Question

Given the function in the picture, find the x-intercepts and the y intercept of the function.

Answer 1

The Solution:

Given the function:

`f(x)=-3\mleft(x-3\mright)^2+3`

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:

`y=f(0)=-3(0-3)^2+3=-3(-3)^2+3=-3(9)+3=-27+3=-24`

So, the y-intercept is:

`(0,-24)`

To find the x-intercept:

`\begin{gathered} f(x)=0 \\ -3(x-3)^2+3=0 \\ -3(x-3)^2=-3 \end{gathered}`

Dividing both sides by -3, we get

`(x-3)^2=1`

Taking the square root of both sides, we get

`\begin{gathered} x-3=\text{ }\sqrt[]{1} \\ x-3=\pm1 \\ x=3\pm1 \end{gathered}`

This means:

`\begin{gathered} x=3+1\text{ or x=3-1} \\ x=4\text{ or x=2} \end{gathered}`

So, the x-intercepts are:

`(4,0)\text{ or (2,0)}`