Question

use transformations of the standard cubic function, f(x) = x³, to graph the function g(x) = x³ + 5. use the graphing tool to graph the function. click to enlarge graph

Explanation:

Step1: Recall Vertical Shift Rule

For a function \( y = f(x) + k \), if \( k>0 \), the graph of \( f(x) \) is shifted up by \( k \) units. Here, \( f(x)=x^3 \) and \( g(x)=x^3 + 5 \), so \( k = 5>0 \).

Step2: Analyze the Transformation

Start with the standard cubic function \( f(x)=x^3 \), which has a point at the origin \((0,0)\), and passes through \((1,1)\), \((-1,-1)\), \((2,8)\), \((-2,-8)\) etc. For \( g(x)=x^3+5 \), each \( y \)-coordinate of \( f(x) \) is increased by 5. So the origin \((0,0)\) on \( f(x) \) moves to \((0,0 + 5)=(0,5)\), \((1,1)\) moves to \((1,1 + 5)=(1,6)\), \((-1,-1)\) moves to \((-1,-1 + 5)=(-1,4)\), \((2,8)\) moves to \((2,8 + 5)=(2,13)\), \((-2,-8)\) moves to \((-2,-8 + 5)=(-2,-3)\). Plot these new points and draw the cubic curve through them, which is the graph of \( g(x)=x^3+5 \), a vertical shift up 5 units of \( f(x)=x^3 \).

Answer:

To graph \( g(x)=x^3 + 5 \), vertically shift the graph of the standard cubic function \( f(x)=x^3 \) upward by 5 units. The key points of \( f(x)=x^3 \) (e.g., \((0,0)\), \((1,1)\), \((-1,-1)\)) are transformed to \((0,5)\), \((1,6)\), \((-1,4)\) etc., and the curve is drawn through these new points.