the graph of quadratic function f(x) has a minimum at (-2,-3) and passes through the point (2,13). The function g(x) is represented by the equation g (x)=-(x+2) (x-3).

Question

how much greater is the y-intercept of g(x) than f(x)

what is the correct answer

Answer 1

To find the y-intercept of function g(x), we need to set x=0 and solve for y:

g(0) = -(0+2)(0-3)
g(0) = -(2)(-3)
g(0) = 6

Therefore, the y-intercept of g(x) is 6.

To find the y-intercept of function f(x), we need to use the vertex form of a quadratic function which is f(x) = a(x-h)^2 + k, where (h,k) is the vertex of the parabola.

Since the minimum of f(x) is (-2,-3), we have:

f(x) = a(x+2)^2 - 3

Using the point (2,13) that f(x) passes through, we can plug in x=2 and y=13 into the equation:

13 = a(2+2)^2 - 3
13 = a(4)^2 - 3
13 = 16a - 3
16a = 16
a = 1

Therefore, the equation of f(x) is f(x) = (x+2)^2 - 3.

To find the y-intercept of f(x), we need to set x=0 and solve for y:

f(0) = (0+2)^2 - 3
f(0) = 4 - 3
f(0) = 1

Therefore, the y-intercept of f(x) is 1.

The difference between the y-intercepts of g(x) and f(x) is:
6 - 1 = 5

Thus, the y-intercept of g(x) is 5 units greater than that of f(x).

Answer 2

the correct answer is

Answer 3

Apologies for the mistake in my previous response. Let's recalculate the correct answer.

The y-intercept of function g(x) is 6.

The y-intercept of function f(x) can be found by plugging in x=0 into the equation f(x) = (x+2)^2 - 3:

f(0) = (0+2)^2 - 3
f(0) = 4 - 3
f(0) = 1

Therefore, the y-intercept of f(x) is 1.

The difference between the y-intercepts of g(x) and f(x) is:
6 - 1 = 5

Therefore, the correct answer is that the y-intercept of g(x) is 5 units greater than that of f(x).