Chapter SH, Problem 9.10EP

To determine

To match the equation $y=-{\left(x+4\right)}^2$ with given graphs.

  

Answer to Problem 9.10EP

The graph of parabola equation $y=-{\left(x+4\right)}^2$ is graph D.

Explanation of Solution

Given information :

Given equation $y=-{\left(x+4\right)}^2$ and its one of the graph among the four graphs.

  

To find the correct graph for the equation $y=-{\left(x+4\right)}^2$ , determine the vertex of the parabola and the y -intercept of the graph.

Solve the equation.

  $\begin{array}{c}y=-{\left(x+4\right)}^2\\ =-\left(x^2+8x+16\right)\\ =-x^2-8x-16\end{array}$

Compare the given parabola equation with the standard form of the quadratic equation.

  $a=-1,b=-8,c=-16$

Write the equation for the axis of the symmetry.

  $x=\frac{-b}{2a}$

Substitute $-8$ for $b$ and $-1$ for $a$ .

  $\begin{array}{c}x=-\frac{\left(-8\right)}{2\left(-1\right)}\\ =-4\end{array}$

So, the equation for the axis of the symmetry is $x=-4$ .

The vertex is equal to $\left(x,y\right)$ . To calculate the value of $y$ , substitute $-4$ for $x$ in the given equation.

  $\begin{array}{c}y=-{\left(-4\right)}^2-8\left(-4\right)-16\\ =-16+32-16\\ =0\end{array}$

So, the vertex of the parabola is at $\left(-4,0\right)$ .

The y -intercept of the graph of the equation is equal to the value of $c$ . So, the y -intercept of the graph of the equation is $-16$ .

Draw the graph for the equation to verify.

  

Among the given graphs, the graph of parabola equation $y=-{\left(x+4\right)}^2$ is graph D.

Therefore, the graph of parabola equation $y=-{\left(x+4\right)}^2$ is graph D.