Question

starting with the graph of a basic function, graph the following function using the techniques of shifting, compressing, stretching, and/or reflecting. be sure to identify at least three key points. find the domain and range of the function.
g(x)=(x + 4)^3+1
complete the table of coordinates that lie on the graph of y = x^3 and the corresponding points that lie on the graph of g(x)=(x + 4)^3+1.
(type ordered pairs. simplify your answers.)
points that lie on the graph of y = x^3 corresponding points that lie on the graph of g(x)=(x + 4)^3+1
(-1,-1) 28
(0,0) 65
(1,1) 126

Explanation:

Step1: Analyze the transformation

The function \(g(x)=(x + 4)^{3}+1\) is a transformation of \(y = x^{3}\). The transformation is a horizontal shift \(4\) units to the left (because of \(x+4\)) and a vertical shift \(1\) unit up (because of \(+1\)).
For a point \((x,y)\) on \(y=x^{3}\), the corresponding point on \(g(x)\) is \((x-4,y + 1)\)

Step2: Find the corresponding points

• For the point \((-1,-1)\) on \(y=x^{3}\):

Substitute \(x=-1\) into \(g(x)\): \(g(-1)=(-1 + 4)^{3}+1=3^{3}+1=27 + 1=28\). The corresponding point is \((-5,0)\) (since \(x=-1-4=-5\) and \(y=-1 + 1=0\))

• For the point \((0,0)\) on \(y=x^{3}\):

Substitute \(x = 0\) into \(g(x)\): \(g(0)=(0 + 4)^{3}+1=64+1=65\). The corresponding point is \((-4,1)\) (since \(x=0-4=-4\) and \(y=0 + 1=1\))

• For the point \((1,1)\) on \(y=x^{3}\):

Substitute \(x = 1\) into \(g(x)\): \(g(1)=(1 + 4)^{3}+1=125+1=126\). The corresponding point is \((-3,2)\) (since \(x=1-4=-3\) and \(y=1 + 1=2\))

Answer:

Points that lie on the graph of \(y = x^{3}\)	Corresponding points that lie on the graph of \(g(x)=(x + 4)^{3}+1\)
\((0,0)\)	\((-4,1)\)
\((1,1)\)	\((-3,2)\)

The domain of \(g(x)\) is \((-\infty,\infty)\) (since for any real - number \(x\), the expression \((x + 4)^{3}+1\) is defined). The range of \(g(x)\) is \((-\infty,\infty)\) (because the cube function \(y = u^{3}\) where \(u=x + 4\) has a range of \((-\infty,\infty)\) and adding \(1\) just shifts the graph vertically, not changing the range).