Question

question
video
examples
write a function in any form that would
match the graph shown below.
answer
attempt 2 out of 2
f(x) =
submit answer
you have up to 8 questions left to raise your
score.

Explanation:

Step1: Identify roots of the function

From the graph, the function crosses the x - axis at \(x=-6\) (with a double root, since it touches the axis there) and \(x = 6\). So the roots are \(x=-6\) (multiplicity 2) and \(x = 6\) (multiplicity 1). The factored form of a polynomial function with roots \(r_1,r_2,\cdots,r_n\) is \(f(x)=a(x - r_1)^{m_1}(x - r_2)^{m_2}\cdots(x - r_n)^{m_n}\), where \(m_i\) is the multiplicity of root \(r_i\) and \(a\) is a leading coefficient. So, \(f(x)=a(x + 6)^{2}(x - 6)\).

Step2: Find the leading coefficient \(a\)

We can use the y - intercept. The y - intercept is the value of the function when \(x = 0\). From the graph, when \(x = 0\), \(y=- 72\) (since the graph passes through \((0,-72)\)). Substitute \(x = 0\) and \(y=-72\) into the function:

$$
LATEXBLOCK0
$$

Step3: Write the function

Substitute \(a=\frac{1}{3}\) into the factored form:

$$
LATEXBLOCK1
$$

Or we can leave it in factored form: \(f(x)=\frac{1}{3}(x + 6)^{2}(x - 6)\)

Answer:

\(f(x)=\frac{1}{3}(x + 6)^{2}(x - 6)\) (or the expanded form \(\frac{1}{3}x^{3}+2x^{2}-12x - 72\))