Question

write a function in any form that would match the graph shown below.
answer attempt 1 out of 3
f(x) =

Explanation:

Step1: Identify x-intercepts

The graph crosses the x - axis at \(x = - 1\) and \(x = 4\). So the roots of the quadratic function are \(x=-1\) and \(x = 4\). In factored form, a quadratic function is \(f(x)=a(x - r_1)(x - r_2)\), where \(r_1\) and \(r_2\) are the roots. So \(f(x)=a(x + 1)(x - 4)\).

Step2: Find the value of 'a'

We know the y - intercept is at \((0,6)\) (from the graph, when \(x = 0\), \(y=6\)). Substitute \(x = 0\) and \(y = 6\) into the equation \(f(x)=a(x + 1)(x - 4)\):

$$
LATEXBLOCK0
$$

Step3: Write the function

Substitute \(a =-\frac{3}{2}\) into the factored form:

$$
LATEXBLOCK1
$$

(We can also expand the factored form. Another way: If we use vertex form or other forms, but factored form with finding 'a' is straightforward here. Alternatively, we can check the y - intercept again. When \(x = 0\), \(f(0)=-\frac{3}{2}(0 + 1)(0 - 4)=-\frac{3}{2}\times(-4)=6\), which matches the graph's y - intercept.)

Answer:

\(f(x)=-\frac{3}{2}(x + 1)(x - 4)\) (or \(f(x)=-\frac{3}{2}x^{2}+\frac{9}{2}x + 6\))