Question

Identify the vertex, intercepts and whether of the graph of the function below opens up or down. Answer as a point (x,y). If an intercept does not exist, answer, “none”. If more than one intercepts exist you can type either intercept.

Answer 1

We were given the function:

`f\mleft(x\mright)=(1)/(2)|4x+12|-2`

We will solve for the vertex as shown below:

`\begin{gathered} f\mleft(x\mright)=(1)/(2)|4x+12|-2 \\ when\colon x=-10 \\ f\mleft(-10\mright)=(1)/(2)|4\cdot-10+12|-2 \\ f(-10)=(1)/(2)|-40+12|-2 \\ f(-10)=(1)/(2)|-28|-2=(1)/(2)|28|-2=14-2 \\ f(-10)=12 \\ \\ when\colon x=-3 \\ f(-3)=(1)/(2)|4\cdot-3+12|-2 \\ f(-3)=(1)/(2)|-12+12|-2 \\ f(-3)=(1)/(2)|0|-2=0-2 \\ f(-3)=-2 \\ \\ when\colon x=0 \\ f(0)=(1)/(2)|4\cdot0+12|-2 \\ f(0)=(1)/(2)|0+12|-2 \\ f(0)=(1)/(2)|12|-2=6-2 \\ f(0)=4 \\ \\ when\colon x=2 \\ f(2)=(1)/(2)|4\cdot2+12|-2 \\ f(2)=(1)/(2)|8+12|-2 \\ f(2)=(1)/(2)|20|-2=10-2 \\ f(2)=8 \end{gathered}`

Plotting this on a graph, we have:

From the graph plotted above, we can deduce that:

Vertex = (-3, -2)

x-intercept = (-2, 0)

y-intercept = (0, 4)

Graph opens up