The Intercepts of a Parabola

Example 1

Find the y- and x-intercepts of the function: f(x) = x² - 4x + 4

Solution

The y-intercept is the point where x = 0. It is the point (0, C). For the given function, it is (0, 4).

To find the x-intercepts, replace f(x) with 0 and then solve for x.

To solve for x

We have one solution, x = 2, of multiplicity 2. This means the parabola touches the x-axis only once.

So, the x-intercept of f(x) = x² - 4x + 4 is (2, 0).

Note that Algebra Solver can easily do all kinds of problems involving parabolas that you enter. Click here for a sample screenshot.

Example 2

Find the y- and x-intercepts of the function: f(x) = x² + 9

Solution

The y-intercept is the point where x = 0. It is the point (0, C). For the given function, it is (0, 9).

To find the x-intercepts, replace f(x) with 0 and then solve for x.

To solve for x:

Subtract 9 from both sides.	-9	= x2
Take the square root of each side.		= x
Simplify.	±3i	= x

Since the solution is two imaginary roots, the function f(x) = x² + 9 has no x-intercepts. That is, it does not cross the x-axis.

Note:

Remember, when you take the square root of each side of an equation you must include both the positive and negative square roots. That is why we need the ± in  =  x.

Remember, taking the square root of a negative number results in an imaginary number, which we indicate with the letter i.