What is the equation of the function?

Steps to Solve

1. Identify the x-intercepts from The Graph

By looking at the graph, we can see the parabola intersects the x-axis at $x = -2$ and $x = 4$. These are the roots of the quadratic equation.

2. Write the equation in factored form

A quadratic equation with roots $r_1$ and $r_2$ can be written in the form $y = a(x - r_1)(x - r_2)$, where $a$ is the leading coefficient. In our case, $r_1 = -2$ and $r_2 = 4$. So the equation becomes $y = a(x - (-2))(x - 4)$ which simplifies to $y = a(x + 2)(x - 4)$.

3. Determine the leading coefficient

From the graph, we can observe that the parabola opens upwards. The coefficient $a$ is positive. We can also check the y coordinate of the vertex, which seems to be around -27. We can test the two possibilities with positive leading coefficients, $a=1$ or $a=3$. If $a = 1$, then $y = (x+2)(x-4) = x^2 - 2x - 8$. The x-coordinate the vertex is $x = \frac{-b}{2a} = \frac{2}{2} = 1$. If $x = 1$, then $y = (1+2)(1-4) = 3(-3) = -9$, which is too high. If $a = 3$, then $y = 3(x+2)(x-4) = 3(x^2 - 2x - 8) = 3x^2 - 6x - 24$. The x-coordinate of the vertex is $x = \frac{-b}{2a} = \frac{6}{6} = 1$. If $x = 1$, then $y = 3(1+2)(1-4) = 3(3)(-3) = -27$, which fits the graph more accurately.

4. Select the correct equation The equation $y = 3(x+2)(x-4)$ matches.