Chapter 3.3, Problem 22AYU

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

$f\left(x\right)={(x-3)}^2-10$

To determine

To graph: The transformation of the function $f(x)={(x-3)}^2-10$.

Answer to Problem 22AYU

Explanation of Solution

Given:

We have to graph the function $f(x)={(x-3)}^2-10$.

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 $f(x)={(x-3)}^2-10$.

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.

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.