Chapter 2.6, Problem 50PPSE

Solution

To graph all of the functions $y=x^2,y=x^2+2,y={\left(x+2\right)}^2,y={\left(x-1\right)}^2-4,\text{and }y=-x^2-2$ on the same window and to make a sketch of the graph of the function $f\left(x\right)=-{\left(x+2\right)}^2+1$ .

Graph:

Graph the functions $y=x^2,y=x^2+2,y={\left(x+2\right)}^2,y={\left(x-1\right)}^2-4,\text{and }y=-x^2-2$on the same window function-by-function:

Graph the function $y=x^2$:

  

Graph the function $y=x^2+2$ along with the graph of $y=x^2$:

  

Graph the function $y={\left(x+2\right)}^2$ along with the graphs of $y=x^2$ and $y=x^2+2$:

  

Graph the function $y={\left(x-1\right)}^2-4$ along with the graphs of $y=x^2$, $y=x^2+2$, and $y={\left(x+2\right)}^2$:

  

Graph the function $y=-x^2+2$ along with the graphs of $y={\left(x-1\right)}^2-4$, $y=x^2$, $y=x^2+2$, and $y={\left(x+2\right)}^2$:

  

Graph the function $f\left(x\right)=-{\left(x+2\right)}^2+1$from the observations:

Observe that the graph of the function $f\left(x\right)=-{\left(x+2\right)}^2+1$ can be obtained from the graph of $y={\left(x+2\right)}^2$ by reflecting it about the $x$ -axis and then by translating the resultant graph vertically up by $1$ unit.

Reflect $y={\left(x+2\right)}^2$ about the $x$-axis:

  

Translate the resultant graph vertically up by $1$

unit:

  

Thus, the graph of $f\left(x\right)=-{\left(x+2\right)}^2+1$ must be the final one obtained in the sequence of transformations.

Check whether the graph of $f\left(x\right)=-{\left(x+2\right)}^2+1$coincides with the graph of the final transformation:

Graph $f\left(x\right)=-{\left(x+2\right)}^2+1$ along with the graph obtained in the final window:

  

Observe that the graph of $f\left(x\right)=-{\left(x+2\right)}^2+1$ coincides with the graph of the final transformation.

Thus, the sketch was correct.