Non-linear graphs

A quadratic function like y = x² graphs as a parabola — a symmetric U-shaped curve. The vertex is the parabola's turning point (minimum if it opens upward, maximum if it opens downward). The axis of symmetry is a vertical line through the vertex, splitting the parabola into two mirror-image halves. Where the parabola crosses the x-axis are its roots (solutions to the equation).

Key terms

Parabola:

The symmetric U-shaped curve a quadratic function graphs as.

Vertex:

A parabola's turning point (minimum or maximum).

Roots:

The x-values where a graph crosses the x-axis.

Questions

1. 1. A quadratic function graphs as a:

   • Parabola
   • Straight line
   • Circle
   • Triangle

2. 2. A parabola is shaped like a:

   • Symmetric U-curve
   • Straight diagonal line
   • A perfect circle
   • A zigzag

3. 3. The vertex of a parabola is its:

   • Turning point
   • Steepest edge
   • Y-intercept only
   • X-axis label

4. 4. A parabola that opens upward has a vertex that is a:

   • Minimum point
   • Maximum point
   • Random point
   • Point with no meaning

5. 5. A parabola that opens downward has a vertex that is a:

   • Maximum point
   • Minimum point
   • A point with no meaning
   • The y-intercept always

6. 6. The roots of a graph are where it crosses the:

   • X-axis
   • Y-axis only
   • Vertex only
   • Nowhere

7. 7. A parabola is symmetric around its:

   • Axis of symmetry, through the vertex
   • Y-axis always, regardless of vertex
   • Top edge only
   • It has no symmetry

8. 8. For y = x², the vertex is at:

   • (0, 0)
   • (1, 1)
   • (2, 2)
   • (−1, −1)

9. 9. For y = x² − 4, the parabola crosses the x-axis at:

   • x = 2 and x = −2
   • x = 4 and x = −4
   • x = 0 only
   • It never crosses the x-axis

10. 10. For y = x² − 4, the y-intercept is:

    • −4
    • 4
    • 0
    • 2

11. 11. A parabola with vertex at (2, −3) that opens upward has a minimum value of:

    • −3
    • 2
    • 3
    • 0

12. 12. For y = −x², the parabola opens:

    • Downward
    • Upward
    • Sideways
    • It doesn't open at all

13. 13. A parabola with no real roots means it:

    • Never crosses the x-axis
    • Always crosses the x-axis twice
    • Is actually a straight line
    • Has an infinite number of roots

14. 14. For y = (x − 3)², the vertex is at:

    • (3, 0)
    • (−3, 0)
    • (0, 3)
    • (0, −3)

15. 15. A ball thrown in the air, modelled by a downward parabola, reaches its highest point at the:

    • Vertex
    • Roots
    • Y-intercept only
    • Starting point always

16. 16. For y = x² − 2x − 3, the roots are found by solving x² − 2x − 3 = 0, giving:

    • x = 3 and x = −1
    • x = −3 and x = 1
    • x = 2 and x = 3
    • x = 0 only

17. 17. For a parabola y = ax² + bx + c, increasing the value of "a" (keeping it positive) makes the parabola:

    • Narrower
    • Wider
    • Turn into a straight line
    • Have no effect on shape

18. 18. A parabola representing profit, with vertex at (50, 2000), suggests maximum profit of $2000 occurs when the input is:

    • 50
    • 2000
    • 0
    • Impossible to determine

19. 19. The axis of symmetry for y = x² − 6x + 5 passes through x =:

    • 3
    • 6
    • 5
    • 0

20. 20. Why might a business use a parabola to model revenue against price, when both very low and very high prices reduce revenue?

    • A parabola naturally captures a value that rises then falls, matching how revenue often peaks at a middle price
    • Parabolas can only ever model straight-line relationships
    • Revenue never has a peak value in real situations
    • Parabolas are unrelated to any real-world scenario

21. 21. Two parabolas have the same roots but different vertex heights. This means they:

    • Cross the x-axis at the same points but reach different maximum/minimum values
    • Are exactly identical in every way
    • Cannot both be valid parabolas
    • Have no roots at all