Question

use transformations to sketch a graph of f\\(f(x) = (x - 1)^3\\)\\(\dots\\)use the graphing tool to graph the function.click to enlarge graph

Explanation:

Step1: Identify the parent function

The parent function here is \( y = x^3 \), which is a cubic function. Its graph passes through the origin \((0,0)\), is symmetric about the origin, and has a point - shape that increases from left to right, with a horizontal inflection point at the origin.

Step2: Determine the transformation

The given function is \( f(x)=(x - 1)^3 \). For a function of the form \( y=(x - h)^3 \), where \( h\) is a constant, the graph of the parent function \( y = x^3 \) is shifted horizontally. If \( h>0 \), the shift is to the right by \( h \) units. In our case, \( h = 1 \), so we shift the graph of \( y=x^3 \) 1 unit to the right.
To sketch the graph:

1. Start with the key points of the parent function \( y = x^3 \). Some key points are \((-2,-8)\), \((-1,-1)\), \((0,0)\), \((1,1)\), \((2,8)\).
2. For each of these points, apply the horizontal shift. To shift a point \((x,y)\) 1 unit to the right, the new \( x \) - coordinate is \( x+1 \) and the \( y \) - coordinate remains the same. So the key points of \( f(x)=(x - 1)^3 \) are:

• For \((-2,-8)\): New \( x=-2 + 1=-1\), so the point is \((-1,-8)\).
• For \((-1,-1)\): New \( x=-1 + 1 = 0\), so the point is \((0,-1)\).
• For \((0,0)\): New \( x=0 + 1=1\), so the point is \((1,0)\).
• For \((1,1)\): New \( x=1 + 1=2\), so the point is \((2,1)\).
• For \((2,8)\): New \( x=2+1 = 3\), so the point is \((3,8)\).

1. Then, plot these new points and draw a smooth curve through them, following the shape of the cubic function (increasing, with a horizontal inflection point at \( x = 1 \) for the transformed function).

Answer:

To graph \( f(x)=(x - 1)^3 \), shift the graph of \( y = x^3 \) 1 unit to the right. Key points of \( f(x) \) are \((-1,-8)\), \((0,-1)\), \((1,0)\), \((2,1)\), \((3,8)\) (among others), and a smooth curve through these points (shifted 1 unit right from \( y = x^3 \)) is the graph of \( f(x)\).