Question

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

Explanation:

Step1: Identify the roots of the quadratic function

From the graph, the parabola intersects the x - axis at \(x=- 4\) and \(x = 8\). So the roots of the quadratic function \(f(x)\) are \(x=-4\) and \(x = 8\). In factored form, a quadratic function with roots \(r_1\) and \(r_2\) is \(f(x)=a(x - r_1)(x - r_2)\). Substituting the roots, we get \(f(x)=a(x + 4)(x - 8)\).

Step2: Find the value of 'a' using the y - intercept

The y - intercept of the parabola is the point where \(x = 0\). From the graph, when \(x = 0\), \(y=-8\). Substitute \(x = 0\) and \(y=-8\) into the equation \(f(x)=a(x + 4)(x - 8)\):

$$
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$$

Step3: Write the function in factored form

Substitute \(a=\frac{1}{4}\) into \(f(x)=a(x + 4)(x - 8)\), we get \(f(x)=\frac{1}{4}(x + 4)(x - 8)\). We can also expand this to standard form:

$$
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$$

Or we can use the vertex form. First, find the vertex of the parabola. The x - coordinate of the vertex of a parabola with roots \(x_1=-4\) and \(x_2 = 8\) is \(x=\frac{x_1 + x_2}{2}=\frac{-4 + 8}{2}=2\). When \(x = 2\), from the graph, \(y=-9\) (we can also calculate it from the factored form: \(f(2)=\frac{1}{4}(2 + 4)(2 - 8)=\frac{1}{4}\times6\times(-6)=-9\)). The vertex form of a parabola is \(f(x)=a(x - h)^{2}+k\), where \((h,k)\) is the vertex. Here, \(h = 2\), \(k=-9\) and \(a=\frac{1}{4}\), so \(f(x)=\frac{1}{4}(x - 2)^{2}-9\).

Answer:

\(\frac{1}{4}(x + 4)(x - 8)\) (or \(\frac{1}{4}x^{2}-x - 8\) or \(\frac{1}{4}(x - 2)^{2}-9\))