Exercise 46 Page 286

To write the quadratic inequality shown on the graph, we need to do two things.

1. Find the equation of the related function.
2. Determine the sign of the inequality.

Equation of the Related Function

We will use the vertex form of a quadratic function to write our related function. In the above equation, $(h,k)$ is the vertex and $a$ is the leading coefficient of the function. Let's consider the given parabola. We see above that the vertex is $(2,\text{-}10).$ Thus, we have that $h=2$ and $k=\text{-}10.$ We can partially write the equation of the function. To find the value of $a,$ we will use one of the given points. For simplicity, let's use $(0,\text{-}6).$ Since this point is on the parabola, we know it satisfies its equation. We will substitute $0$ and $\text{-}6$ for $x$ and $y,$ respectively, and solve for $a.$ Now we can write the complete equation of the parabola.

Sign of the Inequality

To determine the sign of the inequality, we can use a test point. For simplicity, we will use $(0,0).$ Since this point is included in the shaded region, we know it satisfies the inequality.

Notice that the curve is dashed, so our inequality will be strict. With the sign, we can finish writing the quadratic inequality.