Q. 24 - 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 $f\left(x\right)=a(x-h{)}^2+k$.

$f\left(x\right)=x^2-6x-1$

Step-by-Step Solution

Answer

The required graph is shown below:

Step 1. Write the given function in vertex form.

The given function is:

$f\left(x\right)=x^2-6x-1$

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

$$\begin{gathered} f\left(x\right)=x^2-6x-1+9-9 \\ f\left(x\right)=(x^2-6x+9)-1-9 \\ f\left(x\right)=(x-3{)}^2-10 \end{gathered}$$

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 $a=1,h=3,k=-10$. It means the graph of the given function is a parabola that opens up and has its vertex at $(3,-10)$ and its axis of symmetry is the line $x=3$.

The graph of $y=x^2$ shifts 3 units right and 10 units down.

First, plot the graph of $y=x^2$ then shift it 3 units right and then shift the resulted graph 10 units down to get the graph of the function $f\left(x\right)=(x-3{)}^2-10$.

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.