Chapter 8.4, Problem 52E

To determine

To graph function $g(x)=f(x)-6,\text{where}f(x)=5{\left(x-3\right)}^2-1$

Explanation of Solution

Given information :

Given functions $g(x)=f(x)-6,\text{where}f(x)=5{\left(x-3\right)}^2-1$

As we know that the vertex of the graph of $f(x)=m{(x-h)}^2+k\left(\text{where}m\text{is}\text{constant}\right)$ is $(h,k)$ and the axis of symmetry is $x=h$ .

Now compare given function $f(x)=5{\left(x-3\right)}^2-1$ with $f(x)=m{(x-h)}^2+k$ , we get $h=3,k=-1$

Thus, the vertex of $f(x)$ is $(3,-1)$ and the axis of symmetry $x=3$

Given function $g(x)=f(x)-6$ is in the form of $y=f(x)+k$ , where $k=-6$ .

So, the graph of $g(x)$ is a vertical translation of $6$ units down of the graph of $f(x)$ ,

To graph $g(x)$ , subtract $6$ from the $y-$ coordinates of the points on the graph of $f(x)$ .

Thus, the vertex of $g(x)$ is $(3,-7)$ and axis of symmetry remains same $x=3$ .

Graph: The graph of $g(x)=f(x)-6$ is

  

Interpretation: From above graph, we observe that the vertex of $g(x)$ is $(3,-7)$ and axis of symmetry remains same $x=3$ .