Question

(a) Write $x=-4y^2-16y-12$in standard form and (b) use properties of the standard form to graph the equation.

Short answer

Part (a), The standard form is $x=-4(y+2{)}^2+4$

Part (b), The graph is

Step-by-step solution

Part (a) Step 1: Given information

Given function$x=-4y^2-16y-12$

Part (a) Step 2: Rewriting the function

Rewriting, we get

$$\begin{gathered} x=-4y^2-16y-12 \\ x=-4(y^2+4y)-12 \\ x=-4{(y+2)}^2+4 \end{gathered}$$

Part (b) Step 1: Given information

Given the standard form$x=-4(y+2{)}^2+4$

Part (b) Step 2: Identify the constants a, h, k and the other parameters

Calculating, we get

$$\begin{gathered} x=-4(y+2{)}^2+4 \\ a=-4 \\ (h,k)=(4,-2) \end{gathered}$$

Part (b) Step 3: Calculate the x intercept

Calculating, we get

$$\begin{gathered} x=-4(y+2{)}^2+4 \\ x=-4{(0+2)}^2+4 \\ x=-12 \end{gathered}$$

Part (b) Step 4: Calculate the y intercept

Calculating, we get

$$\begin{gathered} x=-4(y+2{)}^2+4 \\ -4{(y+2)}^2+4=0 \\ -4{(y+2)}^2=-4 \\ {(y+2)}^2=1 \\ y=-1,-3 \end{gathered}$$

Part (b) Step 5: Sketching the graph

The graph is