What is the $x$-intercept of the graph of the function $f(x)=x^2-16 x+64 ?$
A. $(-8,0)$
B. $(0,8)$
C. $(8,0)$
D. $(0,-8)$
Answer (1)
Set f ( x ) = 0 to find the x -intercept.
Factor the quadratic equation: x 2 − 16 x + 64 = ( x − 8 ) 2 .
Solve for x : ( x − 8 ) 2 = 0 gives x = 8 .
The x -intercept is ( 8 , 0 ) .
Explanation
Understanding the Problem We are given the function f ( x ) = x 2 − 16 x + 64 and asked to find the x -intercept of its graph. The x -intercept is the point where the graph intersects the x -axis, which occurs when f ( x ) = 0 .
Setting up the Equation To find the x -intercept, we need to solve the equation x 2 − 16 x + 64 = 0 . This is a quadratic equation, and we can solve it by factoring.
Factoring the Quadratic We can factor the quadratic expression as follows: x 2 − 16 x + 64 = ( x − 8 ) ( x − 8 ) = ( x − 8 ) 2
Solving for x Now we set the factored expression equal to zero and solve for x :
( x − 8 ) 2 = 0 Taking the square root of both sides, we get: x − 8 = 0 Adding 8 to both sides, we find: x = 8
Finding the x-intercept The x -intercept is the point where y = f ( x ) = 0 , so the x -intercept is the point ( 8 , 0 ) .
Final Answer Therefore, the x -intercept of the graph of the function f ( x ) = x 2 − 16 x + 64 is ( 8 , 0 ) .
Examples
Understanding x-intercepts is crucial in various real-world applications. For instance, in physics, if you're analyzing the trajectory of a projectile, the x-intercept represents where the projectile lands. Similarly, in business, if you're modeling a cost function, the x-intercept could represent the break-even point where costs equal revenue. Knowing how to find x-intercepts helps in making informed decisions and predictions in these scenarios.
