The Intercepts of a Parabola
Example 1
Find the y and xintercepts of the function: f(x) = x^{2}  4x + 4
Solution
The yintercept is the point where x = 0. It is the point (0, C).
For the given function, it is (0, 4).
To find the xintercepts, replace f(x) with 0 and then solve for x.
Original function.
Substitute 0 for f(x). 
f(x) = x^{2}  4x + 4
0 = x^{2}  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 xaxis only once.
So, the xintercept of f(x) = x^{2}  4x + 4 is (2, 0).
Example 2
Find the y and xintercepts of the function: f(x) = x^{2} + 9
Solution
The yintercept is the point where x = 0. It is the point (0, C).
For the given function, it is (0, 9).
To find the xintercepts, replace f(x) with 0 and then solve for x.
Original function.
Substitute 0 for f(x). 
f(x) = x^{2} + 9
0 = x^{2} + 9 
To solve for x:
Subtract 9 from both sides.

9 
= x^{2} 
Take the square root of each side.


= x 
Simplify. 
Â±3i 
= x 
Since the solution is two imaginary roots, the function f(x) = x^{2}
+ 9 has
no xintercepts. That is, it does not cross the xaxis.
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.
