Equation of a Parabola from a Graph: A Vertex Form Worksheet

Most quadratic practice runs one direction: here is the equation, now graph it. This worksheet runs the other way. You get the picture and have to write the algebra that produced it.

A parabola only hides two numbers once you can see it: the vertex, and how steeply it opens. A labeled graph hands you the vertex for free. From there you only need one more point on the curve to finish the equation. If you'd like a refresher on how vertex form relates to vertex form and intercept form generally, that worksheet is a good place to start before this one.

Start From the Vertex: Writing the Equation of a Parabola

Vertex form is y = a(x − h)² + k. Read the vertex, drop h and k into the equation, and the only thing left is a. Then substitute a second point the curve clearly passes through, and solve.

Watch the sign on h: a vertex at x = −1 goes into the equation as (x + 1)², not (x − 1)². The two examples below show the whole process side by side with the graph — one opening up, one opening down. For a video walkthrough of the same idea, Khan Academy's lesson on graphing a parabola in vertex form is a solid companion.

Worked Example · Opens Upward

Start with the vertex: y = a(x + 1)² − 4.

Substitute (1, 0): 0 = a(1 + 1)² − 4.

Simplify: 0 = 4a − 4, so a = 1.

y = (x + 1)² − 4

Worked Example · Opens Downward

Start with the vertex: y = a(x − 2)² + 3.

Substitute (0, −1): −1 = a(0 − 2)² + 3.

Simplify: −1 = 4a + 3, so a = −1.

y = −(x − 2)² + 3

That's the whole method for finding the equation of a parabola from a graph: read the vertex, then use one more point to solve for a. The practice problems below give you twenty graphs and feature sets to try it on.

Practice Problems: Equation of a Parabola from a Graph

Part A gives you graphs with the vertex marked — write each equation in vertex form. Part B has no graph, only the vertex and one point, the same features you would have read off one. If your quadratic starts in standard form rather than a graph, our vertex and axis of symmetry worksheet covers finding h and k algebraically instead.

Part A — Write the Equation in Vertex Form (Problems 1–12)

1

y =

2

y =

3

y =

4

y =

5

y =

6

y =

7

y =

8

y =

9

y =

10

y =

11

y =

12

y =

Part B — Write an Equation from the Features Given (Problems 13–20)

13. Vertex (4, −2), and the curve passes through (5, 1).
    y =
14. Vertex (−2, −16), and the curve passes through (0, −12).
    y =
15. Vertex (−3, 0), and the curve passes through (−1, −8).
    y =
16. Vertex (5, −4), and the curve passes through (3, 0).
    y =
17. Vertex (0, 6), and the curve passes through (2, −6).
    y =
18. Vertex (2, 8), and the curve passes through (0, 6).
    y =
19. Vertex (1, −8), and the curve passes through (3, 0).
    y =
20. Vertex (2, 0), and the curve passes through (0, 8).
    y =

Answer Key

Every problem is worked below. Each one is the same two moves: write vertex form, then substitute a point to find a.

Part A — Vertex Form

• 1. Vertex (2, −3): y = a(x − 2)² − 3. Sub (0, 1): 1 = 4a − 3, so a = 1. y = (x − 2)² − 3
• 2. Vertex (−1, 4): y = a(x + 1)² + 4. Sub (0, 2): 2 = a + 4, so a = −2. y = −2(x + 1)² + 4
• 3. Vertex (0, −4): y = ax² − 4. Sub (2, 0): 0 = 4a − 4, so a = 1. y = x² − 4
• 4. Vertex (3, 1): y = a(x − 3)² + 1. Sub (1, 5): 5 = 4a + 1, so a = 1. y = (x − 3)² + 1
• 5. Vertex (−2, −1): y = a(x + 2)² − 1. Sub (0, 3): 3 = 4a − 1, so a = 1. y = (x + 2)² − 1
• 6. Vertex (1, 5): y = a(x − 1)² + 5. Sub (3, −3): −3 = 4a + 5, so a = −2. y = −2(x − 1)² + 5
• 7. Vertex (1, −4): y = a(x − 1)² − 4. Sub (0, −3): −3 = a − 4, so a = 1. y = (x − 1)² − 4
• 8. Vertex (3, −1): y = a(x − 3)² − 1. Sub (0, 8): 8 = 9a − 1, so a = 1. y = (x − 3)² − 1
• 9. Vertex (−1, 8): y = a(x + 1)² + 8. Sub (0, 6): 6 = a + 8, so a = −2. y = −2(x + 1)² + 8
• 10. Vertex (−3, −4): y = a(x + 3)² − 4. Sub (0, 5): 5 = 9a − 4, so a = 1. y = (x + 3)² − 4
• 11. Vertex (2, −4): y = a(x − 2)² − 4. Sub (1, −3): −3 = a − 4, so a = 1. y = (x − 2)² − 4
• 12. Vertex (3, 4): y = a(x − 3)² + 4. Sub (0, −5): −5 = 9a + 4, so a = −1. y = −(x − 3)² + 4

Part B — From Features

• 13. y = a(x − 4)² − 2. Sub (5, 1): 1 = a − 2, so a = 3. y = 3(x − 4)² − 2
• 14. y = a(x + 2)² − 16. Sub (0, −12): −12 = 4a − 16, so a = 1. y = (x + 2)² − 16
• 15. y = a(x + 3)². Sub (−1, −8): −8 = 4a, so a = −2. y = −2(x + 3)²
• 16. y = a(x − 5)² − 4. Sub (3, 0): 0 = 4a − 4, so a = 1. y = (x − 5)² − 4
• 17. y = ax² + 6. Sub (2, −6): −6 = 4a + 6, so a = −3. y = −3x² + 6
• 18. y = a(x − 2)² + 8. Sub (0, 6): 6 = 4a + 8, so a = −½. y = −½(x − 2)² + 8
• 19. y = a(x − 1)² − 8. Sub (3, 0): 0 = 4a − 8, so a = 2. y = 2(x − 1)² − 8
• 20. y = a(x − 2)². Sub (0, 8): 8 = 4a, so a = 2. y = 2(x − 2)²

Common Questions

Keep Practicing Quadratics

Once you can find the equation of a parabola from a graph confidently, these related worksheets go further:

• Graphing quadratic functions worksheet — practice going the other direction, from equation to graph.
• Vertex and axis of symmetry worksheet — find h and k algebraically when you're starting from standard form.
• Vertex form and intercept form worksheet — a side-by-side look at both forms and how to convert between them.
• Solving quadratics by factoring worksheet — useful once you need the roots of a quadratic, not just its graph.
• Quadratic formula worksheet — for equations that don't factor cleanly.
• Discriminant and number of roots worksheet — predict how many real roots a quadratic has before you solve it.

Skill level: Algebra I and Algebra II. This worksheet practices writing quadratic functions in vertex form from a graph or from given features, supporting Common Core standards F-IF.C.8a and F-BF.B.3. Free to reproduce for classroom and home instruction.