9.1 – Graphing Quadratic Functions

Ex. 1 Use a table of values to graph the following functions.

a. y = 2x2 – 4x – 5

Ex. 1 Use a table of values to graph the following functions.

a. y = 2x2 – 4x – 5

x y

Ex. 1 Use a table of values to graph the following functions.

a. y = 2x2 – 4x – 5

x y

-2

-1

0

1

2

Ex. 1 Use a table of values to graph the following functions.

a. y = 2x2 – 4x – 5

x y

-2

-1

0

1

2

Ex. 1 Use a table of values to graph the following functions.

a. y = 2x2 – 4x – 5

y = 2(-2)2 – 4(-2) – 5 x y

-2

-1

0

1

2

Ex. 1 Use a table of values to graph the following functions.

a. y = 2x2 – 4x – 5

y = 2(-2)2 – 4(-2) – 5

y = 8 + 8 – 5 = 11x y

-2

-1

0

1

2

Ex. 1 Use a table of values to graph the following functions.

a. y = 2x2 – 4x – 5

y = 2(-2)2 – 4(-2) – 5

y = 8 + 8 – 5 = 11x y

-2 11

-1

0

1

2

Ex. 1 Use a table of values to graph the following functions.

a. y = 2x2 – 4x – 5

y = 2(-2)2 – 4(-2) – 5

y = 8 + 8 – 5 = 11x y

-2 11

-1 1

0 -5

1 -7

2 -5

Ex. 1 Use a table of values to graph the following functions.

a. y = 2x2 – 4x – 5

x y

-2 11

-1 1

0 -5

1 -7

2 -5

Ex. 1 Use a table of values to graph the following functions.

a. y = 2x2 – 4x – 5

x y

-2 11

-1 1

0 -5

1 -7

2 -5

Ex. 1 Use a table of values to graph the following functions.

a. y = 2x2 – 4x – 5

x y

-2 11

-1 1

0 -5

1 -7

2 -5

b. y = -x2 + 4x – 1

x y

-2

-1

0

1

2

b. y = -x2 + 4x – 1

x y

-2 -13

-1 -6

0 -1

1 2

2 3

b. y = -x2 + 4x – 1

x y

-2 -13

-1 -6

0 -1

1 2

2 3

b. y = -x2 + 4x – 1

x y

-2 -13

-1 -6

0 -1

1 2

2 3

3

4

b. y = -x2 + 4x – 1

x y

-2 -13

-1 -6

0 -1

1 2

2 3

3 2

4 -1

b. y = -x2 + 4x – 1

x y

-2 -13

-1 -6

0 -1

1 2

2 3

3 2

4 -1

b. y = -x2 + 4x – 1

x y

-2 -13

-1 -6

0 -1

1 2

2 3

3 2

4 -1

• Axis of symmetry:

• Axis of symmetry: x = - b 2a

• Axis of symmetry: x = - b 2a

• Vertex:

• Axis of symmetry: x = - b 2a

• Vertex: (x, y)

• Axis of symmetry: x = - b 2a

• Vertex: (x, y), where the x = axis of sym.

• Axis of symmetry: x = - b 2a

• Vertex: (x, y), where the x = axis of sym.• Maximum vs. Minimum:

• Axis of symmetry: x = - b 2a

• Vertex: (x, y), where the x = axis of sym.• Maximum vs. Minimum: For ax2 + bx + c,

• Axis of symmetry: x = - b 2a

• Vertex: (x, y), where the x = axis of sym.• Maximum vs. Minimum: For ax2 + bx + c,

–If a is positive, then the vertex is a Minimum.

• Axis of symmetry: x = - b 2a

• Vertex: (x, y), where the x = axis of sym.• Maximum vs. Minimum: For ax2 + bx + c,

–If a is positive, then the vertex is a Minimum.

–If a is negative, then the vertex is a Maximum.

• Axis of symmetry: x = - b 2a

• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,

– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.

Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.

• Axis of symmetry: x = - b 2a

• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,

– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.

Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.

a. -x2 + 2x + 3

• Axis of symmetry: x = - b 2a

• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,

– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.

Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.

a. -x2 + 2x + 3

1) axis of sym.:

• Axis of symmetry: x = - b 2a

• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,

– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.

Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.

a. -x2 + 2x + 31) axis of sym.: x = - b

2a

• Axis of symmetry: x = - b 2a

• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,

– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.

Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.

a. -x2 + 2x + 31) axis of sym.: x = - b = - 2

2a 2(-1)

• Axis of symmetry: x = - b 2a

• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,

– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.

Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.

a. -x2 + 2x + 31) axis of sym.: x = - b = - 2 = -2 = 1

2a 2(-1) -2

• Axis of symmetry: x = - b 2a

• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,

– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.

Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.

a. -x2 + 2x + 31) axis of sym.: x = - b = - 2 = -2 = 1

2a 2(-1) -22) vertex:

• Axis of symmetry: x = - b 2a

• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,

– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.

Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.

a. -x2 + 2x + 31) axis of sym.: x = - b = - 2 = -2 = 1

2a 2(-1) -22) vertex: (x, y)

• Axis of symmetry: x = - b 2a

• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,

– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.

Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.

a. -x2 + 2x + 31) axis of sym.: x = - b = - 2 = -2 = 1

2a 2(-1) -22) vertex: (x, y) = (1,

• Axis of symmetry: x = - b 2a

• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,

– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.

Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.

a. -x2 + 2x + 31) axis of sym.: x = - b = - 2 = -2 = 1

2a 2(-1) -22) vertex: (x, y) = (1, ?)

• Axis of symmetry: x = - b 2a

• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,

– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.

Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.

a. -x2 + 2x + 31) axis of sym.: x = - b = - 2 = -2 = 1

2a 2(-1) -22) vertex: (x, y) = (1, ?)

-x2 + 2x + 3

• Axis of symmetry: x = - b 2a

• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,

– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.

Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.

a. -x2 + 2x + 31) axis of sym.: x = - b = - 2 = -2 = 1

2a 2(-1) -22) vertex: (x, y) = (1, ?)

-x2 + 2x + 3 -(1)2 + 2(1) + 3

• Axis of symmetry: x = - b 2a

• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,

– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.

Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.

a. -x2 + 2x + 31) axis of sym.: x = - b = - 2 = -2 = 1

2a 2(-1) -22) vertex: (x, y) = (1, ?)

-x2 + 2x + 3 -(1)2 + 2(1) + 3-1 + 2 + 3

• Axis of symmetry: x = - b 2a

• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,

– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.

Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.

a. -x2 + 2x + 31) axis of sym.: x = - b = - 2 = -2 = 1

2a 2(-1) -22) vertex: (x, y) = (1, ?)

-x2 + 2x + 3 -(1)2 + 2(1) + 3-1 + 2 + 3 = 4

• Axis of symmetry: x = - b 2a

• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,

– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.

Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.

a. -x2 + 2x + 31) axis of sym.: x = - b = - 2 = -2 = 1

2a 2(-1) -22) vertex: (x, y) = (1, ?)

-x2 + 2x + 3 -(1)2 + 2(1) + 3-1 + 2 + 3 = 4, so (1, 4)

• Axis of symmetry: x = - b 2a

• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,

– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.

Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.

a. -x2 + 2x + 31) axis of sym.: x = - b = - 2 = -2 = 1

2a 2(-1) -22) vertex: (x, y) = (1, ?)

-x2 + 2x + 3 -(1)2 + 2(1) + 3-1 + 2 + 3 = 4, so (1, 4)

3) Max OR Min.?

• Axis of symmetry: x = - b 2a

• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,

– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.

Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.

a. -x2 + 2x + 31) axis of sym.: x = - b = - 2 = -2 = 1

2a 2(-1) -22) vertex: (x, y) = (1, ?)

-x2 + 2x + 3 -(1)2 + 2(1) + 3-1 + 2 + 3 = 4, so (1, 4)

3) Max OR Min.? (1, 4) is a max b/c a is neg.

4) Graph:

4) Graph:

*Plot vertex:

4) Graph:

*Plot vertex: (1, 4)

4) Graph:

*Plot vertex: (1, 4)

*Make a table

based on vertex

4) Graph:

*Plot vertex: (1, 4)

* Make a table

based on vertexx y

1 4

4) Graph:

*Plot vertex: (1, 4)

* Make a table

based on vertexx y

0

1 4

4) Graph:

*Plot vertex: (1, 4)

* Make a table

based on vertexx y

-1

0

1 4

4) Graph:

*Plot vertex: (1, 4)

* Make a table

based on vertexx y

-1

0

1 4

2

4) Graph:

*Plot vertex: (1, 4)

* Make a table

based on vertexx y

-1

0

1 4

2

3

4) Graph:

*Plot vertex: (1, 4)

* Make a table

based on vertexx y

-1 0

0 3

1 4

2 3

3 0

4) Graph:

*Plot vertex: (1, 4)

* Make a table

based on vertexx y

-1 0

0 3

1 4

2 3

3 0

4) Graph:

*Plot vertex: (1, 4)

* Make a table

based on vertexx y

-1 0

0 3

1 4

2 3

3 0

b. 2x2 – 4x – 5

b. 2x2 – 4x – 5

1) axis of sym.:

2) vertex: (x, y) =

3) Max OR Min.?

4) Graph:

b. 2x2 – 4x – 5

1) axis of sym.: x = - b = -(-4) = 4 = 1

2a 2(2) 4

2) vertex: (x, y) = (1, ?)

2x2 – 4x – 5

2(1)2 – 4(1) – 5

2 – 4 – 5 = -7, so (1, -7)

3) Max OR Min.? (1, -7) is a min b/c a is neg.

4) Graph:

*Plot vertex: (1, -7)

* Make a table

based on vertex

x y

-1 1

0 -5

1 -7

2 -5

3 1