Question

if f(x)=x^2, what is g(x)? the graph shows the points: (1, -1), (-1,3), (-3,-1)?

Answer 1

IMAGE OF QUESTION

ANSWER

$g(x)=-(x+1)^2+3$

EXPLANATION

We have that the function graphed g(x) is a transformation of:

$f(x)=x^2$

When the parent quadratic function f(x) is transformed, it takes the following form:

$g(x)=a(x-h)^2+k$

This form also represents the vertex form of a quadratic equation, where (h, k) is the vertex of the function.

This means that we can find the function by using the vertex of the function.

The vertex of a function is the maximum or minimum value of the function; from the given graph, it is a maximum value and it is located at:

$(h,k)=(-1,3)$

Therefore, we can input this into the vertex form of the function:

$\begin{gathered} g(x)=a(x-(-1))^2+3 \\ g(x)=a(x+1)^2+3 \end{gathered}$

Now, we have to find the value of a. To do this, pick any coordinate point from the graph and input it into the function above.

Let us pick:

$(0,2)$

Therefore, we have:

$\begin{gathered} 2=a(0+1)^2+3 \\ 2=a\cdot1+3 \\ 2=a+3 \\ \Rightarrow a=2-3 \\ a=-1 \end{gathered}$

Therefore, the function graphed above is:

$g(x)=-(x+1)^2+3$