Chapter 3.3, Problem 24AYU

In Problems 21-32, 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 $f\left(x\right)=a{(x-h)}^2+k$.]

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

To determine

To graph: The transformation of the function $f(x)=x^2-6x-1$.

Answer to Problem 24AYU

Explanation of Solution

Given:

We have to graph the function $f(x)=x^2-6x-1$.

Graph:

The graph of the given function is

Interpretation:

The graph of $f(x)=a{x}^2,a=1$ is

Now, we can graph the given function by transforming the above graph.

The given function is $f(x)=x^2-6x-1$.

The general form of a quadratic function is $g(x)=a{\left(x-h\right)}^2+k$.

Here, if $a<0$, the graph opens upward otherwise the graph opens downwards.

If $a$ is closer to 0, then the graph is shorter and wider.

If $a$ is large, then the graph is tall and narrow.

Then, the graph of $g(x)$ is the graph of $f(x)$ with $h$ units shifted horizontally and $k$ units shifted vertically.

Here, we can convert the given equation into the standard form by using the square completion technique.

Therefore, let us now add and subtract the square of half of the $x\text{-coefficient}$ in the given equation.

Therefore, we get

$\begin{array}{l}f(x)=x^2-6x-1\\ \Rightarrow x^2-6x+{\left(\frac{6}{2}\right)}^2-{\left(\frac{6}{2}\right)}^2-1\\ \Rightarrow x^2-6x+{\left(3\right)}^2-{\left(3\right)}^2-1\\ \Rightarrow {\left(x-3\right)}^2-10\end{array}$

Thus, the given equation can be written in the standard form as ${\left(x-3\right)}^2-10$.

Since, $a=1$ is positive, the graph opens upwards.

In the given function, we have $h=3$ and $k=-10$, therefore, the graph is the graph of $f(x)=x^2$ with 3 units shifted horizontally right and 10 units shifted vertically downwards.