Answer:
The y-intercept of g(x) is 5 greater than the y-intercept of f(x).
Step-by-step explanation:
It is given that the graph of quadratic function f(x) has a minimum at (-2,-3) and passes through the point (2,13).
The equation of a quadratic function is
[tex]f(x)=a(x-h)^2+k[/tex]
Where, (h,k) is vertex or extreme points and a is stretch factor.
The minimum value of function is (-2,-3), so the vertex is (-2,-3).
[tex]f(x)=a(x-(-2))^2+(-3)[/tex]
[tex]f(x)=a(x+2)^2-3[/tex]
It is given that the function passing through the point (2,13).
[tex]13=a(2+2)^2-3[/tex]
[tex]16=16a[/tex]
[tex]a=1[/tex]
So, the function f(x) is
[tex]f(x)=(x+2)^2-3[/tex]
Substitute x=0, to find the y-intercept.
[tex]f(x)=(0+2)^2-3[/tex]
[tex]f(x)=4-3=1[/tex]
The y-intercept of f(x) is 1.
The given function is
[tex]g(x)=-(x+2)(x-3)[/tex]
Substitute x=0, to find the y-intercept.
[tex]g(x)=-(0+2)(0-3)=6[/tex]
The y-intercept of g(x) is 6.
The difference between y-intercepts is
[tex]6-1=5[/tex]
Therefore y-intercept of g(x) is 5 greater than the y-intercept of f(x).
