Answer:
[tex]f(x)=\dfrac{7}{20}\left(x+2\right)^{2}\left(x-1\right)[/tex].
Step-by-step explanation:
If a graph touches the x-axis at x=c, then (x-c) is a factor of function.
From the given graph it is clear that the graph of function intersect x-axis at x=1 and touch the x-axis at x=-2.
It means (x-1) and (x+2) are not factors of given function but power of (x+2) must be 2.
So, the required function is
[tex]f(x)=a(x+2)^2(x-1)[/tex] ...(1)
where, a is a constant.
From the given figure it is clear that the graph passes through the point (-4,-7). So, substitute x=-4 and f(x)=-7 in the above function.
[tex]-7=a(-4+2)^2(-4-1)[/tex]
[tex]-7=a(-2)^2(-5)[/tex]
[tex]-7=-20a[/tex]
[tex]\dfrac{7}{20}=a[/tex]
Substitute [tex]a=\dfrac{7}{20}[/tex] in (1).
[tex]f(x)=\dfrac{7}{20}(x+2)^2(x-1)[/tex]
Therefore, the required function is [tex]f(x)=\dfrac{7}{20}(x+2)^2(x-1)[/tex].
