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c: Solving & Graphing Quadratic Functions/Equation al Question: How can you solve quadratic equations? -Up : or 9p 2 – 100 d 4 + 4d 3 – 6d 2 – 4d e for x: x 2 + 13x + 6

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Name: Date: Topic: Solving & Graphing Quadratic Functions/Equations Essential Question: How can you solve quadratic equations? Warm-Up : Factor 1. 49p 2 – 100 2. 6d 4 + 4d 3 – 6d 2 – 4d Solve for x: 3. 2x 2 + 13x + 6. Quadratic Function (y = ax 2 + bx + c). y = x 2 – x – 2. - PowerPoint PPT Presentation

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Page 1: Name: Date: Topic:  Solving & Graphing Quadratic Functions/Equations

Name:Date:

Topic: Solving & Graphing Quadratic Functions/EquationsEssential Question: How can you solve quadratic equations?

Warm-Up:Factor1. 49p2 – 100

2. 6d4 + 4d3 – 6d2 – 4d

Solve for x:3. 2x2 + 13x + 6

Page 2: Name: Date: Topic:  Solving & Graphing Quadratic Functions/Equations

Quadratic Function(y = ax2 + bx + c)

y = 3x2

y = x2 + 9

y = x2 – x – 2

Page 3: Name: Date: Topic:  Solving & Graphing Quadratic Functions/Equations

Vocabulary:

1. Quadratic Parent Function

2. Parabola = the graph of a quadratic function is a U-shaped curved.

3. Axis of Symmetry – divide the graph into two halves

Continue

The line of symmetry ALWAYS passes through the vertex.

Page 4: Name: Date: Topic:  Solving & Graphing Quadratic Functions/Equations

y

x

Vertex

Minimum

Vertex

Maximum

4. Vertex• Minimum – lowest point

of the parabola• Maximum – the highest

point of the parabola.

Page 5: Name: Date: Topic:  Solving & Graphing Quadratic Functions/Equations

y = x2

a = 1, b = 0, c = 0

Minimum point (0,0)

Axis of symmetry x=0

y=x2

y = ax2 + bx + c

Page 6: Name: Date: Topic:  Solving & Graphing Quadratic Functions/Equations

Find the line of symmetry of y = 3x 2 – 18x + 7

Finding the Line of SymmetryWhen a quadratic function is in standard form

The equation of the line of symmetry is

y = ax2 + bx + c,

2ba

x

For example…

Using the formula…

18

2 3x 18

6 3

Thus, the line of symmetry is x = 3.

How do I find my vertex?

Finding the VertexWhat is the vertex?

Page 7: Name: Date: Topic:  Solving & Graphing Quadratic Functions/Equations

Another example:

We know the line of symmetry always goes through the vertex.Thus, the line of symmetry gives us the x – coordinate of the vertex.

To find the y – coordinate of the vertex, we need to plug the x – value into the original equation.

STEP 1: Find the line of symmetry

STEP 2: Plug the x – value into the original equation to find the y value.

y = –2x 2 + 8x –3

8 8 22 2( 2) 4ba

x

y = –2(2)2 + 8(2) –3

y = –2(4)+ 8(2) –3

y = –8+ 16 –3

y = 5

Therefore, the vertex is (2 , 5)

Page 8: Name: Date: Topic:  Solving & Graphing Quadratic Functions/Equations

STEP 1: Find the line of symmetry

Let's Graph ONE! Try …

y = 2x 2 – 4x – 1

A Quadratic Function in Standard Form

STEP 2: Find the vertex

STEP 4: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve.

STEP 3: Find the y-intercept.

x 2x 2 – 4x – 1 y (x, y)

Page 9: Name: Date: Topic:  Solving & Graphing Quadratic Functions/Equations

Step 5: Lets Graph it!

y = 2x2 – 4x – 1

A Quadratic Function in Standard Form

y

x

Page 10: Name: Date: Topic:  Solving & Graphing Quadratic Functions/Equations

What happen if we change the value of a and c ?

y=3x2

y=-3x2

y=4x2+3

y=-4x2-2

Page 11: Name: Date: Topic:  Solving & Graphing Quadratic Functions/Equations

Conclusion to Quadratic Function

(y = ax2+bx+c) When a is positive,

When a is negative,

When c is positive When c is negative

the graph concaves downward.

the graph concaves upward.

the graph moves up.

the graph moves down.

Page 12: Name: Date: Topic:  Solving & Graphing Quadratic Functions/Equations

Solving Quadratic Equations

Quadratic Formula

2 4

2

b b acx

a

x2 – 2x – 8 = 0

Method #1:

Hint: Quadratic equation, must equal 0

Page 13: Name: Date: Topic:  Solving & Graphing Quadratic Functions/Equations

x2 – 4x = 21

Example #2 2 4

2

b b acx

a

Page 14: Name: Date: Topic:  Solving & Graphing Quadratic Functions/Equations

Factoringx2 - 2x = 0

Factor in order to solve the equation.

y=x2-2x

Hints: Remember to ask yourself does the function have a GCF.Find the x intercept.

Answer: Two solutions, x=0 and x=2.

Method #2:

Page 15: Name: Date: Topic:  Solving & Graphing Quadratic Functions/Equations

Page 538 (8, 10) Page 546 (43)

Page 549 (a, b, c)

Group Work:

Independent Work:

Group 1:#7, #20

Group 2: #8, #21

Group 5: #11, #24

Group 6: #12, #25

Page 544 – 545

Group 3: #9, #22

Group 4: #10, #23

Group 7: #13, #28

Group 8: #14, #30

Page 16: Name: Date: Topic:  Solving & Graphing Quadratic Functions/Equations

Solve the following equations by factoring:

1. x2 = 25

2. x2 – 8 = - 7x

3. x2 – 12x = -36

4. x2 = 7x

5. m2 – 3m = 10

6. 2r – 8 = - r2

7. 4x2 = 49

8. x2 = 9x – 20

Page 17: Name: Date: Topic:  Solving & Graphing Quadratic Functions/Equations

y = -x2 + 2x + 8

Page 18: Name: Date: Topic:  Solving & Graphing Quadratic Functions/Equations

Find the Solutions

y=x2-4y=x2+2x-15

y=-x2+5

y=-x2-1

Page 19: Name: Date: Topic:  Solving & Graphing Quadratic Functions/Equations

Find the solutions

y=x2+2x+1

y=-x2+4x-1

Page 20: Name: Date: Topic:  Solving & Graphing Quadratic Functions/Equations

HLA#5:

Page 538 (7, 8)Page 544 (2, 3, 4)