Question

write a function in any form that would match the graph shown below.
you have up to 5 questions left to raise your score.

Explanation:

Step1: Identify the roots

The graph touches the x - axis at \(x=-6\) (a double root, since it's a tangent point) and crosses the x - axis at \(x = 8\). So the factored form of the polynomial can be written as \(f(x)=a(x + 6)^2(x - 8)\), where \(a\) is a leading coefficient.

Step2: Find the value of \(a\)

We know that the graph passes through the y - intercept \((0,-40)\). Substitute \(x = 0\) and \(f(x)=-40\) into the equation \(f(x)=a(x + 6)^2(x - 8)\):

$$
LATEXBLOCK0
$$

Step3: Write the function

Substitute \(a=\frac{5}{36}\) into the factored form:
\(f(x)=\frac{5}{36}(x + 6)^2(x - 8)\)
We can also expand this:
First, expand \((x + 6)^2=x^{2}+12x + 36\)
Then, multiply by \((x - 8)\): \((x^{2}+12x + 36)(x - 8)=x^{3}-8x^{2}+12x^{2}-96x + 36x-288=x^{3}+4x^{2}-60x - 288\)
Multiply by \(\frac{5}{36}\): \(f(x)=\frac{5}{36}x^{3}+\frac{5}{9}x^{2}-\frac{25}{3}x - 40\)
Or we can leave it in factored form. Another way is to check the leading coefficient and the roots. Also, if we consider the general cubic function (since the graph has two turning points, it's a cubic function) with roots at \(x=-6\) (double) and \(x = 8\), and y - intercept at \((0,-40)\), the function can also be written as \(f(x)=\frac{5}{36}(x + 6)^2(x - 8)\) or expanded form.

Answer:

\(f(x)=\frac{5}{36}(x + 6)^2(x - 8)\) (or the expanded form \(\frac{5}{36}x^{3}+\frac{5}{9}x^{2}-\frac{25}{3}x - 40\))