Question

write a function in any form that would match the graph shown below.
(graph with x-axis from -10 to 10, y-axis from -100 to 100, curve crossing x-axis at -5, touching x-axis at 4 (double root), y-intercept at -40)

Explanation:

Step1: Identify roots and multiplicity

The graph crosses the x - axis at \(x=-5\) (a single root, multiplicity 1) and touches the x - axis at \(x = 4\) (a double root, multiplicity 2). So the factored form of the polynomial is \(y=a(x + 5)(x - 4)^2\).

Step2: Find the value of \(a\)

We know that the graph passes through the point \((0,-40)\). Substitute \(x = 0\) and \(y=-40\) into the equation:

$$
LATEXBLOCK0
$$

Step3: Write the function

Substitute \(a =-\frac{1}{2}\) into the factored form:

$$y=-\frac{1}{2}(x + 5)(x - 4)^2$$

We can also expand this:

$$
LATEXBLOCK1
$$

Answer:

\(y =-\frac{1}{2}(x + 5)(x - 4)^2\) (or the expanded form \(y=-\frac{1}{2}x^{3}+\frac{3}{2}x^{2}+12x - 40\))