Problem 2 - Algebra Form and Function

Question

The graph of the function \(g(x)\) is a horizontal and/or vertical shift of the graph of \(f(x)=x^{3},\) shown in Figure \(8.19 .\) For each of the shifts described, sketch the graph of \(g(x)\) and find a formula for \(g(x)\). Shifted vertically down 2 units.

Step-by-Step Solution

Answer

Answer: \(g(x) = x^3 - 2\)

Step 1: Identify the Given Function and Desired Shift

The function given is \(f(x) = x^3\), and we are asked to find the function \(g(x)\) after shifting the graph vertically down by 2 units.

Step 2: Apply Vertical Shift

To shift the graph vertically down by 2 units, we need to subtract 2 from the output of the function \(f(x)\). This means that for the new function \(g(x)\), the output will be equal to the output of \(f(x)\) minus 2.

Step 3: Write the New Function

Using the information from Step 2, we can write the new function \(g(x)\) as follows: \(g(x) = f(x) - 2 = x^3 - 2\)

Step 4: Sketch the Graph

To sketch the graph of \(g(x) = x^3 - 2\), we can use the graph of \(f(x) = x^3\) as a reference and move each point on the graph vertically down by 2 units. This will result in a graph that is identical in shape to the original graph, but shifted downwards.

Key Concepts

Horizontal Shift

A horizontal shift in a function's graph involves moving the entire graph left or right along the x-axis.

To achieve this shift, you need to adjust the input values of the function. Typically, horizontal shifts are represented within the function's equation by adding or subtracting a constant from the x variable.

For example, if you have the function \( f(x) = x^3 \), and you want to shift it horizontally by 2 units to the right, you replace \( x \) with \( x - 2 \). This gives you the function \( g(x) = (x - 2)^3 \).

A few key points to remember about horizontal shifts include:

• If the constant is positive (e.g., \( x + 2 \)), the graph shifts to the left.
• If the constant is negative (e.g., \( x - 2 \)), the graph shifts to the right.
• Horizontal shifts do not affect the shape of the graph, only its horizontal position.

It's essential to practice these transformations to understand their effects on a graph.

Vertical Shift

Vertical shifts are more straightforward than horizontal ones because they involve moving the graph up or down along the y-axis without changing the x-values.

To perform a vertical shift, you adjust the output of the function by adding or subtracting a constant.

For the cubic function example given, \( f(x) = x^3 \), shifting vertically down by 2 units means you subtract 2 from the output, giving \( g(x) = x^3 - 2 \).

Here are some critical aspects of vertical shifts:

• Adding a positive constant (e.g., \( +2 \)) shifts the graph upwards.
• Subtracting a constant (e.g., \( -2 \)) shifts the graph downwards.
• The shape and other characteristics of the graph remain unchanged; only the vertical positioning is affected.

This type of transformation is helpful for adjusting the baseline of a graph for better comparison or visual clarity.

Cubic Function

A cubic function is a polynomial of degree three, typically represented in the form \( f(x) = ax^3 + bx^2 + cx + d \). The basic cubic function is \( f(x) = x^3 \), which has a distinct S-like curve.

This curve passes through the origin, and its symmetry doesn't extend to shifted versions unless horizontal shifts are also symmetric.

Characteristics of the cubic function include:

• An inflection point at the origin where the curvature changes direction.
• One local minimum and maximum in its standard symmetric form, but they vary with coefficients.
• The graph continues infinitely in both positive and negative directions along the y-axis.

Understanding transformations like shifts in cubic functions helps in graphing and analyzing more complex polynomials where such transformations are combined with scaling and reflections.