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.

Step3: Graph - making (general steps)

For \(f(x)=-x^{3}\), when \(x = 0\), \(f(0)=0\); when \(x = 1\), \(f(1)=-1\); when \(x=-1\), \(f(-1)=1\). Plot these points and draw a smooth curve through them to get the graph of \(y = f(x)\).
For \(g(x)=-x^{3}+5\), when \(x = 0\), \(g(0)=5\); when \(x = 1\), \(g(1)=4\); when \(x=-1\), \(g(-1)=6\). Plot these points and draw a smooth curve through them. The graph of \(g(x)\) is exactly the graph of \(f(x)\) moved 5 units up.

Answer:

The graph of \(g(x)\) is the graph of \(f(x)\) shifted vertically upwards by 5 units.