Problem 50 - A Graphical Approach to Precalculus with Limits

Question

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. $$f(x)=(x+2)^{3}$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Basic Function

The basic function here is the cube function: \[ y = x^3 \]which is a standard function with a graph that passes through the origin and consists typically of an upward curve in the first quadrant and a downward curve in the third quadrant.

Step 2: Analyze the Transformation

We need to analyze the transformation applied to the basic function. The function given is:\[ f(x) = (x+2)^3 \]The transformation involved here is a horizontal shift. Specifically, the graph of \( y = x^3 \) is shifted horizontally to the left by 2 units because of the \( x + 2 \) term.

Step 3: Plot Key Points After Transformation

Let's consider how key points on the original graph are transformed: 1. The original point (0, 0) moves to (-2, 0) because of the left shift. 2. The original point (1, 1) moves to (-1, 1). 3. The original point (-1, -1) moves to (-3, -1). Plot these new points on the coordinate grid.

Step 4: Draw the Transformed Graph

Using the key points found in Step 3, sketch the graph. The curve should mirror the original shape of \( y = x^3 \), but should now be centered around the point (-2, 0). It will pass through the new points drawn before: (-2, 0), (-1, 1), and (-3, -1). It will still exhibit the characteristic S-shape of the cubic graph.

Key Concepts

Cubic Functions

Cubic functions are polynomial functions of degree three, typically in the form \( y = x^3 \). They are characterized by their distinct graph shape, known as the "S-shape" or sigmoidal curve. This shape arises from how the function values change as the input \( x \) is varied.

• In the first quadrant, the graph rises steeply, indicating a very rapid increase in function value for positive \( x \).


• In the third quadrant, it similarly falls swiftly, representing a rapid decrease for negative \( x \).

At the point \( (0, 0) \), which the graph always passes through, the function changes direction smoothly. This point is crucial because it's the origin point for transformations. Cubic functions are both symmetric around this central point and demonstrate linear-type behavior over small intervals, meaning their slopes change in a consistent, predictable manner.Understanding these properties can help when graphing or transforming cubic functions.

Horizontal Shifts

Horizontal shifts are used to move the graph of a function left or right on the coordinate plane. For a function \( f(x) \), if we see \( f(x + c) \), it implies a horizontal shift of the function to the left by \( c \) units. Conversely, if \( f(x - c) \), the graph shifts to the right.In the case of our given function \( f(x) = (x+2)^3 \), the term \( x+2 \) signifies a left shift by 2 units.

• This means every point on the original graph of \( y = x^3 \) is moved to the left by 2 units.


• Graph transformations like these do not alter the shape of the graph—just its position.

Visualizing horizontal shifts can be thought of as "sliding" the entire graph either left or right while keeping its orientation and shape intact.

Key Points

Key points are specific points on the graph that help outline the structure and position of the graph on the coordinate plane. These are crucial when dealing with transformations, as they can be used to anchor the graph's new position after transformations.For the cubic function \( f(x) = (x+2)^3 \), the key points from the parent function \( y = x^3 \) are:

• \((0, 0)\), which shifts to \((-2, 0)\)


• \((1, 1)\), shifting to \((-1, 1)\)


• \((-1, -1)\), moving to \((-3, -1)\)

By mapping these points post-transformation, you maintain the graph's integrity and ensure its accuracy. The S-curve of the cubic function is defined by these points. Therefore, achieving an accurate transformation means correctly positioning key points before sketching the final graph.