Chapter 4, Problem 22CR

To determine

To graph: $f\left(x\right)=-3{\left(x+1\right)}^2+5$ by using transformations

Explanation of Solution

Given information:

The function $f\left(x\right)=-3{\left(x+1\right)}^2+5$

Graph:

Here, the base function is $x^2$

The graph of the $x^2$ is as follows

  

Now, by shifting the graph of the function $x^2$ to the left side by $1$ units, then the function becomes ${\left(x+1\right)}^2$ and the graph is as follows

  

By compressing the graph of the function ${\left(x+1\right)}^2$ horizontally by $3$ units, then the function becomes $3{\left(x+1\right)}^2$ and the graph of the function is as follows

  

Reflecting the graph of the function $3{\left(x+1\right)}^2$ with respect to $x$ -axis, then the function becomes $-3{\left(x+1\right)}^2$ and the graph of the function is as follows

  

Now, by shifting the graph of the function $-3{\left(x+1\right)}^2$ upward by $5$ units, then the function becomes $-3{\left(x+1\right)}^2+5$ and the graph of the function $f\left(x\right)=-3{\left(x+1\right)}^2+5$ is as follows

  

Interpretation:

The graphs represents the transformations of the function $f(x)=x^2$ to the function $f\left(x\right)=-3{\left(x+1\right)}^2+5$ .