Question

pretest: quadratic relationships
what is the equation of the quadratic function shown in the graph?
graph of a quadratic function on a coordinate plane with x-axis from -10 to 10 and y-axis from -10 to 10
$f(x) = \square (x + \square )(x - \square )$

Explanation:

Step1: Find x-intercepts

The graph crosses the x - axis at \(x=-3\) and \(x = 1\). So the factored form of the quadratic is \(f(x)=a(x + 3)(x - 1)\) (since if \(x=-3\), then \(x + 3=0\); if \(x = 1\), then \(x - 1=0\)).

Step2: Find the value of 'a'

We can use the y - intercept. The y - intercept is the point where \(x = 0\). From the graph, when \(x = 0\), \(y=6\) (since the graph crosses the y - axis at \((0,6)\)).
Substitute \(x = 0\) and \(y = 6\) into \(f(x)=a(x + 3)(x - 1)\):
\(6=a(0 + 3)(0 - 1)\)
\(6=a(3)(-1)\)
\(6=-3a\)
Solve for \(a\): \(a=\frac{6}{-3}=-2\)

Answer:

\(f(x)=-2(x + 3)(x - 1)\)
So the values are \(-2\), \(3\), and \(1\) (in the boxes respectively).