Chapter 3, Problem 6RE

In Problems 6-8, graph each quadratic function using transformations (shifting, compressing, stretching, and/or reflecting).

6. $f\left(x\right) = {(x-2)}^2 + 2$

To determine

To graph: The given function $f(x)={(x-2)}^2+2$, by applying the transformation techniques to the graph of $y=x^2$.

Explanation of Solution

Given:

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

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

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 $g\left(x\right)$ is the graph of $f\left(x\right)$ 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 $h=2$ and $k=2$, therefore, the graph is the graph of $f(x)=x^2$ with 2 units shifted horizontally right and 2 units shifted vertically upwards.