Vertex Formof

Quadratics

November 10, 2014Notes

Objective

relate representations of quadratic functions, such as algebraic, tabular, graphical, and verbal descriptions[6.B]

Essential Question

• What parts of a quadratic function can I determine from the vertex form and how can I graph it?

Vocabulary• parabola: the shape of a quadratic function• vertex: the highest (maximum) or lowest

(minimum) point on a parabola; the turning point

• y-intercept: the point where the graph crosses the y-axis; x = 0

• x-intercepts: the points where the graph crosses the x-axis; y = 0; may be none, 1, or 2

• axis of symmetry: the vertical line that divides a parabola into two equal parts; goes through the vertex

Vertex Form

• f(x) = a(x – h)2 + k

– ‘a’ reflection across the x-axis (PEMDAS order) and/or vertical stretch or compression

– ‘h’ horizontal translation– ‘k’: vertical translation

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k

h

k

Vertex:(h, k)

Axis of symmetry:

x = h

y-intercept:(0, y)

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2

h

k

Vertex:(h, k)

Axis of symmetry:

x = h

y-intercept:(0, y)

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2

h 2

k

Vertex:(h, k)

Axis of symmetry:

x = h

y-intercept:(0, y)

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2

h 2

k 0

Vertex:(h, k)

Axis of symmetry:

x = h

y-intercept:(0, y)

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2

h 2

k 0

Vertex:(h, k) (2, 0)

Axis of symmetry:

x = h

y-intercept:(0, y)

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2

h 2

k 0

Vertex:(h, k) (2, 0)

Axis of symmetry:

x = hx = 2

y-intercept:(0, y)

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2

h 2

k 0

Vertex:(h, k) (2, 0)

Axis of symmetry:

x = hx = 2

y-intercept:(0, y) (0, 4)

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1

h 2

k 0

Vertex:(h, k) (2, 0)

Axis of symmetry:

x = hx = 2

y-intercept:(0, y) (0, 4)

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1

h 2 -3

k 0

Vertex:(h, k) (2, 0)

Axis of symmetry:

x = hx = 2

y-intercept:(0, y) (0, 4)

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1

h 2 -3

k 0 -1

Vertex:(h, k) (2, 0)

Axis of symmetry:

x = hx = 2

y-intercept:(0, y) (0, 4)

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1

h 2 -3

k 0 -1

Vertex:(h, k) (2, 0) (-3, -1)

Axis of symmetry:

x = hx = 2

y-intercept:(0, y) (0, 4)

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1

h 2 -3

k 0 -1

Vertex:(h, k) (2, 0) (-3, -1)

Axis of symmetry:

x = hx = 2 x = -3

y-intercept:(0, y) (0, 4)

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1

h 2 -3

k 0 -1

Vertex:(h, k) (2, 0) (-3, -1)

Axis of symmetry:

x = hx = 2 x = -3

y-intercept:(0, y) (0, 4) (0, 8)

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1 y= -3(x+2)2+4

h 2 -3

k 0 -1

Vertex:(h, k) (2, 0) (-3, -1)

Axis of symmetry:

x = hx = 2 x = -3

y-intercept:(0, y) (0, 4) (0, 8)

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1 y= -3(x+2)2+4

h 2 -3 -2

k 0 -1

Vertex:(h, k) (2, 0) (-3, -1)

Axis of symmetry:

x = hx = 2 x = -3

y-intercept:(0, y) (0, 4) (0, 8)

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1 y= -3(x+2)2+4

h 2 -3 -2

k 0 -1 4

Vertex:(h, k) (2, 0) (-3, -1)

Axis of symmetry:

x = hx = 2 x = -3

y-intercept:(0, y) (0, 4) (0, 8)

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1 y= -3(x+2)2+4

h 2 -3 -2

k 0 -1 4

Vertex:(h, k) (2, 0) (-3, -1) (-2, 4)

Axis of symmetry:

x = hx = 2 x = -3

y-intercept:(0, y) (0, 4) (0, 8)

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1 y= -3(x+2)2+4

h 2 -3 -2

k 0 -1 4

Vertex:(h, k) (2, 0) (-3, -1) (-2, 4)

Axis of symmetry:

x = hx = 2 x = -3 x = -2

y-intercept:(0, y) (0, 4) (0, 8)

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1 y= -3(x+2)2+4

h 2 -3 -2

k 0 -1 4

Vertex:(h, k) (2, 0) (-3, -1) (-2, 4)

Axis of symmetry:

x = hx = 2 x = -3 x = -2

y-intercept:(0, y) (0, 4) (0, 8) (0, -8)

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1 y= -3(x+2)2+4 y= 2(x+3)2+1

h 2 -3 -2

k 0 -1 4

Vertex:(h, k) (2, 0) (-3, -1) (-2, 4)

Axis of symmetry:

x = hx = 2 x = -3 x = -2

y-intercept:(0, y) (0, 4) (0, 8) (0, -8)

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1 y= -3(x+2)2+4 y= 2(x+3)2+1

h 2 -3 -2 -3

k 0 -1 4

Vertex:(h, k) (2, 0) (-3, -1) (-2, 4)

Axis of symmetry:

x = hx = 2 x = -3 x = -2

y-intercept:(0, y) (0, 4) (0, 8) (0, -8)

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1 y= -3(x+2)2+4 y= 2(x+3)2+1

h 2 -3 -2 -3

k 0 -1 4 1

Vertex:(h, k) (2, 0) (-3, -1) (-2, 4)

Axis of symmetry:

x = hx = 2 x = -3 x = -2

y-intercept:(0, y) (0, 4) (0, 8) (0, -8)

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1 y= -3(x+2)2+4 y= 2(x+3)2+1

h 2 -3 -2 -3

k 0 -1 4 1

Vertex:(h, k) (2, 0) (-3, -1) (-2, 4) (-3, 1)

Axis of symmetry:

x = hx = 2 x = -3 x = -2

y-intercept:(0, y) (0, 4) (0, 8) (0, -8)

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1 y= -3(x+2)2+4 y= 2(x+3)2+1

h 2 -3 -2 -3

k 0 -1 4 1

Vertex:(h, k) (2, 0) (-3, -1) (-2, 4) (-3, 1)

Axis of symmetry:

x = hx = 2 x = -3 x = -2 x = -3

y-intercept:(0, y) (0, 4) (0, 8) (0, -8)

What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1 y= -3(x+2)2+4 y= 2(x+3)2+1

h 2 -3 -2 -3

k 0 -1 4 1

Vertex:(h, k) (2, 0) (-3, -1) (-2, 4) (-3, 1)

Axis of symmetry:

x = hx = 2 x = -3 x = -2 x = -3

y-intercept:(0, y) (0, 4) (0, 8) (0, -8) (0, 19)

To Graph from Vertex Form:

1. Identify the vertex and axis of symmetry and graph.

2. Find the y-intercept and graph along with its reflection across the axis of symmetry.

3. Make a table (with the vertex in the middle) to calculate at least 5 points on the parabola.

Example 1 (Left Side): y = (x + 3)2 - 1

h:_______ k:_______ vertex:__________axis of symmetry: ___________ y-int: ________

x y

-3 -1 (-3, -1)x = -3

y = (x + 3)2 – 1y = (0 + 3)2 – 1y = (3)2 – 1y = 9 – 1y = 8

(0, 8)

-3 -1-4 0-5 3

-2 0-1 3

y = -3(x – 2)2 + 4y = -3(0 – 2)2 + 4y = -3(-2)2 + 4y = -3 • 4 + 4y = -8

Example 2 (Left Side): y = -3(x – 2)2 + 4

h:_______ k:_______ vertex:__________axis of symmetry: ___________ y-int: ________

2 4 (2, 4)x = 2 (0, -8)

2 4 1 1 0 -8

3 1 4 -8

x y

Assignment

1. f(x) = x2 – 2

2. g(x) = -(x – 4)2

3. h(x) = (x + 1)2 – 3

4. j(x) = (x + 2)2 + 2

Reflection

1. How do you know if a parabola will open upward or downward?

2. When does the parabola have a maximum point?

3. When does the parabola have a minimum point?