Question

Please help with explanation each graph shown is a translation of the graph of f(x)=x^2 wrote the function in vertex form

Answer 1

Given:

The graph of a function.

To find:

The vertex form of the function.

Solution:

The graph is a U-shaped curve or a parabola with vertex at $(-3,-1)$.

The parent function for vertical parabolas is $f(x)=x^2$ with vertex at (0,0).

It means the graph of $f(x)=x^2$  translated 3 units left and 1 unit down the get the given graph.

The vertex form of a parabola is:

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

Where, a is a constant and (h,k) is vertex.

The vertex of the given parabola is at $(-3,-1)$. So, $h=-3,k=-1.$.

$y=a(x-(-3))^2+(-1)$

$y=a(x+3)^2-1$            ...(i)

The graph passes through point (-2,0). Putting $x=-2,y=0$, we get

$0=a(-2+3)^2-1$

$1=a(1)^2$

$1=a$

Putting $a=1$ in (i).

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

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

Therefore, the vertex form of the given graph is $y=(x+3)^2-1$.