Question

graph the given functions, f and g, in the same rectangular coordinate system. describe how the graph of g is related to the graph of f.

$f(x)=-x^{3}$
$g(x)=-x^{3}+5$

Explanation:

Step1: Recall function - transformation rules

For functions \(y = f(x)\) and \(y=f(x)+k\), when \(k>0\), the graph of \(y = f(x)+k\) is a vertical - shift of the graph of \(y = f(x)\) upwards by \(k\) units.

Step2: Identify the relationship between \(f(x)\) and \(g(x)\)

Given \(f(x)=-x^{3}\) and \(g(x)=-x^{3}+5\). Here, \(k = 5>0\). So the graph of \(g(x)\) is a vertical shift of the graph of \(f(x)\) upwards by 5 units.

To graph \(f(x)=-x^{3}\), we can find some points:
When \(x=-2\), \(f(-2)=-(-2)^{3}=8\); when \(x = - 1\), \(f(-1)=-(-1)^{3}=1\); when \(x = 0\), \(f(0)=0\); when \(x = 1\), \(f(1)=-1\); when \(x = 2\), \(f(2)=-8\).
To graph \(g(x)=-x^{3}+5\), we use the same \(x\) - values:
When \(x=-2\), \(g(-2)=-(-2)^{3}+5=8 + 5=13\); when \(x=-1\), \(g(-1)=-(-1)^{3}+5=1 + 5=6\); when \(x = 0\), \(g(0)=0 + 5=5\); when \(x = 1\), \(g(1)=-1+5=4\); when \(x = 2\), \(g(2)=-8 + 5=-3\).

Answer:

The graph of \(g(x)=-x^{3}+5\) is the graph of \(f(x)=-x^{3}\) shifted vertically upwards by 5 units.