Question

write a function in any form that would match the graph shown below.

Explanation:

Step1: Identify x - intercepts

The graph crosses the x - axis at \(x=- 2\) and \(x = 2\). So, the function has factors \((x + 2)\) and \((x - 2)\). A cubic function of the form \(y=a(x + 2)(x - 2)(x - h)\) can be considered. Since the graph has a y - intercept above the origin, assume \(h = 0\) for simplicity.

Step2: Find the value of \(a\)

The y - intercept of the graph is at \(y = 8\). Substitute \(x = 0\) and \(y=8\) into \(y=a(x + 2)(x - 2)x\). We get \(8=a(0 + 2)(0 - 2)(0)\), which simplifies to \(8=a\times2\times(-2)\times0\). Let's consider the general cubic \(y=a(x + 2)(x - 2)x=ax(x^{2}-4)=ax^{3}-4ax\). Substituting \(x = 0,y = 8\) doesn't work with this form. Let's try a cubic of the form \(y=a(x + 2)(x - 2)(x - k)\). If we assume the function is \(y=a(x + 2)(x - 2)x=ax^{3}-4ax\). When \(x = 0,y = 0\) is wrong. Let's use the fact that the function is a cubic and assume \(y=a(x + 2)(x - 2)(x - 0)=a(x^{3}-4x)\). Substitute a non - zero point. Let's say we assume the function passes through \((1,6)\) (by estimating from the graph). Substitute \(x = 1\) and \(y = 6\) into \(y=a(x^{3}-4x)\), we have \(6=a(1^{3}-4\times1)=a(1 - 4)=-3a\). Solving for \(a\) gives \(a=- 2\).

Step3: Write the function

The function is \(f(x)=-2x^{3}+8x\).

Answer:

\(f(x)=-2x^{3}+8x\)