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Fractional Quadratics

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Fractional Quadratics

The school play’s lighting team is testing parabolic light patterns. Explore how fractional coefficients change a quadratic, then use your results to help plan the final design. Give exact answers where possible and show working for algebra questions.

Read the quadratic

For y = ax² + bx + c, the coefficient a controls opening direction and width. The axis of symmetry is x = −b/(2a), and the vertex lies on that axis.

1.The lighting team compares y = (1/2)x² with y = x². Which statement describes y = (1/2)x²?

It is wider and opens upwards.
It is narrower and opens upwards.
It is wider and opens downwards.
It has the same width and opens upwards.

2.A spotlight follows y = −(3/4)x² + (1/2)x − 1. Find its axis of symmetry.

3.For y = (1/3)x² + x − 2, identify the y-intercept and explain how you know.

Analyse the play’s light patterns

The table gives values for the quadratic y = (1/2)x² − x. Use it to spot the symmetry before finding the intercepts.

−1/2

4.Use the table to identify the axis of symmetry and vertex of y = (1/2)x² − x.

5.Factor y = (1/2)x² − x and find both x-intercepts. Show how you set y to zero.

6.The quadratic y = −(2/5)x² + (3/4)x + 1 is used for a second light pattern. State whether it opens up or down, whether it is wider or narrower than y = x², and find its axis of symmetry.

7.Charlie says the vertex of y = (1/3)x² + x − 2 has x-coordinate 3/2. Is Charlie correct? Find the vertex to check.

TrueFalse

8.If the coefficient a in y = ax² + bx + c is halved while b and c stay fixed, must the axis of symmetry stay the same? Explain using x = −b/(2a).

9.Talia wants a quadratic with axis of symmetry x = 2 and y-intercept −3. Which equation meets both conditions?

y = (1/2)x² − 2x − 3
y = (1/2)x² + 2x − 3
y = −(1/2)x² − 2x + 3
y = x² − 2x + 3

10. Sketch y = (1/2)x² − x using the table values. Label the vertex, axis of symmetry and both x-intercepts. Your sketch should show the correct opening and relative width.