphs ……………………………………………………………………………………………….. 126

End of Module Review (Sample Regents Exam Questions) ………………………………………………………….. 130

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Name ______________________ Multiplying Polynomial ExpressionsShow all your work! Opening Exercise

Write expressions for the areas of the two rectangles in the figures given below.

Now write an expression for the area of this rectangle:

8

2 z 2 z

z

8

2 z

z

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Page 6:  · Web viewNature of the Roots of Quadratic Equations . 98 Graphing Quadratic Functions from the Vertex Graphing Quadratic Functions from the Standard Graphing Cubic, Square Root,

Multiplying Polynomial ExpressionsLearning Target: I can multiply polynomial expressions.

Example: Multiply Two Binomials

Fill in the table to identify the partial products of (x+2)( x+5). Then write the product of (x+2)( x+5) in standard form.

Multiply Two Binomials—Without the Aid of a Table

Multiply: (x+2)( x−5)

Example: The Difference of Squares

Find the product of (x+2)( x−2). Use the distributive property to distribute the first binomial over the second.

With the use of a table:

Without the use of a table:

x +¿ 5x

+¿

2

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Example: The Square of a Binomial

To square a binomial, like (x+3)2, multiply the binomial by itself.

Square the binomial.

A. (a+6)2

B. (5−w)2

C. (a+b)2

D. (a−b)2

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Exercises: Multiply each pair of binomials

1. (x+1)(x−7)

2. ( x+9 )(x+2)

3. (x−5)(x−3)

4. (x+ 152 )(x−1)

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Name ______________________ Multiplying Polynomial ExpressionsShow all your work! Problem Set

1. Multiply:a. (n−5)(n+5)

b. (4− y)(4+ y )

c. (k+10)2

d. (4+b)2

2. Find the dimensions:The measure of a side of a square is x units. A new square is formed with each side 6 units longer than the original square’s side. Write an expression to represent the area of the new square.

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3. In the accompanying diagram, the width of the inner rectangle is represented by x−3 and its length by x+3. The width of the outer rectangle is represented by 3 x+4 and its length by 3 x−4.

a. Find the area of the larger rectangle.

b. Find the area of the smaller rectangle.

c. Express the area of the pink shaded region as a polynomial in terms of x. (Hint: You will have to add or subtract polynomials to get your final answer.)

3 x+4

3 x – 4

x+3x−3

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Name ______________________ Multiplying Polynomial ExpressionsShow all your work! Exit Ticket

When you multiply two terms by two terms, you should get four terms. Why is the final result when you multiply two binomials sometimes only three terms? Give an example of how your final result can end up with only two terms.

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Name ______________________ Factoring GCF and Difference of SquaresShow all your work! Opening Exercise

The total area of this rectangle is represented by 3a2+3a. Find expressions for the dimensions of the total rectangle.

3a2+3a square units

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Factoring GCF and Difference of Squares

Learning Target: I can factor polynomial expressions by using a greatest common factor or difference of squares.

Factor each by factoring out the Greatest Common Factor:

1. 10ab+5 a

2. 3 g3h– 9 g2h+12h

3. 6 x2 y3+9 x y4+18 y5

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4. Factor the following examples of the difference of perfect squares.

a. t 2−25

b. 4 x2−9

c. 16h2−36k2

d. 4−b2

e. x4−4

f. x6−25

g. a2−b2

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Factor each of the following differences of squares completely:

5. 9 y2−100 z2

6. a4−b6

7. r 4−16 s4 (Hint: This one will factor twice.)

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Name ______________________ Factoring GCF and Difference of SquaresShow all your work! Problem Set

For each of the following factor out the greatest common factor:

a. 6 y2+18

b. 27 y2+18 y

c. 21b−15 a

d. 14c2+2c

e. 3 x2−27

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Name ______________________ Factoring GCF and Difference of SquaresShow all your work! Exit Ticket

Factor completely: x6−81 x2

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Name ______________________ Factoring TrinomialsShow all your work! Opening Exercises

1A. List the factor pairs of 24.

________ x ________ = 24 ________ x ________ = 24

________ x ________ = 24 ________ x ________ = 24

________ x ________ = 24 ________ x ________ = 24

________ x ________ = 24 ________ x ________ = 24

1B. Which factor pair of 24 adds up to – 11?

1C. Factor x2 – 11x + 24.

2A. List the factor pairs of –32.

2B. Which factor pair of –32 adds up to –14?

2C. Factor x2 – 14x – 32.

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Factoring Trinomials

Learning Target: I can factor trinomial expressions and understand that factoring reverses multiplication.

Exercises

1. Use a table to aid in finding the product of (x+7)(x+3) .

2. Factor: x2+10x+21

3. Factor: x2+8 x+7

4. Factor: m2+m−90

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5. Factor: k 2−13k+40

6. Factor: −100+99 v+v2

7. Factor Completely: 2 x3−50x

8. Factor Completely: –16 t 2+32t+48

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Name ______________________ Factoring TrinomialsShow all your work! Problem Set

4. Factor these trinomial as the product of two binomials and check your answer by multiplying: a. x2+3x+2

b. x2−8 x+15

c. x2+8 x+15

Factor completely: d. 4m2−4 n2

e. −2 x3−2x2+112 x

f. y8−81x4

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2. The parking lot at Gene Simon’s Donut Palace is going to be enlarged, so that there will be anadditional 30 ft. of parking space in the front and 30 ft. on the side of the lot. Write an expression interms of x that can be used to represent the area of the new parking lot. Explain how your solution is demonstrated in the area model.

x 30

30

x

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Name ______________________ Factoring TrinomialsShow all your work! Exit Ticket

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Name ______________________ Factoring Quadratic Expressions with a Leading CoefficientShow all your work! Opening Exercises

1. Simplify: 4n(n2 +2x – 5)

2. Simplify: (x + 7)(x – 3)

3. Factor x2 + 5x + 6

4. Factor x2 + x – 6

5. Factor x2 – x – 6

6. Factor x2 – 5x + 6

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Factoring Quadratic Expressions with a Leading Coefficient

Learning Target: I can factor quadratic expressions with a leading coefficient.

1. Multiply the binomials: (2 x+3)(1 x+5)

2. Factor the trinomial: 2 x2+13 x+15

3. Factor: 3 x2−x−4

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Exercises 1–6

Factor the expanded form of these quadratic expressions. Pay particular attention to the negative/positive signs.

1. 3 x2−2x−8

2. 3 x2+10 x−8

3. 3 x2+x−14

4. 2 x2−21 x−36

5. −2 x2+3 x+9

6. r2+64

r+ 916

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Name ______________________ Factoring Quadratic Expressions with a Leading CoefficientShow all your work! Problem Set

Factor the following quadratic expressions.

1. 3 x2−2x−5

2. –2 x2+5x−2

3. 5 x2+19 x−4

4. 4 x2−12x+9 [This one is tricky, but look for a special pattern.]

5. 3 x2−13 x+12

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Name ______________________ Factoring Quadratic Expressions with a Leading CoefficientShow all your work! Exit Ticket

Factor completely.

1. x2 + 10x – 24

2. x2 – 81

3. 2x2 + 5x + 2

4. 2x2 + 6x + 4

5. 6x2 – 9x

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Name ______________________ The Zero Product PropertyShow all your work! Opening Exercise

Consider the equation a ∙b ∙ c ∙ d=0. What values of a, b, c, and d would make the equation true?

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The Zero Product Property

Learning Target: I can use the zero product property to solve equations.

Exercises 1–4

Find values of c and d that satisfy each of the following equations. (There may be more than one correct answer.)

1. cd=0

2. (c−5)d=2

3. (c−5)d=0

4. (c−5)(d+3)=0

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Example 1

For each of the related questions below use what you know about the Zero-Product Property to find the answers.

a. The area of a rectangle can be represented by the expression, x2+2x−3. Write each dimension of this rectangle as a binomial, and then write the area in terms of the product of the two binomials.

b. Suppose the area of the rectangle is 21 square units. Rewrite the equation so that it is equal to zero and solve.

c. What are the actual dimensions of the rectangle?

d. If a smaller rectangle, which can fit inside the first rectangle, has an area that can be expressed by the equation x2−4 x−5. What are the dimensions of the smaller rectangle?

e. What value for xwould make the smaller rectangle have an area of 13 that of the larger?

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Exercises 5–8

Solve. Show your work:

5. x2−11 x+19=– 5

6. 7 x2+x=0

7. 7 r2−14 r=– 7

8. 2d2+5d−12=0

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Name ______________________ The Zero Product PropertyShow all your work! Problem Set

1. x2−11 x+19=– 5

2. 7 x2+2 x=0

3. b2+5b−35=3b

4. 7 r2−14 r=– 7

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Name ______________________ The Zero Product PropertyShow all your work! Exit Ticket

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Name ______________________ Quadratic EquationsShow all your work! Opening Exercise

Solve each quadratic equation.

1. y2+6 y=0

2. a2−10a=24

3. x2=25

4. 5n2=45

5. (x−3)2=4

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Quadratic Equations

Learning Target: I can solve quadratic equations.

Example 1

A physics teacher put a ball at the top of a ramp and let it roll down toward the floor. The class determined that the height of the ball could be represented by the equation, h=–16 t 2+4, where the height is measured in feet from the ground and time in seconds.

a. What do you notice about the structure of the quadratic expression in this problem? How can this structure help us when we apply this equation?

b. In the equation, explain what the 4 represents.

c. Use the equation to determine the time it takes the ball to reach the floor.

d. Now consider the two solutions for t . Which one is reasonable? Does the final answer make sense based on this context? Explain.

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Exercises: Solve each equation.

4. 3 x2−9=0

5. (x−3)2=1

6. 4 (x−3)2=1

7. 2(x−3)2=12

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8. Peter is a painter and he wonders if he would have time to catch a paint bucket dropped from his ladder before it hits the ground. He drops a bucket from the top of his 9-foot ladder. The height, h, of the bucket during its fall can be represented by the equation, h=–16 t 2+9, where the height is measured in feet from the ground, and the time since the bucket was dropped is measured in seconds. After how many seconds does the bucket hit the ground? Do you think he could catch the bucket before it hits?

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Name ______________________ Quadratic EquationsShow all your work! Problem Set

9. Factor completely: 15 x2−40 x−15

Solve:

10. 4 x2=9

11. 3 y2−8=13

12. Mischief is a Toy Poodle who competes with her trainer in the agility course. Within the course, Mischief must leap through a hoop. Mischief’s jump can be modeled by the equation h=−16 t2+12 t, where h is the height of the leap in feet and t is the time since the leap, in seconds. At what values of t does Mischief start and end the jump?

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Name ______________________ Quadratic EquationsShow all your work! Exit Ticket

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Name ______________________ Creating and Solving Quadratic EquationsShow all your work! Opening Exercise

The length of a rectangle is 5 in. more than twice a number. The width is 4 in. less than the same number. If the perimeter of the rectangle is 26 in., find the unknown number.

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Creating and Solving Quadratic Equations

Learning Target: I can interpret word problems to create equations in one variable and solve them by factoring.

Example 1

The length of a rectangle is 5 in. more than twice a number. The width is 4 in. less than the same number. If the area of the rectangle is 15, find the unknown number.

Example 2

A picture has a height that is 43 its width. It is to be enlarged so that the ratio of height to width

remains the same but the area is 192 sq in. What are the dimensions of the enlargement?

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Exercises

Solve the following problems. Be sure to indicate if a solution is to be rejected based on the contextual situation.

1. The length of a rectangle 4cm more than 3 times its width. If the area of the rectangle is 15 cm2, find the width.

2. The ratio of length to width in a rectangle is 2 :3. Find the length of the rectangle when the area is 150 sq in.

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3. Karen wants to plant a garden and surround it with decorative stones. She has enough stones to enclose a rectangular garden with a perimeter of 68 ft., and she wants the garden to cover 240 sq ft. What will the length and width of her garden be?

4. Find two consecutive odd integers whose product is 99. [Note: There are two different pairs of consecutive odd integers and only an algebraic solution will be accepted.]

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Name ______________________ Creating and Solving Quadratic EquationsShow all your work! Problem Set

1. The length of a rectangle is 2cm less than its width. If the area of the rectangle is 35 cm2, find the width.

2. The ratio of length to width (measured in inches) in a rectangle is 4 :7. Find the length of the rectangle if the area is known to be 700sq in.

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3. A student is painting an accent wall in his room where the length of the room is 3 ft. more than its width. The wall has an area of 130 ft2. What are the length and the width, in feet?

4. Find two consecutive even integers whose product is 80. (There are two pairs and only an algebraic solution will be accepted.)

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Name ______________________ Creating and Solving Quadratic EquationsShow all your work! Exit Ticket

1. The perimeter of a rectangle is 54 cm. If the length is 2 cm more than a number, and the width is 5 cm less than twice the same number, what is the number?

2. A plot of land for sale has a width of 𝑥 ft. and a length that is 8 ft. less than its width. A farmer will only purchase the land if it measures 240 ft2. What value for 𝑥 will cause the farmer to purchase the land?

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Name ______________________ Exploring the Symmetry in Graphs of Quadratic FunctionsShow all your work! Opening Exercise

Learning Target: I can identify key features of the graphs of quadratic functions.

Use the graphs of quadratic functions A and B to fill in the table.

Graph By=−x2−4 x

Graph Ay=x2−4 x+3

x f (x)−1 8

2 −1

4 3

x f (x)

−3 3

−2 4

1 −5

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Exploring the Symmetry in Graphs of Quadratic Functions

Use your graphs and tables of values from the previous page to fill in the blanks or answer the questions for each below:

Graph A Graph B

1 x-intercepts

2 Vertex

3 Sign of the leading coefficient

4 Vertex represents a minimum or maximum?

5 Points of Symmetry

Find f (−1) and f (5)

Is f (7) greater than or less than 8? Explain

Find f (−1) and f (−3)

f (2)=−12 Predict the value for f (−6) and explain your answer.

6 Increasing and Decreasing Intervals

On what intervals of the domain is the function depicted by the graph increasing?

On what intervals of the domain is the function depicted by the graph decreasing.

On what intervals of the domain is the function depicted by the graph increasing?

On what intervals of the domain is the function depicted by the graph decreasing.

7 Average Rate of Change on an Interval

What is the average rate of change for the following intervals:

[−1 ,0 ]:

[0 ,1]:

[0 ,3 ]:

[1,3]:

What is the average rate of change for the following intervals:

[-5, -4]:

[-4, -3]:

[-4, -1]:

[-3, -1]:

Understanding the symmetry of quadratic functions and their graphs (Look at row 5 in the chart and the tables).

a. What patterns do you see in the tables of values you made next to Graph A and Graph B?

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Finding the vertex and axis of symmetry (Look at rows 1 and 2 of the chart.)

b. How can we know the x-coordinate of the vertex by looking at the x-coordinates of the zeros (or any pair of symmetric points)?

Understanding end behavior (Look at rows 3 and 4 of the chart.)

c. What happens to the y-values of the functions as the x-values increase to very large numbers? What about as the x-values decrease to very small numbers (in the negative direction)?

d. How can we know whether a graph of a quadratic function will open up or down?

Identifying intervals on which the function is increasing (or decreasing) (Look at row 6 in the chart).

e. Is possible to determine the exact intervals that a quadratic function is increasing or decreasing just by looking at a graph of the function?

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Computing average rate of change on an interval (Look at row 7 in the chart).

f. Explain why the average rate of change over the interval [1,3] for Graph A was zero.

g. How are finding the slope of a line and finding the average rate of change on an interval of a quadratic function similar? How are they different?

Finding a unique quadratic function

h. Can you graph a quadratic function if you don’t know the vertex? Can you graph a quadratic function if you only know the x-intercepts?

i. Remember that we need to know at least two points to define a unique line. Can you identify a unique quadratic function with just two points? Explain.

j. What is the minimum number of points you would need to identify a unique quadratic function? Explain why?

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Exploratory Exercise 3

Below you see only one side of the graph of a quadratic function. Complete the graph by plotting three additional points of the quadratic function. Explain how you found these points then fill in the table on the right.

a. What are the coordinates of the x-intercepts?

b. What are the coordinates of the y-intercept?

c. What are the coordinates of the vertex? Is it a minimum or a maximum?

d. If we knew the equation for this curve, what would the sign of the leading coefficient be?

e. Verify that the average rate of change for interval, –3≤ x≤ –2, [– 3 , –2] is 5. Show your steps.

x f (x)

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Name ______________________ Exploring the Symmetry in Graphs of Quadratic FunctionsShow all your work! Problem Set

1. Is it possible for the graphs of two DIFFERENT quadratic functions to each have x=– 3 as its line of symmetry and both have a maximum at y=5? Explain and support your answer with a sketch of the graphs.

2. Use the symmetric properties of quadratic functions to sketch the graph of the function below, given these points:

(1, 3)

(2,- 3)

(0, 5)

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Name ______________________ Exploring the Symmetry in Graphs of Quadratic FunctionsShow all your work! Exit Ticket

1. If possible, find the equation for the axis of symmetry for the graph of a quadratic function with the given pair of coordinates. If not possible, explain why.

a. (3,10) (15,10)

b. (−2,6) (6,4)

2. The point (4,−2) is the vertex of the graph of a quadratic function. The points (8,6) and (2,0) also fall on the graph of the function. Complete the graph of this quadratic function by first finding two additional points on the graph. (If needed, make a table of values on your own paper.) Then, answer the questions on the right.

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Name ______________________ Graphing Quadratic Functions from Factored FormShow all your work! Opening Exercise

Identify the following key features of the graph of the quadratic function y=x2+4 x−12.

1. The x-intercepts.

2. The y-intercept.

3. The axis of symmetry.

4. The vertex.

5. The coordinates of 4 different points on the graph.

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Graphing Quadratic Functions from Factored Form

Learning Target: I can use the key features of a quadratic to graph the function.

Example 1

Consider the equation y=x2+6 x−40.

a. Given this quadratic equation, find the point(s) where the graph crosses the x-axis.

b. How can we write a corresponding quadratic equation if we are given a pair of roots?

c. Use the symmetrical nature of the graph of a quadratic function to find the vertex for the graph.

d. Find the y-intercept (where the graph crosses the y-axis and where x=0).

e. What else can we say about the graph based on our knowledge of the symmetrical nature of the graph of a quadratic function? Can we determine the coordinates of any other points?

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f. Plot the points you know for this equation and connect them to show the graph of the equation y=x2+6 x−40.

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Exercise 1: Graph the following functions and identify the vertex, x-intercept, and y-intercept.a. f (x)=−(x+2)(x−5)

b. g ( x )=x2−5 x−24

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Example 2

A science class designed a ball launcher and tested it by shooting a tennis ball straight up from the top of a 15-story building. They determined that the motion of the ball could be described by the function:

h (t )=−16 t 2+144 t+160

wheret represents the time the ball is in the air in seconds and h, the height, in feet, of the ball above the ground.

a. Graph the function.

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b. Using the graph, at what time does the ball hit the ground?

c. Over what domain is the ball rising? Over what domain is the ball falling?

d. Using the graph, what is the maximum height the ball reaches?

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Exercise 2: Graph the following functions and identify the vertex, x-intercept, and y-intercept.

a. f (x)=5(x−2)(x−3)

b. t ( x )=x2+8 x−20

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c. p ( x )=−6 x2+42 x−60

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Name ______________________ Graphing Quadratic Functions from Factored FormShow all your work! Exit Ticket

Graph the following function, and identify the key features of the graph: ℎ(𝑥) = −3(𝑥 − 2)(𝑥 + 2).

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Name ______________________ Interpreting Quadratic Functions from Graphs and TablesShow all your work! Opening Exercises

1. What are the roots of the equation x2−10 x+21=0?

2. One of the roots of the equation x2+3x−18=0 is 3. What is the other root?

3. Which equation has roots of –3 and 5?

1)2)3)4)

4. Write a quadratic equation whose roots are –5 and 7.

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Interpreting Quadratic Functions from Graphs and Tables

Learning Target: I can answer questions by interpreting quadratic functions from graphs and tables.

Example

The table below represents the value of Andrew’s stock portfolio, with V representing the value of the portfolio, in hundreds of dollars, and t is the time, in months, since he started investing. Answer the following questions based on the table of values:

t (months) V (t ) (hundreds of dollars)

2 3254 3856 4058 38510 32512 22514 8516 −9518 −315

a. What kind of function do you think this table represents? How do you know?

b. Assuming this data is in fact quadratic, how much did Andrew invest in his stock initially? Explain how you arrived at this answer.

c. What is the maximum value of his stock and how long did it take to reach the maximum value?

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d. If the pattern continues to follow the quadratic trend shown above, do you advise Andrew to sell or keep his stock portfolio? Explain why.

e. How fast is Andrew’s stock value decreasing between [10 ,12]? Find another two-month interval where the average rate of change is faster than [10 ,12] and explain why.

f. Are there other two-month intervals where the rate of change is same as [10 ,12]? Explain your answer.

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Name ______________________ Interpreting Quadratic Functions from Graphs and TablesShow all your work! Problem Set

Pettitte and Ryu each threw a baseball into the air. The vertical height of Pettitte’s baseball is represented by the graph P(t) below. P represents the vertical distance of the baseball from the ground in feet and trepresents time in seconds.

The vertical height of Ryu’s baseball is represented by the table values R(t ) below. R(t ) represents the vertical distance of the baseball from the ground in feet and t represents time in seconds.

Use the above functions to answer the following questions.

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t R(t )0 860.5 981 1021.5 982 862.5 663 383.52 0

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a. Whose baseball reached the highest? Explain your answer.

b. Whose ball reached the ground fastest? Explain your answer.

c. Pettitte claims that his ball reached its maximum faster than Ryu’s? Is his claim correct or incorrect? Explain your answer.

d. Find P(0) and R(0) values and explain what it means in the problem. What conclusion can you make based on these values? Did they throw the ball from the same place? Explain your answer.

e. Ryu claims that he can throw the ball higher than Pettitte. Is his claim correct or incorrect? Explain your answer.

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Name ______________________ Interpreting Quadratic Functions from Graphs and TablesShow all your work! Exit Ticket

A toy company is manufacturing a new toy and trying to decide on a price that will result in a maximum profit. The graph below represents profit (𝑃) generated by each price of a toy (𝑥). Answer the questions based on the graph of the quadratic function model.

a. If the company wants to make a maximum profit, what should the price of a new toy be?

b. What is the minimum price of a toy that will produce profit for the company? Explain your answer.

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Name ______________________ Completing the SquareShow all your work! Opening Exercise

Factor each of the following expressions.

STANDARD FORM FACTORED FORM

x2+12 x+36

x2−12 x+36

x2+20 x+100

x2−3 x+ 94

x2+100 x+2,500

x2+8 x+3

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Completing the Square

Learning Target: I can rewrite quadratic expressions by completing the square.

Example 1

Rewrite the following standard form quadratic expressions as perfect squares.

STANDARD FORM FACTORED FORM

x2+12 x+36

x2−12 x+36

x2+20 x+100

x2−3 x+ 94

x2+100 x+2,500

x2+8 x+3

Example 2

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Find an expression equivalent to x2+8 x+3 that includes a perfect square binomial.

Exercises

Rewrite each expression by completing the square:

1. a2– 4a+15

2. n2−2n−15

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3. c2+20c−40

4. y2−3 y+10

5. k 2+7 k+6

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Name ______________________ Completing the SquareShow all your work! Problem Set

Rewrite each expression by completing the square:

1. q2+12q+32

2. m2−4m−5

3. x2−7 x+6.5

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Name ______________________ Completing the SquareShow all your work! Exit Ticket

Rewrite the expression r2+4 r+3, first by factoring, and then by completing the square. Which way is easier? Explain why you think so.

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Name ______________________ Completing the Square with Leading CoefficientsShow all your work! Opening Exercise

Rewrite each expression by completing the square.

1. z2−5 z+8

2. x2+0.6 x+1

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Completing the Square with Leading Coefficients

Learning Target: I can rewrite quadratic expressions with leading coefficients by completing the square.

Example

Now complete the square for:

2 x2+16 x+3

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Name ______________________ Completing the Square with Leading CoefficientsShow all your work! Problem Set

Rewrite each expression by completing the square.

1. 3 x2+12 x−8

2. 4 p2−12 p+13

3.12

y2+3 y−4

4. −2 x2+8 x+5

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Name ______________________ Completing the Square with Leading CoefficientsShow all your work! Exit Ticket

Complete the square: a x2+x+3.

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Name ______________________ Solving Quadratics by Completing the SquareShow all your work! Opening Exercise

What is the solution of x2+4 x+4=25?

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Solving Quadratics by Completing the Square

Learning Target: I can solve quadratic equations by completing the square.

Essential Understanding Completing a perfect square trinomial allows us to factor the completed trinomial as the square of a binomial. Completing the square allows us to solve quadratic equations where factoring is either impossible or inefficient. Completing the square also allows us to rewrite quadratic equations in vertex form making it easier to study key features of quadratics and their graphs.

1. Use an area model to show how x2+8 x−4=0 is solved by completing the square.

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Example 1

Solve for x:

12=x2+6 x

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Example 2

Now try this one:

4 x2−40 x+94=0

Example 3

Brian correctly used a method of completing the square to solve the equation x2+7x−11=0. Brian’s first step was to rewrite the equation as x2+7x=11. He then added a number to both sides of the equation. Which number did he add?

1) 3)

2) 4) 49

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Exercises

Solve each equation by completing the square.

1. x2−2 x=12

2. 2 p2+8 p=7

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Name ______________________ Solving Quadratics by Completing the SquareShow all your work! Problem Set

Solve each equation by completing the square.

1. p2−3 p=8

2. 2q2+8q=3

Name ______________________ Solving Quadratics by Completing the Square

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Show all your work! Exit Ticket

Solve the following quadratic equation both by factoring and by completing the square:

x2−4 x=12

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Name ______________________ Simplifying RadicalsShow all your work! Opening Exercise

12=¿¿ 62=¿¿ 112=¿¿

22=¿¿ 72=¿¿ 122=¿¿

32=¿¿ 82=¿¿ 132=¿¿

42=¿¿ 92=¿¿ 142=¿¿

52=¿¿ 102=¿¿ 152=¿¿

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Simplifying Radicals

Learning Target: I can simplify radical expressions.

1. Express √12 in simplest radical form.

2. Express √50 in simplest radical form.

3. When √72 is expressed in simplest a√b form, what is the value of a?

4. When 5√20 is expressed in simplest radical form, the result is k √5. What is the value of k?

5. What is √324

expressed in simplest radical form?

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6. Express 7√90 in simplest radical form.

7. Express −3√48 in simplest radical form.

8. Theo determined that the correct length of the hypotenuse of the right triangle in the accompanying diagram is √20 . Fiona found the length of the hypotenuse to be 2√5 . Is Fiona’s answer also correct? Justify your answer.

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Name ______________________ Simplifying RadicalsShow all your work! Problem Set

1. Simplify √20

2. Simplify √300

3. Simplify √48

4. Simplify 5√162

5. Simplify √82

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Name ______________________ Simplifying RadicalsShow all your work! Exit Ticket

Express √75 in simplest radical form.

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Name ______________________ Quadratic FormulaShow all your work! Opening Exercise

1. Solve for n: n2−4n=21

2. Solve for p: p2−9=0

3. Solve for m. Express the solution in simplest radical form: m2−20=0

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Quadratic Formula

Learning Target: I can solve equations using the quadratic formula.

1. Solve x2+2x−8=0 using the quadratic formula.

2. Solve x2+4=4 x using the quadratic formula.

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3. Solve 3 x2−10 x+5=0

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Name ______________________ Quadratic FormulaShow all your work! Problem Set

Solve the following equations using the quadratic formula:

1. x2−2 x+1=0

2. 3b2+4b+8=0

3. 2 t2+7 t−4=0

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4. q2−2q –1=0

5. m2−4=3

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Name ______________________ Quadratic FormulaShow all your work! Exit Ticket

Solve using the quadratic formula: 3 x2+6x+8=6.

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Name ______________________ Nature of the Roots of Quadratic EquationsShow all your work! Opening Exercise

1. Use the quadratic formula to solve 9 x2+3x−4=0. Express the solution in simplest radical form.

2. What are the roots of the equation 8 x2+ x=−4?

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The Discriminant and the Nature of the Roots of Quadratic Equations

Learning Target: I can use the discriminant to determine the nature of the roots of quadratic equations.

1. The roots of the equation 9 x2+3x−4=0 are

1) imaginary2) real, rational, and equal3) real, rational, and unequal4) real, irrational, and unequal

2. Which number is the discriminant of a quadratic equation whose roots are real, unequal, and irrational?

1) 02)3) 74) 4

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3. Use the discriminant to determine all values of k that would result in the equation x2−kx+4=0 having equal roots (only one solution).

For Exercises 4–7, without solving, determine the number of real solutions for each quadratic equation:

4. p2+7 p+33=8−3 p

5. 7 x2+2 x+5=0

6. 2 y2+10 y= y2+4 y−3

7. 4 z2+9=−4 z

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8. State whether the discriminant of each quadratic equation is positive, negative, or equal to zero on the line below the graph.

Identify which graph above matches the discriminants below:

Discriminant A:(−2 )2−4 (1 ) (2 )

Graph #: ______

Discriminant B:(−4 )2−4 (−1 ) (−4 )

Graph #: ______

Discriminant C:(−4 )2−4 (1 ) (0 )

Graph #: ______

Discriminant D:(−8 )2−4 (−1 ) (−13 )

Graph #: ______

_____________________ ______________________ _______________________ ______________________

Graph #1 Graph #2 Graph #3 Graph #4

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Name ______________________ Nature of the Roots of Quadratic EquationsShow all your work! Problem Set

Without solving, determine the number of real solutions for each quadratic equation.

1. b2−4 b+3=0

2. 2n2+7=−4 n+5

3. 4.

5. 6.

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Name ______________________ Nature of the Roots of Quadratic EquationsShow all your work! Exit Ticket

How many zeros does the function f ( x )=3 x2+6 x+2 have? Explain how you know.

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Name ______________________ Graphing Quadratic Functions from the Vertex FormShow all your work! Opening Exercise

On the same set of axes graph and label y=x2, y=(x−2)2, and y=(x+2)2 on the interval −3≤x ≤3.

The graph of y=(x−2)2 is a translation the graph of y=x2 _____ units to the ________.

The graph of y=(x+2)2 is a translation the graph of y=x2 _____ units to the ________.

The vertex of the graph of y=x2 is __________.

The vertex of the graph of y=(x−2)2is __________.

The vertex of the graph of y=(x+2)2is __________.

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Graphing Quadratic Functions from the Vertex Form

Learning Target: I can graph quadratic functions from the vertex form.

On the same set of axes graph and label y=x2, y=x2−3, and y=x2+3.

The graph of y=x2−3 is a translation the graph of y=x2 _____ units ________.

The graph of y=x2+3 is a translation the graph of y=x2 _____ units ________.

The vertex of the graph of y=x2 is __________.

The vertex of the graph of y=x2−3 is __________.

The vertex of the graph of y=x2+3 is __________.

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Exercises

1. Without graphing, state the vertex for each of the following quadratic equations:a. y= (x−5 )2+3

b. y=x2−2.5

c. y= (x+4 )2

2. Write a quadratic equation whose graph will have the given vertex: a. (1.9 ,−4)

b. (0 ,100)

c. (−2 , 32 )

Lesson Summary

When graphing a quadratic equation in vertex form, y=a(x – h)2+k ,(h , k ) are the coordinates of the vertex.

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Name ______________________ Graphing Quadratic Functions from the Vertex FormShow all your work! Problem Set

1. Find the vertex of the graphs of the following quadratic equations.

a. y=2 ( x−5 )2+3.5

b. y=−( x+1 )2−8

2. Write quadratic equations representing the function with the following vertex. Use a leading coefficient other than 1.

a. (100 ,200)

b. (−34 ,−6)

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Name ______________________ Graphing Quadratic Functions from the Vertex FormShow all your work! Exit Ticket

Write two different equations representing quadratic functions whose graphs have vertices at (4.5,−8).

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Name ______________________ Graphing Quadratic Functions from the Standard FormShow all your work! Opening Exercise

A high school baseball player throws a ball straight up into the air for his math class. The math class was able to determine that the relationship between the height of the ball and the time since it was thrown could be modeled by the function, h ( t )=−16 t 2+96 t+6, where t represents the time (in seconds) since the ball was thrown and h, the height (in feet) of the ball above the ground.

a. What does the domain of the function represent in this context?

b. What does the range of this function represent?

c. At what height does the ball get thrown?

d. After how many seconds does the ball hit the ground?

e. What is the maximum height that the ball reaches while in the air? How long will the ball take to reach its maximum height?

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Graphing Quadratic Functions from the Standard Form

Learning Target: I can graph quadratic functions from the standard form.

Graph h (t )=−16 t2+96 t+6 in an appropriate interval based on the context described above. Identify the vertex, axis of symmetry, x- and y-intercepts, and maximum of the graph.

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Exercises

1. a. What is the y-intercept of the graph of n ( x )=x2−6 x+5?

b. What are the x-intercepts (roots) of the graph of n ( x )=x2−6 x+5?

c. What is the equation of the axis of symmetry of the graph of n ( x )=x2−6 x+5?

d. What is the minimum value of the graph of n ( x )=x2−6 x+5?

e. What is the vertex of the graph of n ( x )=x2−6 x+5?

f. Graph the equation n ( x )=x2−6 x+5.

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2. a. What is the y-intercept of the graph off ( x )=12

x2+5 x+6?

b. What are the x-intercepts (roots) of the graph off ( x )=12

x2+5 x+6?

c. What is the equation of the axis of symmetry of the graph off ( x )=12

x2+5 x+6?

d. What is the minimum value of the graph off ( x )=12

x2+5 x+6?

e. What is the vertex of the graph off ( x )=12

x2+5 x+6?

f. Graph the equationf ( x )=12

x2+5 x+6.

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3. A student throws a bag of chips to her friend. Unfortunately, her friend does not catch the chips and the bag hits the ground. The distance from ground (height) for the bag of chips is modeled by the function h (t )=−16 t 2+32t+4 ,where h( t) is the height (distance from the ground) of the chips and t is the number of seconds the chips are in the air.

a. Graph h( t).

b. From what height are the chips being thrown? Tell how you know.

c. What is the maximum height the bag of chips reaches while airborne? Tell how you know.

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d. How many seconds after the bag was thrown did it hit the ground?

e. What is the average rate of change of height for the interval from 0 to ½ second? What does that number represent in terms of the context?

f. Based on your answer to part (e), what is the average rate of change for the interval from 1.5 to 2 sec?

Name ______________________ Graphing Quadratic Functions from the Standard FormShow all your work! Problem Set

1. a. What is the y-intercept of the graph off (x)=x2−2x−15?

b. What are the x-intercepts (roots) of the graph off (x)=x2−2x−15?

c. What is the equation of the axis of symmetry of the graph of f (x)=x2−2x−15?

d. What is the minimum value of the graph off (x)=x2−2x−15?

e. What is the vertex of the graph off (x)=x2−2x−15?

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f. Graph the equationf (x)=x2−2 x−15.

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2. Graph the following equation f ( x )=−x2+2 x+15 and identify the roots and the vertex.

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Name ______________________ Graphing Quadratic Functions from the Standard FormShow all your work! Exit Ticket

Graph g ( x )=x2+10 x−7

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Name ______________________ Graphing Cubic, Square Root, and Cube Root FunctionsShow all your work! Opening Exercises

1. Evaluate x2 when x=7.

2. Evaluate √ x when x=81.

3. Evaluate x3 when x=5.

4. Evaluate 3√ x when x=27.

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Graphing Cubic, Square Root, and Cube Root Functions

Learning Target: I can graph cubic, square root, and cube root functions.

Exercise 1

Use your graphing calculator to create a data table for the functions y=x2 and y=√x for a variety of x-values, Use both negative and positive numbers and round decimal answers to the nearest hundredth.

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x y=x2 y=√x

-4

-3

-2

-1

0

1

2

3

4

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Exercise 2

A. Create the graphs of y=x2 and y=√x on the same set of axes.

B. What is the domain of the function y=√x?

C. What is the range of the function y=√x?

D. How are the graphs of y=√x and y=x2 related?

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Exercise 3

A. Create data tables for y=x3 and = y= 3√x and graph both functions on the same set of axes. Round decimal answers to the nearest hundredth.

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x y=x3 y= 3√x

−8

−2

−1

0

1

2

8

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B. What is the domain of the function ¿ 3√ x ?

C. What is the range of the function ¿ 3√ x ?

D. How are the graphs of y=x3 and y= 3√x related?

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Name _________________________ Interpreting Quadratic Functions and Their GraphsShow all your work!

1. An arch is built so that it is 6 feet wide at the base. Its shape can be represented by a parabola with the equation y=−2x2+12x , where y is the height of the arch.

a Graph the parabola from x=0 to x=6 on the grid below.

b Determine the maximum height, y, of the arch.

2. The height, y, of a ball tossed into the air can be represented by the equation y=−x2+10 x+3, where x is the elapsed time. What is the equation of the axis of symmetry of this parabola?

1)2)3)4)

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3. Amy tossed a ball in the air in such a way that the path of the ball was modeled by the equation y=−x2+6 x. In the equation, y represents the height of the ball in feet and x is the time in seconds.

a Graph y=−x2+6 x for 0≤ x≤6 on the grid provided below.

b At what time, x, is the ball at its highest point?

4. A model rocket is launched from ground level. Its height, h meters above the ground, is a function of time t seconds after launch and is given by the equation h=−4.9 t 2+68.6 t. What would be the maximum height, to the nearest meter, attained by the model?

1) 2432) 2423) 2414) 240

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5. An architect is designing a museum entranceway in the shape of a parabolic arch represented by the equation y=−x2+20 x, where 0≤ x≤20 and all dimensions are expressed in feet. On the accompanying set of axes, sketch a graph of the arch and determine its maximum height, in feet.

6. For which quadratic equation is the axis of symmetry x=3?

1)

2)

3)

4)

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7. Tom throws a ball into the air. The ball travels on a parabolic path represented by the equation

h=−8 t 2+40 t, where h is the height, in feet, and t is the time, in seconds.

a On the accompanying set of axes, graph the equation from t=0 to t=5 seconds, including all integral values of t from 0 to 5.

b What is the value of t at which h has its greatest value?

8. The equationW=120 I−12 I 2 represents the power (W), in watts, of a 120-volt circuit having a resistance of 12 ohms when a current (I ) is flowing through the circuit. What is the maximum power, in watts, that can be delivered in this circuit?

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Name __________________________ Key Features of Quadratic GraphsShow all your work!

1. Which type of graph is shown in the diagram below?

1) absolute value2) exponential3) linear4) quadratic

2. Consider the graph of the equation y=a x2+bx+c , when a≠0. If a is multiplied by 3, what is true of the graph of the resulting parabola?

1) The vertex is 3 units above the vertex of the original parabola.

2) The new parabola is 3 units to the right of the original parabola.

3) The new parabola is wider than the original parabola.

4) The new parabola is narrower than the original parabola.

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3. The diagram below shows the graph of y=−x2−c.

Which diagram shows the graph of ?1)

2)

3)

4)

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4. The equation y=x2+3 x−18 is graphed on the set of axes below.

Based on this graph, what are the roots of the equation y=x2+3 x−18?

1) and 62) 0 and 3) 3 and 4) 3 and

5. A swim team member performs a dive from a 14-foot-high springboard. The parabola below shows the path of her dive.

Which equation represents the axis of symmetry?1)2)3)4)

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6. The graph of a quadratic equation is shown in the accompanying diagram. The scale on the axes is a unit scale. Write an equation of this graph in standard form.

7. Greg is in a car at the top of a roller-coaster ride. The distance, d, of the car from the ground as the car descends is determined by the equation ¿144−16 t2, where t is the number of seconds it takes the car to travel down to each point on the ride. How many seconds will it take Greg to reach the ground?

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1. If the quadratic formula is used to find the roots of the equation x2−6 x−19=0, the correct roots are

(1) 3±2√7 (3) 3±4√14

(2) −3±2√7 (4) −3±4√14

2. The graphs below represent functions defined by polynomials. For which function are the zeros of the polynomials 2 and –3?

3. For which function defined by a polynomial are the zeros of the polynomial –4 and –6?

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(1) y=x2−10 x−24(2) y=x2+10 x+24(3) y=x2+10 x−24(4) y=x2−10 x+24

4. The length of the shortest side of a right triangle is 8 inches. The lengths of the other two sides are represented by consecutive odd integers. Which equation could be used to find the lengths of the other sides of the triangle?

(1) 82+( x+1 )=x2

(2) x2+82=( x+1 )2

(3) 82+( x+2 )2=x2

(4) x2+82=( x+2 )2

5. Solve 8m2+20m=12 for m by factoring.

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6. On the set of axes below, graph the function represented by y= 3√x−2 for the domain −6≤ x≤10.

7. Ryker is given the graph of the function y=12

x2−4. he wants to find the zeros of the function, but

is unable to read them exactly from the graph. Find the zeros in simplest radical form.

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