Warm Up for Lesson 3.5

1)Solve: x2 – 8x – 20 = 0

2) Sketch the graph of the equation y = 2x – 4

Graphing Quadratic FunctionsVertex Form

Section 3.5

Essential Question: How do I analyze and graph quadratic functions in vertex form?

The graph of a quadratic is called a parabola. We can use a table of values to graph any quadratic function.

x y-2-1012

4

4101

-8 -6 -4 -2 2 4 6 8

8

6

4

2

-2

-4

-6

-8

(1). y = x2

Characteristic of: y = x2 (a). Domain:(b). Range: (c). Vertex:(d). Axis of symmetry:(e). Opens:(f). Max or Min:(g). x-intercept: (h). y-intercept:(j). Increasing:(k). Decreasing:

All reals or -∞ < x < ∞y 0 or 0 ≤ y < ∞(0, 0)

x = 0upward

Min (0, 0)(0, 0)

x < 0 or -∞ < x < 0 x > 0 or 0 < x < ∞

We can shift the graph of y = x2 both horizontally and vertically.

2. y = (x – 2)2 + 3 is created by shifting the parent graph (y = x2) 2 units _____ and 3 units _____.

3. y = (x + 4)2 – 5 is created by shifting the parent graph (y = x2) 4 units _____ and 5 units _____.

4. y = -x2 is created by reflecting the parent graph (y = x2) about the ____ axis.

right up

left down

x

-8 -6 -4 -2 2 4 6 8

8

6

4

2

-2

-4

-6

-8

(5). Graph: y = (x + 2)2 – 1

V = (-2, -1)

x y

-2 -103

30

-10

-4-3

Identify each characteristic of: y = (x + 2)2 – 1 (a). Domain:(b). Range: (c). Vertex:(d). Axis of symmetry:(e). Opens:(f). Max or Min:(g). x-intercept:(h). y-intercept:*(i). Extrema:(j). Increasing:(k). Decreasing:*(l). Rate of change (0 ≤ x ≤ 4):

All reals or -∞ < x < ∞y -1 or -1 ≤ y < ∞(-2, -1)

x = -2upward

Min (-3, 0) and (-1, 0)(0, 3)Min value = -1x > -2 or -2 < x < ∞x < -2 or -∞ < x < -2

Estimated Rate of change over the interval (0 ≤ x ≤ 4): y = (x + 2)2 – 1

y = (0 + 2)2 – 1 = 4 – 1 = 3

y = (4 + 2)2 – 1 = 36 – 1 = 35

(0, 3)

(4, 35)

40353

m 84

32

-8 -6 -4 -2 2 4 6 8

8

6

4

2

-2

-4

-6

-8

(6). Graph: y = -(x – 4)2

V = (4, 0)

x y

4 032

56

-1-4

-1-4

Identify each characteristic of: y = -(x – 4)2 (a). Domain:(b). Range: (c). Vertex:(d). Axis of symmetry:(e). Opens:(f). Max or Min:(g). x-intercept:(h). y-intercept:*(i). Extrema:(j). Increasing:(k). Decreasing:*(l). Rate of change (4≤ x ≤ 5):

All reals or -∞ < x < ∞y ≤ 0 or -∞ < y ≤ 0(4, 0)

x = 4downward

Max (4, 0) (0, -16)Max value = 0x < 4 or -∞ < x < 4x > 4 or 4 < x < ∞

Estimated Rate of change over the interval (4 ≤ x ≤ 5): y = -(x – 4)2

y = -(4 – 4 )2 = -(0)2 = 0

y = -(5 – 4 )2 = -(1)2 = -1

(4, 0)

(5, -1)

54)1(0

m 11

1

Vertex Form for Quadratic Functions: y = a(x – h)2 + k

Vertex has coordinates (h, k).

• If a > 0, the parabola opens up (vertex min) If a < 0, the parabola opens down (vertex max)

• To find x-intercept (zeros), let y = 0.

• To find y-intercept, let x = 0.

• Axis of symmetry: x = h

• Extrema: max or min value is y-coordinate of vertex.

• Interval of increasing and decreasing: look at x- coordinate of vertex.

• Rate of change:

Note: since the graph is not linear, the rate of change will vary and will NOT have a constant value.

2 1

2 1

y ymx x

-8 -6 -4 -2 2 4 6 8

8

6

4

2

-2

-4

-6

-8

(7). Graph: y = ½(x + 3)2 – 4

V = (-3, -4)

x y

-3 -4-5-7

-11

-24

-24

Identify each characteristic of: y = ½(x + 3)2 – 4(a). Domain:(b). Range: (c). Vertex:(d). Axis of symmetry:(e). Opens:(f). Max or Min:(g). x-intercept: (h). y-intercept: (i). Extrema:(j). Increasing:(k). Decreasing:(l). Rate of change (-7 ≤ x ≤ -5):

All reals or -∞ < x < ∞y -4 or -4 ≤ y < ∞(-3, -4)

x = -3upward

Min

Min value = -4x > -3 or -3 < x < ∞ x < -3 or -∞ < x <-3

(-0.17, 0) and (-5.82, 0)(0, 0.5)

**x-intercepts: y = ½(x + 3)2 – 4 0 = ½(x + 3)2 – 4 4 = ½(x + 3)2

8 = (x + 3)2 2)3(8 x

38 x

x 83

83.583

17.083

**y-intercepts: y = ½(x + 3)2 – 4 y = ½(0 + 3)2 – 4 y = ½(3)2 – 4 y = 4.5 – 4 y = 0.5

Estimated Rate of change over the interval (-7 ≤ x ≤ -5): y = ½(x + 3)2 – 4

y = ½(-7 + 3)2 – 4 = ½(16) – 4 = 4

y = ½(-5 + 3)2 – 4 = ½(4) – 4 = -2

(-7, 4)

(-5, -2)

)5(7)2(4

m 32

6

Without graphing, find the x and y intercepts for the graph of:

(8). y = 7(x – 6)2 – 14

x-int: 14)6(70 2 x2)6(714 x

2)6(2 x

62 x

x 26

41.726 x

59.426 x

(7.41, 0) and (4.59, 0)

Without graphing, find the x and y intercepts for the graph of:

(8). y = 7(x – 6)2 – 14

y-int: 14)60(7 2 y

(0, 238)

14)36(7 y

14252 y

238y