Q. 27 - Precalculus Enhanced with Graphing Utilities

Question

Graph the function $f$ by starting with the graph of $y=x^2$ and using transformations (shifting, compressing, stretching, and/or reflection). Verify your results using a graphing utility.

Hint: If necessary, write $f$ in the form $f\left(x\right)=a(x-h{)}^2+k$.

$f\left(x\right)=3x^2+6x$

Step-by-Step Solution

Answer

The required graph is shown below:

Step 1. Write the given function in vertex form.

The given function is:

$$\begin{gathered} f\left(x\right)=-x^2-2x \\ f\left(x\right)=-(x^2+2x) \end{gathered}$$

Add and subtract the square of half of the coefficient of x inside the parenthesis.

$$\begin{gathered} f\left(x\right)=-(x^2+2x+1-1) \\ f\left(x\right)=-(x^2+2x+1)+1 \\ f\left(x\right)=-(x+1{)}^2+1 \end{gathered}$$

Step 2. Determine the transformations used.

In the function $f\left(x\right)=a(x-h{)}^2+k$, $a$ is a constant and $(h,k)$ is the vertex.

In the given function $a=-1,h=-1,k=1$. It means the graph of the given function is a parabola that opens down and has its vertex at $(-1,1)$ and its axis of symmetry is the line $x=-1$.

The graph of $y=x^2$ reflects along the x-axis, shifts 1 unit left and 1 unit up.

First, plot the graph of $y=x^2$ then reflect it along the x-axis to get the graph of $y=-x^2$, then shift it 1 unit left to get the graph of the function . After that shift the resulting graph 1 unit up to get the graph of the function $f\left(x\right)=-(x+1{)}^2+1$.

Step 3. Draw the graph.

The required graph is shown below:

Using a graphing utility, we get a graph that is the same as the above graph.