Vertex Form Graphs | Math Coach Amy

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Key Ideas Easy Easy 2

y = a(x − h)² + k

vertex (h, k) · axis x = h

What to Remember

Vertex form

y = a(x − h)² + k has vertex (h, k).

The axis of symmetry is x = h.

Opens up or down

If a is positive, the vertex is a minimum. If a is negative, the vertex is a maximum. The min/max value is k.

Y-intercept

Plug in x = 0. Domain of a quadratic is all real numbers. Range starts or ends at k.

Shifts, then the graph

Under the parentheses, write Right or Left. Under k, write Up or Down.

Sketch the parabola, then list the axis, vertex, min or max, y-intercept, domain, and range.

Level 1

Easy

a = 1. Read (h, k) from the equation.

Example

Given y = (x − 2)² + 3. Sketch the graph and list the following. Give the domain and range in interval notation.a. Axis of symmetryb. Vertexc. Minimum or maximumd. Y-intercepte. Domainf. Range

y =	(x − 2)²	+ 3
	Right 2	Up 3

a. Axis of symmetry (must be an equation) x = 2

b. Vertex (2, 3)

c. Minimum or maximum (only y-value) min 3

d. Y-intercept (plug x = 0 into the equation)

y = (0 − 2)² + 3 = 7, (0, 7)

e. Domain (forever wide): (−∞, ∞)

f. Range (how low can you go?) [3, ∞)

Answer: a. Axis of Symmetry: x = 2b. Vertex: (2, 3)c. Minimum: 3d. Y-intercept: (0, 7)e. Domain: (−∞, ∞)f. Range: [3, ∞)

Given y = (x + 1)² − 4. Sketch the graph and list the following. Give the domain and range in interval notation.a. Axis of symmetryb. Vertexc. Minimum or maximumd. Y-intercepte. Domainf. Range

Show Answer

y =	(x + 1)²	− 4
	Left 1	Down 4

a. Axis of symmetry (must be an equation) x = −1

b. Vertex (−1, −4)

c. Minimum or maximum (only y-value) min −4

d. Y-intercept (plug x = 0 into the equation)

y = (0 + 1)² − 4 = −3, (0, −3)

e. Domain (forever wide): (−∞, ∞)

f. Range (how low can you go?) [−4, ∞)

Answer: a. Axis of Symmetry: x = −1b. Vertex: (−1, −4)c. Minimum: −4d. Y-intercept: (0, −3)e. Domain: (−∞, ∞)f. Range: [−4, ∞)

Given y = (x − 3)² + 5. Sketch the graph and list the following. Give the domain and range in interval notation.a. Axis of symmetryb. Vertexc. Minimum or maximumd. Y-intercepte. Domainf. Range

Show Answer

y =	(x − 3)²	+ 5
	Right 3	Up 5

a. Axis of symmetry (must be an equation) x = 3

b. Vertex (3, 5)

c. Minimum or maximum (only y-value) min 5

d. Y-intercept (plug x = 0 into the equation)

y = (0 − 3)² + 5 = 14, (0, 14)

e. Domain (forever wide): (−∞, ∞)

f. Range (how low can you go?) [5, ∞)

Answer: a. Axis of Symmetry: x = 3b. Vertex: (3, 5)c. Minimum: 5d. Y-intercept: (0, 14)e. Domain: (−∞, ∞)f. Range: [5, ∞)

Given y = (x + 2)². Sketch the graph and list the following. Give the domain and range in interval notation.a. Axis of symmetryb. Vertexc. Minimum or maximumd. Y-intercepte. Domainf. Range

Show Answer

y =	(x + 2)²
	Left 2

a. Axis of symmetry (must be an equation) x = −2

b. Vertex (−2, 0)

c. Minimum or maximum (only y-value) min 0

d. Y-intercept (plug x = 0 into the equation)

y = (0 + 2)² = 4, (0, 4)

e. Domain (forever wide): (−∞, ∞)

f. Range (how low can you go?) [0, ∞)

Answer: a. Axis of Symmetry: x = −2b. Vertex: (−2, 0)c. Minimum: 0d. Y-intercept: (0, 4)e. Domain: (−∞, ∞)f. Range: [0, ∞)

Given y = (x − 4)² − 1. Sketch the graph and list the following. Give the domain and range in interval notation.a. Axis of symmetryb. Vertexc. Minimum or maximumd. Y-intercepte. Domainf. Range

Show Answer

y =	(x − 4)²	− 1
	Right 4	Down 1

a. Axis of symmetry (must be an equation) x = 4

b. Vertex (4, −1)

c. Minimum or maximum (only y-value) min −1

d. Y-intercept (plug x = 0 into the equation)

y = (0 − 4)² − 1 = 15, (0, 15)

e. Domain (forever wide): (−∞, ∞)

f. Range (how low can you go?) [−1, ∞)

Answer: a. Axis of Symmetry: x = 4b. Vertex: (4, −1)c. Minimum: −1d. Y-intercept: (0, 15)e. Domain: (−∞, ∞)f. Range: [−1, ∞)

Given y = (x + 5)² + 2. Sketch the graph and list the following. Give the domain and range in interval notation.a. Axis of symmetryb. Vertexc. Minimum or maximumd. Y-intercepte. Domainf. Range

Show Answer

y =	(x + 5)²	+ 2
	Left 5	Up 2

a. Axis of symmetry (must be an equation) x = −5

b. Vertex (−5, 2)

c. Minimum or maximum (only y-value) min 2

d. Y-intercept (plug x = 0 into the equation)

y = (0 + 5)² + 2 = 27, (0, 27)

e. Domain (forever wide): (−∞, ∞)

f. Range (how low can you go?) [2, ∞)

Answer: a. Axis of Symmetry: x = −5b. Vertex: (−5, 2)c. Minimum: 2d. Y-intercept: (0, 27)e. Domain: (−∞, ∞)f. Range: [2, ∞)

Level 2

Easy 2

a is negative. The vertex is a maximum.

Example

Given y = −2(x + 1)² + 6. Sketch the graph and list the following. Give the domain and range in interval notation.a. Axis of symmetryb. Vertexc. Minimum or maximumd. Y-intercepte. Domainf. Range

y =	−2(x + 1)²	+ 6
	Left 1	Up 6

a. Axis of symmetry (must be an equation) x = −1

b. Vertex (−1, 6)

c. Minimum or maximum (only y-value) max 6

d. Y-intercept (plug x = 0 into the equation)

y = −2(0 + 1)² + 6 = −2 · 1 + 6 = 4, (0, 4)

e. Domain (forever wide): (−∞, ∞)

f. Range (how high can you go?) (−∞, 6]

Answer: a. Axis of Symmetry: x = −1b. Vertex: (−1, 6)c. Maximum: 6d. Y-intercept: (0, 4)e. Domain: (−∞, ∞)f. Range: (−∞, 6]

Given y = −3(x − 2)² + 9. Sketch the graph and list the following. Give the domain and range in interval notation.a. Axis of symmetryb. Vertexc. Minimum or maximumd. Y-intercepte. Domainf. Range

Show Answer

y =	−3(x − 2)²	+ 9
	Right 2	Up 9

a. Axis of symmetry (must be an equation) x = 2

b. Vertex (2, 9)

c. Minimum or maximum (only y-value) max 9

d. Y-intercept (plug x = 0 into the equation)

y = −3(0 − 2)² + 9 = −3 · 4 + 9 = −3, (0, −3)

e. Domain (forever wide): (−∞, ∞)

f. Range (how high can you go?) (−∞, 9]

Answer: a. Axis of Symmetry: x = 2b. Vertex: (2, 9)c. Maximum: 9d. Y-intercept: (0, −3)e. Domain: (−∞, ∞)f. Range: (−∞, 9]

Given y = −(x + 4)² + 5. Sketch the graph and list the following. Give the domain and range in interval notation.a. Axis of symmetryb. Vertexc. Minimum or maximumd. Y-intercepte. Domainf. Range

Show Answer

y =	−(x + 4)²	+ 5
	Left 4	Up 5

a. Axis of symmetry (must be an equation) x = −4

b. Vertex (−4, 5)

c. Minimum or maximum (only y-value) max 5

d. Y-intercept (plug x = 0 into the equation)

y = −(0 + 4)² + 5 = −16 + 5 = −11, (0, −11)

e. Domain (forever wide): (−∞, ∞)

f. Range (how high can you go?) (−∞, 5]

Answer: a. Axis of Symmetry: x = −4b. Vertex: (−4, 5)c. Maximum: 5d. Y-intercept: (0, −11)e. Domain: (−∞, ∞)f. Range: (−∞, 5]

Given y = −2(x − 3)². Sketch the graph and list the following. Give the domain and range in interval notation.a. Axis of symmetryb. Vertexc. Minimum or maximumd. Y-intercepte. Domainf. Range

Show Answer

y =	−2(x − 3)²
	Right 3

a. Axis of symmetry (must be an equation) x = 3

b. Vertex (3, 0)

c. Minimum or maximum (only y-value) max 0

d. Y-intercept (plug x = 0 into the equation)

y = −2(0 − 3)² = −2 · 9 = −18, (0, −18)

e. Domain (forever wide): (−∞, ∞)

f. Range (how high can you go?) (−∞, 0]

Answer: a. Axis of Symmetry: x = 3b. Vertex: (3, 0)c. Maximum: 0d. Y-intercept: (0, −18)e. Domain: (−∞, ∞)f. Range: (−∞, 0]

Given y = −4(x + 2)² + 8. Sketch the graph and list the following. Give the domain and range in interval notation.a. Axis of symmetryb. Vertexc. Minimum or maximumd. Y-intercepte. Domainf. Range

Show Answer

y =	−4(x + 2)²	+ 8
	Left 2	Up 8

a. Axis of symmetry (must be an equation) x = −2

b. Vertex (−2, 8)

c. Minimum or maximum (only y-value) max 8

d. Y-intercept (plug x = 0 into the equation)

y = −4(0 + 2)² + 8 = −4 · 4 + 8 = −8, (0, −8)

e. Domain (forever wide): (−∞, ∞)

f. Range (how high can you go?) (−∞, 8]

Answer: a. Axis of Symmetry: x = −2b. Vertex: (−2, 8)c. Maximum: 8d. Y-intercept: (0, −8)e. Domain: (−∞, ∞)f. Range: (−∞, 8]

Given y = −(x − 1)² − 3. Sketch the graph and list the following. Give the domain and range in interval notation.a. Axis of symmetryb. Vertexc. Minimum or maximumd. Y-intercepte. Domainf. Range

Show Answer

y =	−(x − 1)²	− 3
	Right 1	Down 3

a. Axis of symmetry (must be an equation) x = 1

b. Vertex (1, −3)

c. Minimum or maximum (only y-value) max −3

d. Y-intercept (plug x = 0 into the equation)

y = −(0 − 1)² − 3 = −1 − 3 = −4, (0, −4)

e. Domain (forever wide): (−∞, ∞)

f. Range (how high can you go?) (−∞, −3]

Answer: a. Axis of Symmetry: x = 1b. Vertex: (1, −3)c. Maximum: −3d. Y-intercept: (0, −4)e. Domain: (−∞, ∞)f. Range: (−∞, −3]

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