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 the graph of \(x^3\) one unit left and three units down.

Step 1: Identify the Parent Function

The parent function of \(g(x) = (x+1)^3 - 3\) is \(f(x) = x^3\). This is because the expression inside the parentheses, once simplified, resembles the standard form of a cubic function \(x^3\).

Step 2: Determine Horizontal Translation

The function \((x+1)^3\) indicates a horizontal shift. Since \(x+1\) includes \(+1\), it means the graph of \(y = x^3\) is shifted one unit to the left.

Step 3: Determine Vertical Translation

The \(-3\) outside the cubic expression \((x+1)^3\) represents a vertical downward shift. This means the entire graph will be moved three units down.

Step 4: Combine Translations and Sketch

Taking into account both translations, the graph of \(g(x) = (x+1)^3 - 3\) is the graph of \(f(x) = x^3\) but shifted one unit to the left and three units down. Sketch the graph with these transformations applied.

Key Concepts

Understanding Cubic Functions

Cubic functions are polynomial functions where the highest degree of the variable is three. They are generally expressed in the form \(f(x) = ax^3 + bx^2 + cx + d\). The most basic form of a cubic function is \(f(x) = x^3\), where the graph typically passes through the origin and exhibits a distinctive "S" shape.

This basic cubic graph increases to the right and decreases to the left of the origin. It is symmetric about the origin, making it an odd function. Plotting this function helps us visualize transformations like translations, which we can use to modify its shape and position.

Horizontal Translation Explained

A horizontal translation involves moving the graph of a function left or right along the x-axis. For the function \(g(x) = (x+1)^3 - 3\), the expression \((x+1)^3\) indicates that the parent graph \(f(x) = x^3\) will move horizontally.

The \(+1\) inside the parentheses suggests that the entire graph shifts one unit to the left. In general, for \(f(x + h)\), the graph shifts \(h\) units to the left if \(h\) is positive. Conversely, it shifts to the right if \(h\) is negative. This movement allows us to explore different x-intercepts by repositioning the graph without altering its shape.

Vertical Translation Insights

Vertical translation involves moving the graph up or down along the y-axis. In the equation \((x+1)^3 - 3\), the \(-3\) indicates a vertical translation. Specifically, it means the graph will shift downward by three units.

For a function \(f(x) + k\), the graph moves down \(k\) units when \(k\) is negative and up \(k\) units when it is positive. This shift affects the y-intercept of the graph and is an essential tool for adjusting the height of the curve while keeping its shape consistent.

Graph Sketching Techniques

Graph sketching is the process of drawing a function based on its algebraic expression and transformations. To sketch \(g(x) = (x+1)^3 - 3\), we start with the parent graph \(f(x) = x^3\), known for its "S" shape passing through the origin.

First, apply the horizontal shift by moving the entire graph one unit to the left, as indicated by \(x+1\). Then, shift the graph three units down due to \(-3\). Carefully plotting a few points could help verify the graph's new position.

• Determine significant points like the y-intercept and zeroes.
• Draw the transformed curve smoothly.
• Check for symmetry, which should remain intact due to the nature of cubic functions.

This approach ensures the sketched graph correctly represents the function, complete with all translations.