Problem 50 - A Graphical Approach to College Algebra

Question

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$f(x)=(x+2)^{3}$$

Step-by-Step Solution

Answer

Shift the graph of \(y = x^3\) 2 units to the left.

Step 1: Identify the Basic Graph

The given function is \(f(x) = (x+2)^3\). The basic graph we start with is \(g(x) = x^3\), which is a cubic function.

Step 2: Recognize the Transformation

The function \(f(x) = (x+2)^3\) involves a horizontal shift. The \(+2\) inside the parentheses indicates a shift to the left by 2 units. This is a horizontal transformation.

Step 3: Apply the Horizontal Shift

Starting from the basic graph \(g(x) = x^3\), shift every point on this graph 2 units to the left. For example, the point (0,0) on \(g(x)\) becomes (-2,0) on \(f(x)\).

Step 4: Sketch the Transformed Graph

Draw the transformed graph based on the horizontal shift. The graph should maintain the same cubic shape as \(x^3\), but all points will be moved 2 units left.

Key Concepts

Cubic Function

A cubic function is one of the most basic forms of polynomial equations, characterized by the highest exponent being three. The general form is given by \( f(x) = ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants and \( a eq 0 \). In the simplest case, where \( b = c = d = 0 \), the function is \( g(x) = x^3 \). This basic cubic graph exhibits a few distinct features that are vital to understanding its behavior and transformations:

• Points of intersection: The graph typically crosses the x-axis at the origin (0,0).
• Symmetry: Cubic functions are symmetric with respect to the origin, meaning they exhibit rotational symmetry of 180 degrees around the origin.
• End behavior: As \( x \) approaches infinity, \( f(x) \) approaches infinity as well, and as \( x \) approaches negative infinity, \( f(x) \) also moves towards negative infinity.

The graph of \( g(x) = x^3 \) is a perfect starting point for exploring transformations, including horizontal shifts.

Horizontal Shift

A horizontal shift is one type of transformation used to move the graph of a function left or right along the x-axis. This transformation is included in the equation by adding or subtracting a constant value inside the function's expression, usually inside the input. For the given function \( f(x) = (x+2)^3 \), we see a horizontal shift.
Typically, the rule of thumb is as follows:

• When you add a positive number inside the parentheses (e.g., \( x+2 \)), it shifts the graph to the left.
• When you subtract a number (e.g., \( x-2 \)), it shifts the graph to the right.

Understanding this principle helps you predict how graphs will transform without having to plot numerous points. In this case, the \( x+2 \) in the function causes a shift of the classic cubic function \( x^3 \) two units to the left.

Graph Sketching

Graph sketching involves drawing the graph of a function by understanding and applying transformations. In the context of the cubic function and its transformations, sketching serves as an essential tool in visualizing how modifications to equations result in changes to their graphs.
To sketch the graph of \( f(x) = (x+2)^3 \), follow these steps:

1. Begin with the basic function: Start your sketch by drawing the graph of \( g(x) = x^3 \). Make sure to mark key features like the origin and the symmetry across the origin.
2. Apply the transformation: Shift every point on this graph 2 units left. You are essentially relocating the origin of this graph to \(-2, 0\). Use this shift to guide your sketching of the entire curve.
3. Connect the points: After shifting, ensure the cubic curve retains its original shape. This includes its points of intersection with axes and its symmetry. The cubic function will still rise to positive infinity as \( x \) becomes larger, and fall to negative infinity as \( x \) becomes smaller.

Through careful application of these steps, you can accurately sketch the graph and gain a clearer understanding of transformations.