Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of \(g(x)=(x+1)^{3}-3\) using translations.

Step-by-Step Solution

Answer

Shift \(x^3\) left by 1 unit and down by 3 units to graph \(g(x)=(x+1)^3-3\).

Step 1: Identify the Parent Function

The parent function for the given function is \( f(x) = x^3 \). This is a basic cubic function with a graph that passes through the origin (0,0) and has an 'S' shape.

Step 2: Determine Horizontal Translation

The function \( g(x) = (x+1)^3 - 3 \) includes \( (x+1) \), which indicates a horizontal translation. The graph of the function is shifted 1 unit to the left compared to the parent function \( f(x) = x^3 \).

Step 3: Determine Vertical Translation

The function also includes \( -3 \), which indicates a vertical translation downward. The entire graph of \( (x+1)^3 \) is shifted 3 units down.

Step 4: Sketch the Translated Graph

Begin by sketching the basic cubic graph of \( x^3 \). Then, move the graph 1 unit to the left and 3 units down to get the graph of \( g(x) = (x+1)^3 - 3 \). Ensure the graph maintains its cubic 'S' shape.

Key Concepts

Cubic Functions

Cubic functions offer a unique and fascinating aspect of algebraic graphing due to their distinctive shape and properties. A basic cubic function is often represented as \( f(x) = x^3 \). This function produces a graph known for its serpentine 'S' shape, smoothly increasing or decreasing through the origin (0,0). The cubic curve is symmetric around the origin, giving it an appealing balance.

When analyzing cubic functions, remember these key characteristics:

• The function is odd, meaning \( f(-x) = -f(x) \), which results in its symmetrical behavior.
• It has no maximum or minimum point, as its ends head infinitely upwards and downwards.
• The slope changes direction at a point, defined as an inflection point. For the function \( f(x) = x^3 \), this occurs at the origin.

Understanding these properties helps you anticipate the behavior of transformations applied to cubic functions.

Horizontal Translation

Horizontal translation shifts the graph left or right along the x-axis without altering its shape. In the function \( g(x) = (x+1)^3 - 3 \), the expression inside the cube, \((x+1)\), indicates a horizontal shift. Specifically, the "+1" moves the graph 1 unit to the left.

How do we interpret this shift? A positive value inside the parentheses (adding 1 here) causes the graph to move in the opposite direction (left). Thus, if you encounter a function like \((x-h)^3\), you should shift the graph \( h \) units to the right if \( h \) is positive and to the left if \( h \) is negative.

Remember this pattern of opposite direction when finding the new position for the graph. It ensures that even when dealing with complex translations, you'll know exactly which way the graph shifts.

Vertical Translation

Vertical translation involves shifting the graph up or down along the y-axis. In our function \( g(x) = (x+1)^3 - 3 \), the "-3" specifies a vertical shift. This means the entire graph of \( (x+1)^3 \) is moved 3 units downward.

Vertical translations are, in essence, easier to comprehend since they move in the direction you'd anticipate: positive values lift the graph upwards, while negative values bring it down.

• Consider this rule simple: "+" goes up, "-" goes down.

By applying a vertical translation, the graph retains its original shape; only its position changes. This transformation allows adjustments to where the graph intersects the y-axis, which can be especially useful when matching real-world data.