The Graphs of Mixed Functions (Day 1)

In this unit, we will remember how to graph some old functions and discover how to graph lots of new functions.

Example 1: Graph and label the parent functions f(x) = x2 and g(x) =√ xon the same set of axes by creating a table of values. Round all decimals to the nearest tenth.

What are the roots of f(x)?

What is the domain of f(x) = x2? What is the range of f(x) = x2?

What is the domain of g(x) =√ x ? What is the range of g(x) =√ x ?

What are the solutions of the system of f(x) and g(x)?

x f(x)x g(x)

Example 2: Graph and label the parent functions f(x) = x3 and g(x) =3√ xon the same

set of axes by creating a table of values for each. Round to the nearest tenth.

What are the solutions of f(x) and g(x)?

What is the domain of f(x) = x3? What is the range of f(x) = x3?

What is the domain of g(x) =3√ x ? What is the range of g(x) =

3√ x ?

Try on your own:3. The graph of the function f (x)=√x+4 is shown.

The domain of the function is(1) {x|x>0} (3) {x|x>− 4 }

(2) {x|x≥0} (4) {x|x≥− 4}

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x yx y

The Translations of Mixed Functions (Day 2)

Recall: Graph y = x

Example 1: Graph y = |x|

1. How are the graphs of y = |x| and y = x2 related?

2. How are the graphs different?

3. What is the domain of y = |x|? 4. What is the range of y = |x|?

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x y

x y

Graph the following on your calculator and state the vertex, domain, and range of each.

5. f(x) = |x| 6. g(x) = |x|+3 7. h(x) = |x−2|+4

Vertex: Vertex: Vertex:

D: D: D:

R: R: R:

Now, let’s come up with the rules for translating equations:

Try on your own:Graph the following on your calculator and state the vertex, domain, and range of each.

8. f(x) = x2 9. g(x) = x2 – 5 10. h(x) = (x + 3)2 + 2Vertex: Vertex: Vertex:

D: D: D:

R: R: R:

Rules for Writing Equations with Translations

When the graph is shifted ___________ or ___________ the ______ variable is affected in the equation. (add or subtract this # to the inside).

o Left or Right Shifts, always use the OPPOSITE sign of the shifted # on the inside.

When graph is shifted _______ or ___________ the ______ variable is changed in the equation by just _____________ or ______________ the shifted # to the outside (end) of the equation.

o Up or Down Shifts always use the SAME sign of the shifted # on

the outside.

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The Translations of Mixed Functions Continued (Day 3)

Recall: Graph the parent function f(x) = x3.

1. On the same set of axes above, graph g(x) = x3 + 4 without a calculator.

2. On the same set of axes above, graph h(x) = (x – 5)3 + 2 without a calculator.

Recall: Graph the parent function f(x) = 3√ x .

3. On the same set of axes above, graph g(x) = 3√ x+4 without a calculator.

4. On the same set of axes above, graph h(x) = 3√ x+4 – 3 without a calculator.

Give an equation for the image of the graph of y = x2 under the transformation. Name the vertex and axis of symmetry (AOS) of the image.

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5. 3 units right, 4 units down 6. 6 units left, 2 units up

__________________________ __________________________

vertex: __________ vertex: __________

AOS: ________ AOS: ________

7. If the graph y = (x + 3)2 is translated down 8, what is the new equation?

What is the name of this function?

8. If the graph y = | x−5 |+2 is translated left 3 and down 9, what is the new equation?

What is the parent function called?

Try on your own:9. If the graph y =

3√ x+1 – 4 is translated right 6 and up 7, what is the new equation?

What is the parent function called?

10. Which equation represents the function shown in the accompanying graph?

(1) f ( x )=|x|+1 (3) f ( x )=| x+1 |

(2) f ( x )=|x|−1 (4) f ( x )=| x−1 |

Stretching & Shrinking of Mixed Functions (Day 4)

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Fill out the table for each of the 3 absolute value functions below. Then graph and label each of the functions on the grid below. Use different colors!

Observations: y = k|x|

Example 1: Graph the following functions on the same set of axes below:

f(x) = √ x

Reflections & Stretch/Compressions

If k is greater than ____, then the graph is _________________________________________________

If k is a fraction smaller than _____, the graph is ___________________________________________

If k has a ___________________, the graph is _______________________________________________

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Xg(x) = 4

|x| h(x) = 12

|x|

j(x) = -2|x|

-2-1012

g(x) = 3√ x

Describe what happened to the graph of f(x) to become g(x).

Example 2: Graph the following functions on the same set of axes below:

f(x) = x2

g(x) = -2(x – 4)2 + 1

Describe what happened to the graph of f(x) to become g(x).

3. The graph of a parent function g(x) = x2 has been translated 3 units to the right, vertically stretched by a factor of 4, and moved 2 units up. Write the formula for the function that defines the transformed graph.

Try on your own:4. The graph of an parent function f(x) = |x| has been translated 6 units to the left,

vertically shrunk by a factor of 12 , reflected over the x-axis, and moved down 5

units. Write the formula for the function that defines the transformed graph.

Completing the Square & Vertex Form of a Quadratic (Day 5)

Recall: Which is a step when you solve 2x2 + 12x + 1 = 0 by completing the square?

(1) (x + 3)2 = 4

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(2) (x + 3)2 = 172

(3) (x + 12)2 = 1

(4) (x + 12) = 172

Vertex Form of a Quadratic Equation: y = (𝑥 − ℎ)2 + kWe have already seen in this unit how valuable this form is as we can tell what the vertex and axis of symmetry are, any translations that are made, the stretch or compression of the graph, and whether it looks like a smile or a frown (open up or down).

Standard Form of a Quadratic Equation:

Example 1: When we have a quadratic equation in standard form, we can complete the square with it to get into vertex form!

A quadratic function is defined by g(x) = 2x2 + 12x + 1. Write this in the completed-square (vertex) form and show all the steps.

g(x) = 2x2 + 12x + 1 Steps:

Example 2: Get the quadratic equation f(x) = 2x2 + 4x + 1 into vertex form and then answer the follow-up questions.

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a. What is the vertex of the function?

b. What is the axis of symmetry?

c. What is the domain of the function?

d. What is the range of the function?

e. What transformations would turn the parent function f(x) = x2 into g(x)?

Example 3: TRY b. and d. WITHOUT A CALCULATOR!!!Draw lines matching the graph of the function to the equation of the function.

a. y = 𝒙𝟐 + 𝟔𝒙 + 5 g(x)

b. y = 𝟑(𝒙 − 𝟐)𝟐 – 𝟏 h(x)

c. y = 𝟐𝒙𝟐 + 𝟖𝒙 k(x)

d. y = −(𝒙 − 𝟒)𝟐 + 𝟑 p(x)

Quadratic Functions & the Second Difference (Day 6)

Recall: For the following table of values:

a) Determine if the pattern of terms are arithmetic or geometric. Why?b) Complete the missing part of the tables. c) Determine what function the terms form when graphed.

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1. 2.

Example 1: Now let’s look at another table of values from the equation y = x2 – 8x + 11.

a) Is the function linear? Why?

b) Is the function exponential? Why?

We know a function is quadratic if the terms have a _______________________________________

Example 2: Look at the next three tables, fill in any missing parts, and graph the points on the corresponding grid. Then determine what type of function the points

form. 1.

a) What type of function is this? Why?

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x y

1 62 18

3 54

4 162

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x y Common Difference

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3

4

5

6

7

x y1 -82 -53 -24 15 467

2. a) What type of function is this? Why?

3. a) What type of function is this? Why?

Creating Equations of Functions Using Points (Day 7)

Investigate:

How many different quadratic functions can you draw through the points (0, 4) & (1, 9)?

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x y1 92 43 14 05 167

x y1 32 63 124 245 4867

How many different quadratic functions can you draw through the points (0, 4), (1, 9) and (-3, 1)?

How many different quadratics can you draw through the points (0, 4), (1, 9), (-3, 1) and (2, 5)?

Example 1: Write the equation of the quadratic function that goes through the following points: (0, 4) (1, 9) (-3, 1)

Example 2: Write the equation of the exponential function that goes through the following points: (-3, 16) (-1, 4) (2, 0.5)

Regression Equations: Requirements _____ points are needed to use Linear Regression

______ points are needed to use ANY OTHER Regression Equation (curves)

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Example 3: Write the equation of the quadratic function that goes through the following points (-4, -6) (2, 15) (-1,0)

Example 4: Write the equation of the given functions:

a) b)

Example 5: Louis dropped a watermelon from the roof of a tall building. As it was falling, Amanda and Martin were on the ground with a stopwatch.

a. Write a quadratic function to model the above table of data relating the height of the watermelon (distance in feet from the ground) to the number of seconds that

had passed.

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b. How do we know this data will be represented by a quadratic function?

c. Do we need to use all five data points to write the equation?

Try on your own:

Example 6: Write the equation of the quadratic function that goes through the following points: (-9, -28) (-4, 7) (1, -8)

Piecewise & Step Functions (Day 8)

Piecewise Function: Functions that are made up of “pieces” of different functions defined over specific intervals.

To graph a piecewise function:

1. Create a table of values for each equation separately.

VERY IMPORTANT: Only use x values that are SPECIFICALLY defined for each function.

2. Plot correct points (if more are plotted than are required, you WILL LOSE POINTS)

3. Correct endpoints for each function (open or closed circles)

VERY IMPORTANT: Make sure you have the correct end point for each situation.

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Example 1: Graph the following piecewise function.

{y=x2−50≤x<3y=2x−2 x≥3

Example 2: What is the value of f(2) when f ( x )=¿ {3 x2+x−1 , x≥1¿ ¿¿¿(1) 4 (3) 11(2)7 (4) 13

Step Function: Functions that increase or decrease from one constant to another. It looks like a staircase when graphed!

Example 2: Graph the following step function.

f ( x )={ 20≤x<242≤∧x<46 4≤x<686≤x<8

Why are both Examples 1 & 2 functions?? ________________________________________________

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Example 3: Given the piecewise function , what is the average rate of change over the interval ?

(1) (3)

(2) 0 (4) 4

Try on your own:Example 4: A function is graphed below on the set of axes to the right. Which function is related to the

graph?

(1) f ( x )=¿ {x2 , x<1 ¿ ¿¿¿ (3) f ( x )=¿ {x2 , x<1¿ ¿¿¿

(2)f ( x )=¿ {x2 , x<1 ¿ ¿¿¿

(4) f ( x )=¿ {x2 , x<1 ¿¿¿¿

Piecewise & Step Function Applications (Day 9)The reason piecewise and step functions are so important is because there are many real world examples that are graphically represented this way:

Example 1: During a snowstorm, a meteorologist tracks the amount of accumulating snow. For the first three hours of the storm, the snow fell at a constant rate of one inch per hour. The storm then stopped for two hours and then started again at a constant rate of one-half inch per hour for the next four hours.

a) On the grid below, draw and label a graph that models the accumulation of snow over time using the data the meteorologist collected.

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b) Write the equations for the different “pieces” of the function and the intervals over which they were graphed.

c) If the snowstorm started at 6 p.m., how much snow would have accumulated by midnight?

Example 2: The per pound price of lobster varies with the weight of the lobster. Generally, the greater the weight of the lobster, the more you pay per pound for it. Cook's Lobster House has a lobster pricing structure given below:

where w is the weight of the lobster, in pounds, and p is the price per pound for the lobster.

a) Graph this function on the axes provided.

b) Marty ordered a lobster that weighed in at pounds. How much did he pay for his lobster? Show the work that leads to your answer.

Price Per Pound ($)

Weight (pounds)

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c) Why is this considered a step function?

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