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.

Short answer

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

Step-by-step solution

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.

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.

Write the New Function

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

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.