Question

use transformations to sketch a graph of f. f(x) = (x + 3)^3 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 where it increases from left to right, with a relatively "flat" region near the origin and steeper regions as \(|x|\) increases.

Step2: Determine the transformation

For the function \( f(x)=(x + 3)^3 \), we use the transformation rule for horizontal shifts. The general form for a horizontal shift of a function \( y = f(x) \) is \( y=f(x - h) \), where if \( h>0 \), the graph shifts \( h \) units to the right, and if \( h < 0 \), the graph shifts \(|h|\) units to the left.

In our function \( f(x)=(x+3)^3=f(x-(- 3)) \), we have \( h=-3 \). So, the graph of \( y = x^3 \) is shifted 3 units to the left.

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 \((x,y)\) 3 units to the left, we replace \( x \) with \( x - 3 \). So:

• For the point \((-2,-8)\) on \( y = x^3 \), the new point on \( y=(x + 3)^3 \) is \((-2-3,-8)=(-5,-8)\).
• For the point \((-1,-1)\), the new point is \((-1 - 3,-1)=(-4,-1)\).
• For the point \((0,0)\), the new point is \((0-3,0)=(-3,0)\).
• For the point \((1,1)\), the new point is \((1-3,1)=(-2,1)\).
• For the point \((2,8)\), the new point is \((2-3,8)=(-1,8)\).

1. Then, plot these new points \((-5,-8)\), \((-4,-1)\), \((-3,0)\), \((-2,1)\), \((-1,8)\) and draw a smooth curve through them, following the general shape of a cubic curve (increasing from left to right, symmetric about the point \((-3,0)\) in terms of the shifted graph's behavior). The end - behavior of the graph of \( f(x)=(x + 3)^3 \) is the same as the parent function \( y = x^3 \): as \( x

ightarrow+\infty \), \( f(x)
ightarrow+\infty \) and as \( x
ightarrow-\infty \), \( f(x)
ightarrow-\infty \).

Answer:

To sketch the graph of \( f(x)=(x + 3)^3 \), shift the graph of \( y = x^3 \) 3 units to the left. Key points of the shifted graph can be found by shifting key points of \( y=x^3 \) 3 units left (e.g., the point \((0,0)\) on \( y = x^3 \) moves to \((-3,0)\) on \( f(x)=(x + 3)^3 \)), and then drawing a smooth cubic - shaped curve through the shifted points.