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Exploring Quadratic Functions
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Exploring Quadratic Functions
At a stargazing night, Amira and Evie use quadratic models to investigate a telescope dish and a model rocket. Show working where useful. Give exact values or round to 2 decimal places when needed.
Read the models
A parabola can be written in expanded form y = ax² + bx + c, or vertex form y = a(x − h)² + k. Its axis of symmetry is x = h, and its turning point is (h, k).
1.Amira models a telescope dish cross-section with y = 0.5(x − 2)² + 1. State its turning point and axis of symmetry.
2.The model y = −2(x + 1)² + 8 describes a parabola. Which statement is correct?
- It opens upwards and has a minimum of 8.
- It opens downwards and has a maximum of 8.
- It opens downwards and has a minimum of −1.
- It opens upwards and has a maximum of −1.
3.Evie expands the dish model y = 0.5(x − 2)² + 1 into the form y = ax² + bx + c. Find a, b and c.
4.For y = x² − 6x + 5, find the axis of symmetry and the coordinates of the turning point.
5.The same quadratic y = x² − 6x + 5 crosses the x-axis at two points. Find both x-intercepts by factorising.
6.A model rocket’s height is h = −4t² + 16t + 1, where h is in metres and t is in seconds. Find its initial height, the time it reaches maximum height, and that maximum height.
7.For the rocket model in Question 6, calculate when the rocket returns to its launch height. Solve h = 1 and state which solution describes the later time.
8.At the stargazing site, a parabolic reflector has a vertex at (0, 0) and passes through the point (2, 3). Find its equation in vertex form.
9.A parabola has turning point (−2, 5), opens downwards, and passes through (0, 1). Find its equation in vertex form.
10.True or false? Every quadratic function has two distinct real x-intercepts. Explain your choice using an example or a condition on the discriminant.
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