The Parabola — Given the Intercepts

Experiment with functions and relations using digital tools, making and testing conjectures and generalising emerging patterns.

Worked example

e.g.

\text{given } (0;-2),\ (-2;0) \text{ and } (2;0)

Write the equation in the form $y = a(x - p)(x - q)$ $y = a(x - p)(x - q)$
Substitute the two $x$ intercepts. Watch the signs $y = a(x - 2)(x + 2)$
Substitute the other coordinate to calculate $a$ $-2 = a(0-2)(0+2) \ \therefore \ a = \tfrac{1}{2}$
Substitute $a$ back in and expand to the form $y = ax^{2} + q$ $y = \tfrac{1}{2}x^{2} - 2$

Determine the equation of the parabola with:

x \text{ intercepts } -1, 3 \text{ and passing through } (0;-3)

Answer

y = x^{2} - 2x - 3

x \text{ intercepts } -3, 3 \text{ and passing through } (0;-9)

Answer

y = x^{2} - 9

x \text{ intercepts } 0, 4 \text{ and passing through } (2;-4)

Answer

y = x^{2} - 4x

x \text{ intercepts } -1, 1 \text{ and passing through } (0;3)

Answer

y = -3x^{2} + 3

x \text{ intercepts } 2, 5 \text{ and passing through } (0;10)

Answer

y = x^{2} - 7x + 10

x \text{ intercepts } -4, 2 \text{ and passing through } (0;-8)

Answer

y = x^{2} + 2x - 8