Question

In two or more complete sentences, compare the number of x-intercepts in the graph of f(x) = x2 to the number of x-intercepts in the graph of g(x) = -x2 -5. Be sure to include the transformations that occurred between the parent function f(x) and its image g(x).

Answer 1

Part (1):
To get the x-intercepts, we will need to equate the equation to zero and solve for x.
For the parent function, we have:
x² = 0
x = 0
Therefore, the function intersects the x-axis at one point which is the origin.

For the image function we have:
-x² - 5 = 0
x² = -5
x = √-5 which does not exist in real numbers
This means that the image function does not intersect the x-axis at all.

Based on the above, the parent function has one x-intercept while the image one doesn't have any x-intercepts.

Part (2):
I have attached the image containing the two functions , the parent one as well as the image.
As we can note, two changes occurred to the parent function in order to form the image:
1- The parabola was flipped to be pointing downwards instead of pointing upwards
2- The vertex was translated 5 units downwards to be at (0,-5) instead of (0,0)

Answer 2

x^2 only touches the x axis when x=0 and it increases without bound on either side of 0.

-x^2-5 never touches the x axis reaching a maximum at (0,-5) and decreasing without bound on either side of 0

g(x) is f(x) inverted and moved down five units...