Question

write a function in any form that would match the graph shown below.
answer
attempt 1 out of 2
f(x) =
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Explanation:

Step1: Identify roots and multiplicity

The graph touches the x - axis at \(x = - 2\) (so multiplicity 2) and crosses at \(x = 3\) (multiplicity 1). So the factored form starts as \(f(x)=a(x + 2)^{2}(x - 3)\).

Step2: Find the value of \(a\)

Use the y - intercept \((0,6)\). Substitute \(x = 0\) and \(y=6\) into the equation:
\(6=a(0 + 2)^{2}(0 - 3)\)
\(6=a(4)(- 3)\)
\(6=-12a\)
Solve for \(a\): \(a=\frac{6}{-12}=-\frac{1}{2}\)

Step3: Write the function

Substitute \(a =-\frac{1}{2}\) into the factored form:
\(f(x)=-\frac{1}{2}(x + 2)^{2}(x - 3)\)
We can also expand it:
First, expand \((x + 2)^{2}=x^{2}+4x + 4\)
Then, \((x^{2}+4x + 4)(x - 3)=x^{3}-3x^{2}+4x^{2}-12x + 4x-12=x^{3}+x^{2}-8x - 12\)
Multiply by \(-\frac{1}{2}\): \(f(x)=-\frac{1}{2}x^{3}-\frac{1}{2}x^{2}+4x + 6\)

Answer:

\(f(x)=-\frac{1}{2}(x + 2)^{2}(x - 3)\) (or the expanded form \(f(x)=-\frac{1}{2}x^{3}-\frac{1}{2}x^{2}+4x + 6\))