Mark is graphing a parabola that has an x-intercept of –4 and a y-intercept of 16. Which of the following functions is he graphing?
A. f(x)=x^2−8x+16
B. f(x)=x^2+x−4
C. f(x)=x^2+8x+16
D. f(x)=−4x^2+16
To find which of the given functions matches the parabola that Mark is graphing, we need to think about how x-intercepts and y-intercepts relate to the graphs of functions.
The x-intercepts of a function are the points where the function intersects the x-axis, meaning that the y-coordinate of those points is 0. Likewise, the y-intercepts are the points where the function intersects the y-axis, meaning that the x-coordinate of those points is 0.
Here, we know that Mark's parabola has an x-intercept of -4, which means that the function will have a factor of (x+4) in it. Additionally, the y-intercept is 16, which means that the function will have a factor of 16 as well.
Looking at the given options:
A. f(x) = x^2 - 8x + 16
B. f(x) = x^2 + x - 4
C. f(x) = x^2 + 8x + 16
D. f(x) = -4x^2 + 16
If we try to factor the function based on the x-intercept and y-intercept information, we can see that the correct function is C. f(x) = x^2 + 8x + 16. This function can be factored as (x + 4)(x + 4), which gives x-intercept of -4 and y-intercept of 16.