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Math Coach Amy
Amy Ferguson Moncure

H Algebra 2/Trig · Unit 2

Vertex Form from a Point

Plug the vertex into vertex form, then use the given point to find a.

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Key Ideas Easy · Integer a Medium · Fractional a Hard · Fractional Vertex

y = a(x − h)² + k

plug in the point to find a

Start with Vertex Form
Vertex (3, 8).

The formula is

y = a(x − h)² + k

Inside the parentheses, move left and right, opposite of the inside.

Right 3 is (x − 3)².

Outside the parentheses, move up and down. It is what it is.

Up 8 is +8.

y = a(x − 3)² + 8

Use the given point
Start with y = a(x − 3)² + 8. The point is (4, 10).

y = a(x − 3)² + 8

Substitute (4, 10).

10 = a(4 − 3)² + 8

10 = a(1)² + 8

10 = a + 8

10 − 8 = a

a = 2

y = 2(x − 3)² + 8

Easy

Integer Values of a

a is a positive or negative integer.

Example
Write an equation in vertex form for the parabola with vertex (2, 5) that passes through (4, −3).

The formula is

y = a(x − h)² + k

Vertex (2, 5).

Inside the parentheses, move left and right, opposite of the inside.

Right 2 is (x − 2)².

Outside the parentheses, move up and down. It is what it is.

Up 5 is +5.

y = a(x − 2)² + 5

Substitute the point (4, −3).

−3 = a(4 − 2)² + 5

−3 = a(2)² + 5

−3 = 4a + 5

−3 − 5 = 4a

−8 = 4a

a = −84

a = −2

y = −2(x − 2)² + 5

Answer: y = −2(x − 2)² + 5

1
Write an equation in vertex form for the parabola with vertex (1, 2) that passes through (2, 5).
Show Answer

The formula is

y = a(x − h)² + k

Vertex (1, 2).

Inside the parentheses, move left and right, opposite of the inside.

Right 1 is (x − 1)².

Outside the parentheses, move up and down. It is what it is.

Up 2 is +2.

y = a(x − 1)² + 2

Substitute the point (2, 5).

5 = a(2 − 1)² + 2

5 = a(1)² + 2

5 − 2 = a

3 = a

a = 3

y = 3(x − 1)² + 2

Answer: y = 3(x − 1)² + 2

2
Write an equation in vertex form for the parabola with vertex (0, 6) that passes through (2, 2).
Show Answer

The formula is

y = a(x − h)² + k

Vertex (0, 6).

Inside the parentheses, move left and right, opposite of the inside.

The vertex is on the y-axis, so the squared part is x².

Outside the parentheses, move up and down. It is what it is.

Up 6 is +6.

y = ax² + 6

Substitute the point (2, 2).

2 = a(2)² + 6

2 = 4a + 6

2 − 6 = 4a

−4 = 4a

a = −44

a = −1

y = −x² + 6

Answer: y = −x² + 6

3
Write an equation in vertex form for the parabola with vertex (−2, −1) that passes through (0, 7).
Show Answer

The formula is

y = a(x − h)² + k

Vertex (−2, −1).

Inside the parentheses, move left and right, opposite of the inside.

Left 2 is (x + 2)².

Outside the parentheses, move up and down. It is what it is.

Down 1 is −1.

y = a(x + 2)² − 1

Substitute the point (0, 7).

7 = a(0 + 2)² − 1

7 = a(2)² − 1

7 = 4a − 1

7 + 1 = 4a

8 = 4a

a = 84

a = 2

y = 2(x + 2)² − 1

Answer: y = 2(x + 2)² − 1

4
Write an equation in vertex form for the parabola with vertex (3, 4) that passes through (5, −8).
Show Answer

The formula is

y = a(x − h)² + k

Vertex (3, 4).

Inside the parentheses, move left and right, opposite of the inside.

Right 3 is (x − 3)².

Outside the parentheses, move up and down. It is what it is.

Up 4 is +4.

y = a(x − 3)² + 4

Substitute the point (5, −8).

−8 = a(5 − 3)² + 4

−8 = a(2)² + 4

−8 = 4a + 4

−8 − 4 = 4a

−12 = 4a

a = −124

a = −3

y = −3(x − 3)² + 4

Answer: y = −3(x − 3)² + 4

5
Write an equation in vertex form for the parabola with vertex (−5, 0) that passes through (−4, 4).
Show Answer

The formula is

y = a(x − h)² + k

Vertex (−5, 0).

Inside the parentheses, move left and right, opposite of the inside.

Left 5 is (x + 5)².

Outside the parentheses, move up and down. It is what it is.

The vertex is on the x-axis, so add 0.

y = a(x + 5)²

Substitute the point (−4, 4).

4 = a(−4 + 5)²

4 = a(1)²

a = 4

y = 4(x + 5)²

Answer: y = 4(x + 5)²

Medium

Fractional Values of a

a is a positive or negative fraction.

Example
Write an equation in vertex form for the parabola with vertex (0, 1) that passes through (2, 3).

The formula is

y = a(x − h)² + k

Vertex (0, 1).

Inside the parentheses, move left and right, opposite of the inside.

The vertex is on the y-axis, so the squared part is x².

Outside the parentheses, move up and down. It is what it is.

Up 1 is +1.

y = ax² + 1

Substitute the point (2, 3).

3 = a(2)² + 1

3 = 4a + 1

3 − 1 = 4a

2 = 4a

a = 24

a = 12

y = 12x² + 1

Answer: y = 12x² + 1

1
Write an equation in vertex form for the parabola with vertex (1, 2) that passes through (3, 4).
Show Answer

The formula is

y = a(x − h)² + k

Vertex (1, 2).

Inside the parentheses, move left and right, opposite of the inside.

Right 1 is (x − 1)².

Outside the parentheses, move up and down. It is what it is.

Up 2 is +2.

y = a(x − 1)² + 2

Substitute the point (3, 4).

4 = a(3 − 1)² + 2

4 = a(2)² + 2

4 = 4a + 2

4 − 2 = 4a

2 = 4a

a = 24

a = 12

y = 12(x − 1)² + 2

Answer: y = 12(x − 1)² + 2

2
Write an equation in vertex form for the parabola with vertex (0, 5) that passes through (2, 3).
Show Answer

The formula is

y = a(x − h)² + k

Vertex (0, 5).

Inside the parentheses, move left and right, opposite of the inside.

The vertex is on the y-axis, so the squared part is x².

Outside the parentheses, move up and down. It is what it is.

Up 5 is +5.

y = ax² + 5

Substitute the point (2, 3).

3 = a(2)² + 5

3 = 4a + 5

3 − 5 = 4a

−2 = 4a

a = −24

a = −12

y = −12x² + 5

Answer: y = −12x² + 5

3
Write an equation in vertex form for the parabola with vertex (−2, 1) that passes through (1, 4).
Show Answer

The formula is

y = a(x − h)² + k

Vertex (−2, 1).

Inside the parentheses, move left and right, opposite of the inside.

Left 2 is (x + 2)².

Outside the parentheses, move up and down. It is what it is.

Up 1 is +1.

y = a(x + 2)² + 1

Substitute the point (1, 4).

4 = a(1 + 2)² + 1

4 = a(3)² + 1

4 = 9a + 1

4 − 1 = 9a

3 = 9a

a = 39

a = 13

y = 13(x + 2)² + 1

Answer: y = 13(x + 2)² + 1

4
Write an equation in vertex form for the parabola with vertex (2, 0) that passes through (4, 6).
Show Answer

The formula is

y = a(x − h)² + k

Vertex (2, 0).

Inside the parentheses, move left and right, opposite of the inside.

Right 2 is (x − 2)².

Outside the parentheses, move up and down. It is what it is.

The vertex is on the x-axis, so add 0.

y = a(x − 2)²

Substitute the point (4, 6).

6 = a(4 − 2)²

6 = a(2)²

6 = 4a

a = 64

a = 32

y = 32(x − 2)²

Answer: y = 32(x − 2)²

5
Write an equation in vertex form for the parabola with vertex (−4, 2) that passes through (−1, −1).
Show Answer

The formula is

y = a(x − h)² + k

Vertex (−4, 2).

Inside the parentheses, move left and right, opposite of the inside.

Left 4 is (x + 4)².

Outside the parentheses, move up and down. It is what it is.

Up 2 is +2.

y = a(x + 4)² + 2

Substitute the point (−1, −1).

−1 = a(−1 + 4)² + 2

−1 = a(3)² + 2

−1 = 9a + 2

−1 − 2 = 9a

−3 = 9a

a = −39

a = −13

y = −13(x + 4)² + 2

Answer: y = −13(x + 4)² + 2

Hard

Fractional Vertex

One or both parts of the vertex are fractions. a is an integer.

Example
Write an equation in vertex form for the parabola with vertex (12, 4) that passes through (52, 8).

The formula is

y = a(x − h)² + k

Vertex (12, 4).

Inside the parentheses, move left and right, opposite of the inside.

Right 12 is (x − 12)².

Outside the parentheses, move up and down. It is what it is.

Up 4 is +4.

y = a(x − 12)² + 4

Substitute the point (52, 8).

8 = a(52 − 12)² + 4

8 = a(2)² + 4

8 = 4a + 4

8 − 4 = 4a

4 = 4a

a = 44

a = 1

y = (x − 12)² + 4

Answer: y = (x − 12)² + 4

1
Write an equation in vertex form for the parabola with vertex (12, 3) that passes through (32, 7).
Show Answer

The formula is

y = a(x − h)² + k

Vertex (12, 3).

Inside the parentheses, move left and right, opposite of the inside.

Right 12 is (x − 12)².

Outside the parentheses, move up and down. It is what it is.

Up 3 is +3.

y = a(x − 12)² + 3

Substitute the point (32, 7).

7 = a(32 − 12)² + 3

7 = a(1)² + 3

7 − 3 = a

4 = a

a = 4

y = 4(x − 12)² + 3

Answer: y = 4(x − 12)² + 3

2
Write an equation in vertex form for the parabola with vertex (2, 12) that passes through (4, −72).
Show Answer

The formula is

y = a(x − h)² + k

Vertex (2, 12).

Inside the parentheses, move left and right, opposite of the inside.

Right 2 is (x − 2)².

Outside the parentheses, move up and down. It is what it is.

Up 12 is +12.

y = a(x − 2)² + 12

Substitute the point (4, −72).

−72 = a(4 − 2)² + 12

−72 = a(2)² + 12

−72 = 4a + 12

−72 − 12 = 4a

−4 = 4a

a = −44

a = −1

y = −(x − 2)² + 12

Answer: y = −(x − 2)² + 12

3
Write an equation in vertex form for the parabola with vertex (12, 32) that passes through (32, 72).
Show Answer

The formula is

y = a(x − h)² + k

Vertex (12, 32).

Inside the parentheses, move left and right, opposite of the inside.

Right 12 is (x − 12)².

Outside the parentheses, move up and down. It is what it is.

Up 32 is +32.

y = a(x − 12)² + 32

Substitute the point (32, 72).

72 = a(32 − 12)² + 32

72 = a(1)² + 32

72 − 32 = a

2 = a

a = 2

y = 2(x − 12)² + 32

Answer: y = 2(x − 12)² + 32

4
Write an equation in vertex form for the parabola with vertex (23, 1) that passes through (53, −2).
Show Answer

The formula is

y = a(x − h)² + k

Vertex (23, 1).

Inside the parentheses, move left and right, opposite of the inside.

Right 23 is (x − 23)².

Outside the parentheses, move up and down. It is what it is.

Up 1 is +1.

y = a(x − 23)² + 1

Substitute the point (53, −2).

−2 = a(53 − 23)² + 1

−2 = a(1)² + 1

−2 − 1 = a

−3 = a

a = −3

y = −3(x − 23)² + 1

Answer: y = −3(x − 23)² + 1

5
Write an equation in vertex form for the parabola with vertex (−12, 52) that passes through (12, 32).
Show Answer

The formula is

y = a(x − h)² + k

Vertex (−12, 52).

Inside the parentheses, move left and right, opposite of the inside.

Left 12 is (x + 12)².

Outside the parentheses, move up and down. It is what it is.

Up 52 is +52.

y = a(x + 12)² + 52

Substitute the point (12, 32).

32 = a(12 + 12)² + 52

32 = a(1)² + 52

32 − 52 = a

−1 = a

a = −1

y = −(x + 12)² + 52

Answer: y = −(x + 12)² + 52

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