Q. 29 - 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 $f\left(x\right)=a(x-h{)}^2+k$.

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

Step-by-Step Solution

Answer

The required graph is shown below:

Step 1. Write the given function in vertex form.

The given function is:

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

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

$$\begin{gathered} f\left(x\right)=\frac{1}{2}(x^2+2x+1-1)-1 \\ f\left(x\right)=\frac{1}{2}(x^2+2x+1)-0.5-1 \\ f\left(x\right)=\frac{1}{2}(x+1{)}^2-1.5 \end{gathered}$$

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 $a=\frac{1}{2},h=-1,k=-1.5$. It means the graph of the given function is a parabola that opens up and has its vertex at $(-1,-1.5)$ and its axis of symmetry is the line $x=-1$.

The graph of $y=x^2$ vertically compressed by factor 2, shifts 1 unit left and 1.5 units down.

First, plot the graph of $y=x^2$ then vertically compressed by factor 2 to get the graph of $y=\frac{1}{2}{x}^2$, then shift it 1 unit left to get the graph of the function $y=\frac{1}{2}(x+1{)}^2$. After that shift the resulting graph 1.5 units down to get the graph of the function $f\left(x\right)=\frac{1}{2}(x+1{)}^2-1.5$.

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.