The function is V shaped, this means that it is an absolute value function.
[tex]f(x)=|x|[/tex]
I'll use the vertex form to determine the equation of the function:
[tex]f(x)=a|x-h|+k[/tex]
Where
a is the coefficient that multiplies tha absolute value
(h, k) are the coordinate of the vertex, note that the x-coordinate (h) appears in the function with the opposite sign.
The value of "a" indicates the direction of the function.
If "a" is positive, the V opens upwards
If "a" is negative, the V opens downwards.
For this graph the V opens downwards so we'll expect "a" to be negative.
Now, the first step is to determine the coordinates of the vertex, looking at the graph, the maximum point of the function is at the point (1, 3) and that is the vertex.
Let's replace these values in the formula and we get that:
[tex]f(x)=a|x-1|+3[/tex]
Now using one of the roots of the function, i.e. the coordinates of one of the x-intercepts, you can calculate the value of a.
So use (-2, 0) and replace the coordinates in the formula:
[tex]\begin{gathered} 0=a|-2-1|+3 \\ 0=a|-3|+3 \\ 0=3a+3 \\ -3=3a \\ -\frac{3}{3}=\frac{3a}{3} \\ a=-1 \end{gathered}[/tex]
Then the equation for the graph is
[tex]f(x)=-|x-1|+3[/tex]
