← Back to Graphing Quadratic Functions

Graphing Quadratic Functions - Worksheet 9 - 9th Grade

Hard

Graph quadratics and identify key features (Hard)

18 problems · 20-25 minutes · Skills: quadratics, graphing, parabolas

About This Worksheet

This hard-level worksheet contains 18 problems designed for 9th Grade students practicing graphing quadratic functions. Hard worksheets feature challenging problems including multi-step questions, missing values, and real-world applications to push students further.

The parabola is the most important non-linear curve in algebra. Quadratic functions model projectile motion, maximizing revenue, minimizing cost, and the shape of satellite dishes — making them one of the most applied mathematical objects students will ever study.

  • Identify the vertex, axis of symmetry, and direction of opening from y = ax² + bx + c
  • Plot a parabola by creating a table of values
  • Identify x-intercepts (zeros), y-intercept, and the maximum or minimum value
  • Convert between standard form and vertex form to reveal key features of the parabola
  • Determine the domain and range of a quadratic function from its graph

How to Solve: Worked Example

Example Problem: Find the vertex of y = -3x² + 24x - 52 and say which way it opens.
Solution Steps:
  1. x = -b/(2a) = -24/(2 · -3) = 4.
  2. Substitute: y = -3(16) + 24(4) - 52 = -48 + 96 - 52 = -4.
  3. Because a = -3 is negative, the parabola opens downward.
Answer: (4, -4), opens down
Tip: The sign of a decides the direction: positive opens up (minimum), negative opens down (maximum).
1
Find the vertex and state whether y = -3x² + 24x - 52 opens up or down.
2
Find the vertex and state whether y = -2x² - 4x - 4 opens up or down.
3
Find the vertex and state whether y = 2x² + 12x + 19 opens up or down.
4
Find the vertex and state whether y = 2x² - 4x + 0 opens up or down.
5
Find the vertex and state whether y = -2x² + 8x - 2 opens up or down.
6
Find the vertex and state whether y = 2x² + 16x + 31 opens up or down.
7
Find the vertex and state whether y = 2x² - 4x + 5 opens up or down.
8
Find the vertex and state whether y = 3x² + 12x + 18 opens up or down.
9
Find the vertex and state whether y = 3x² - 12x + 17 opens up or down.
10
Find the vertex and state whether y = -3x² - 18x - 30 opens up or down.
11
Find the vertex and state whether y = 2x² - 16x + 27 opens up or down.
12
Find the vertex and state whether y = -2x² - 12x - 14 opens up or down.
13
Find the vertex and state whether y = 2x² - 4x + 4 opens up or down.
14
Find the vertex and state whether y = 2x² + 0x + 5 opens up or down.
15
Find the vertex and state whether y = 3x² + 6x + 1 opens up or down.
16
Find the vertex and state whether y = 3x² - 6x + 1 opens up or down.
17
Find the vertex and state whether y = 3x² - 18x + 33 opens up or down.
18
Find the vertex and state whether y = -3x² + 24x - 54 opens up or down.

Answer Key

1. (4, -4), opens down2. (-1, -2), opens down3. (-3, 1), opens up4. (1, -2), opens up5. (2, 6), opens down6. (-4, -1), opens up7. (1, 3), opens up8. (-2, 6), opens up9. (2, 5), opens up10. (-3, -3), opens down11. (4, -5), opens up12. (-3, 4), opens down13. (1, 2), opens up14. (0, 5), opens up15. (-1, -2), opens up16. (1, -2), opens up17. (3, 6), opens up18. (4, -6), opens down

Frequently Asked Questions

How do I download this graphing quadratic functions - worksheet 9 worksheet?

Click the "Download PDF / Print" button at the top of the worksheet. You can save it as a PDF or print it directly. The answer key will be on a separate page.

How do I find worksheets at different difficulty levels?

Each subtopic has 10 worksheets: Worksheets 1-3 are Easy, 4-7 are Medium, and 8-10 are Hard. Start with Easy worksheets to build confidence, then progress to harder ones.

What skills does this quadratic functions worksheet cover?

This worksheet focuses on quadratics, graphing, parabolas. It is designed for 9th Grade students and provides structured practice through varied problem types with a worked example to guide students before they begin.

Can I use this worksheet in my classroom?

Yes. All worksheets on K12Worksheets are free for classroom use. You can print copies for your entire class. With 10 worksheets per subtopic at three difficulty levels, you have plenty of options for differentiated instruction and individualized practice.

How many problems are on this worksheet?

This worksheet contains 18 problems. It typically takes 20-25 minutes to complete. An answer key with all solutions is included so parents and teachers can check work quickly.