Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Access the Desmos guide PDF for tips on solving problems with the Desmos graphing calculator.

Activity

For questions 1 - 2, write an equation in vertex form of the quadratic function that has a vertex at (5, 8) and also goes through the point $(- 3, 6)$ .

1.

Find the value of $a$ for vertex form.

$y = a (x - h)^{2} + k$

$6 = a (- 3 - 5)^{2} + 8$

$6 = a (- 8)^{2} + 8$

$- 2 = 64 a$

$a = \frac{- 2}{64} = \frac{- 1}{32}$

2.

Write the equation in vertex form.

Compare your answer:

$y = \frac{- 1}{32} (x - 5)^{2} + 8$

For questions 3 and 4, write an equation in vertex form of the quadratic function that has a vertex at $(- 4, 3)$ and also goes through the point $(2, 5)$ .

3.

Find the value of $a$ for vertex form.

$y = a (x - h)^{2} + k$

$5 = a (2 - (- 4))^{2} + 3$

$5 = a (6)^{2} + 3$

$2 = 36 a$

$a = \frac{2}{36} = \frac{1}{18}$

4.

Write the equation in vertex form.

Compare your answer:

$y = \frac{1}{18} (x + 4)^{2} + 3$

For questions 5 - 6, write an equation in vertex form of the quadratic function that has a vertex at $(6, - 2)$ and also goes through the point $(3, 1)$ .

5.

Find the value of $a$ for vertex form.

$y = a (x - h)^{2} + k$

$1 = a (3 - 6)^{2} + (- 2)$

$1 = a (- 3)^{2} - 2$

$3 = 9 a$

$a = \frac{3}{9} = \frac{1}{3}$

6.

Write the equation in vertex form.

Compare your answer:

$y = \frac{1}{3} (x - 6)^{2} - 2$

Video: Writing an Quadratic Equation from a Vertex and a Point

Watch the following video to learn more about writing a quadratic equation from a vertex and a point.

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Self Check

Which of the following is the quadratic equation of the quadratic with a vertex of

(4, -2)

that also goes through the point

(6, 5)

?

$y=\frac{4}{7}(x-6)^2-5$
$y=\frac{7}{4}(x-6)^2-5$
$y=\frac{7}{4}(x-4)^2-2$
$y=\frac{4}{7}(x-4)^2-2$

Additional Resources

Creating Vertex Form from a Vertex and a Point

Write the quadratic in vertex form that has a vertex at $(3, 4)$ and goes through the point $(1, 2)$ .

First start with the vertex form of a quadratic function.

$y = a (x - h)^{2} + k$

Step 1 - Substitute the vertex in at $(h, k)$ .

$y = a (x - 3)^{2} + 4$

Step 2 - Substitute the other point at $(x, y)$ .

$2 = a (1 - 3)^{2} + 4$

Step 3 - Solve for $a$ .

$2 = a (1 - 3)^{2} + 4$

$2 = a (- 2)^{2} + 4$

$2 = a (4) + 4$

$2 - 4 = a (4) + 4 - 4$

$- 2 = a (4)$

$- \frac{2}{4} = \frac{a (4)}{4}$

$a = - \frac{1}{2}$

Step 4 - Substitute the value of $a$ value and the vertex $(h, k)$ into the vertex form.

$y = - \frac{1}{2} (x - 3)^{2} + 4$

Try it

Try It: Creating Vertex Form from a Vertex and a Point

Write the quadratic in vertex form that has a vertex at $(2, 5)$ and goes through the point $(3, 7)$ .

Compare your answer:

$y = 2 (x - 2)^{2} + 5$

First, start with the vertex form of a quadratic function:

$y = a (x - h)^{2} + k$

Step 1 - Substitute the vertex at $(h, k)$ :

$y = a (x - 2)^{2} + 5$

Step 2 - Substitute the other point at $(x, y)$ :

$7 = a (3 - 2)^{2} + 5$

Step 3 - Solve for $a$ :

$7 = a (1)^{2} + 5$

$7 - 5 = a (1)^{2} + 5 - 5$

$a = 2$

Step 4 - Substitute the value of $a$ and the vertex $(h, k)$ into the vertex form:

$y = 2 (x - 2)^{2} + 5$

9.9.3 Vertex Form Given a Vertex and Point

Activity

Video: Writing an Quadratic Equation from a Vertex and a Point

Additional Resources

Creating Vertex Form from a Vertex and a Point

Try It: Creating Vertex Form from a Vertex and a Point