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# 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.

 Original function. Substitute 0 for f(x). f(x) = x2 - 4x + 4 0 = x2 - 4x + 4

To solve for x

 Factor. Set each factor equal to 0. Solve each equation. 0 x - 2 x = (x - 2)(x + 2)= 0 or x - 2 = 0 = 2 or x = 2

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). 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.

 Original function. Substitute 0 for f(x). f(x) = x2 + 9 0 = x2 + 9

To solve for x:

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

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