Lesson 7

Explore

 

You’ve just discovered some facts about quadratic functions. You’re now going to use this information and your knowledge of factoring to solve quadratic equation questions.

 

As you saw in Discover, it is straightforward to find the x-intercepts of a quadratic function in factored form. Is it as simple to find the factored form of a quadratic given its x-intercepts?

 

Suppose you are given a quadratic function with x-intercepts of x = −1 and x = 3. Write two quadratic functions that match this information.

 

Since you know that x = −1 and x = 3 are x-intercepts, you can find the (x − r) and (x − s) factors of f(x) = a(x − r)(x − s) quite easily.

 

 

The equations show that f(x) = a(x + 1)(x − 3) has the required x-intercepts.

 

To find more solutions, you can change the value of a. Options here are endless. You might choose 2 or or −3, giving the following functions with the same x-intercepts:

 

f(x) = 2(x2 − 2x − 3)

 

f(x) = (x2 − 2x − 3)

 

f(x) = −3(x2 − 2x − 3 )

 

The first two functions open upwards, and the third function opens downwards.

 

Self-Check 1

 

Suppose the height of a kicked ball is given as a function of distance, h = −0.0218d2 + 1.308d, where the distance and the height are in metres. How far will the ball go if it hits the ground before being caught? Answer