The Solution:
Given the function:
[tex]f(x)=-3\mleft(x-3\mright)^2+3[/tex]We are asked to find the y-intercepts of the above function.
Recall:
x-intercepts of a function are the values of x for which the function is zero. That is if y=f(x) then the x-intercept is the value of x when y=f(x)=0
Similarly,
y-intercepts of a function are the values of y for which x is zero.
So, in this case, the y-intercept is:
[tex]y=f(0)=-3(0-3)^2+3=-3(-3)^2+3=-3(9)+3=-27+3=-24[/tex]So, the y-intercept is:
[tex](0,-24)[/tex]To find the x-intercept:
[tex]\begin{gathered} f(x)=0 \\ -3(x-3)^2+3=0 \\ -3(x-3)^2=-3 \end{gathered}[/tex]Dividing both sides by -3, we get
[tex](x-3)^2=1[/tex]Taking the square root of both sides, we get
[tex]\begin{gathered} x-3=\text{ }\sqrt[]{1} \\ x-3=\pm1 \\ x=3\pm1 \end{gathered}[/tex]This means:
[tex]\begin{gathered} x=3+1\text{ or x=3-1} \\ x=4\text{ or x=2} \end{gathered}[/tex]So, the x-intercepts are:
[tex](4,0)\text{ or (2,0)}[/tex]