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 up 3 units.

Short answer

Answer: The formula for function g(x) is g(x) = x^3 + 3. Its graph is the same cubic shape as the original function f(x)=x^3, but shifted 3 units upward.

Step-by-step solution

Find the formula of the shifted function g(x)

To find the formula of the function \(g(x)\), we need to add the vertical shift to the output of the function \(f(x)\). In our case, we need to shift the graph upwards by 3 units. Therefore, the formula for \(g(x)\) is given as follows: g(x) = f(x) + 3 Now, we need to substitute the given function \(f(x)=x^3\): g(x) = x^3 + 3 The formula for the function \(g(x)\) is \(g(x)=x^3 +3\).

Sketch the graph of g(x)

To sketch the graph of the function \(g(x)=x^3 + 3\), we'll follow these steps: 1. Draw the graph of the original function, \(f(x)=x^3\). 2. Shift the graph of the original function vertically upward by 3 units. The graph of the function \(f(x)=x^3\) is a cubic polynomial that increases steeply for positive x-values and decreases steeply for negative x-values. The graph passes through the origin (0,0). To create the graph of the function \(g(x)=x^3 + 3\), we need to shift this original graph 3 units upwards. Doing so, the cubic shape of the graph remains the same, but all the y-values are increased by 3. The graph of the function \(g(x)=x^3 + 3\) is a cubic function similar to the graph of \(f(x)=x^3\) but shifted 3 units upwards.

Key concepts

Cubic Function

A cubic function is a type of polynomial function of degree three. In general, it is expressed in the form \( f(x) = ax^3 + bx^2 + cx + d \), where \( a eq 0 \). This function can have up to three real roots and up to two turning points. In a cubic function, the graph tends to show an S-shaped curve, typically originating from the origin when in the standard form of \( f(x) = x^3 \).
The behavior of cubic functions is interesting because it can represent various types of curves depending on the coefficients. Specifically, for positive leading coefficients, as \( x \) becomes positive, \( y \) will increase without bounds and vice versa for negative values.
Cubic functions exhibit certain symmetry, known as point symmetry, about their inflection point. This makes the graph of \( f(x) = x^3 \) symmetric about the origin, providing an easily recognizable shape for transformations.

Graph Transformation

Graph transformation involves modifying the basic shape or position of a graph. In the context of this exercise, the specific transformation is a vertical shift. When you perform a vertical shift, you adjust the graph's position up or down along the y-axis.

• Vertical Shift: When you shift a graph vertically, every point moves up or down by the same amount. For a function \( f(x) \), if you add a constant \( k \), resulting in \( g(x) = f(x) + k \), the entire graph of \( f(x) \) is moved up \( k \) units if \( k > 0 \), or down \( k \) units if \( k < 0 \). In our exercise, the graph of \( f(x) = x^3 \) is shifted upwards by 3 units, resulting in \( g(x) = x^3 + 3 \).

This type of transformation retains the shape of the original graph but changes its position. For example, the original graph of \( f(x) = x^3 \), which passes through the origin, is moved so that it now intersects the y-axis at the point (0,3).
Remember that while the shape and "steepness" (slope behavior) of the graph remain unchanged in a vertical shift, the actual values of the function increase or decrease by the amount of the shift.

Function Notation

Function notation provides a concise way to express the relationship between input values and their corresponding outputs in a function. The notation \( f(x) \) is read as "f of x," and it represents the output of the function when \( x \) is the input.

• Input and Output: The expression inside the parentheses, \( x \), is known as the input, variable, or argument of the function. It is what you "plug into" the function. The result of the function's rule, \( f(x) \), is the output.
• Function Transformation: When we alter the function notation, such as introducing \( g(x) \), it indicates a new function derived from modifying original functions. This is crucial in graph transformations, like vertical shifts.

In the exercise, the transformation is described using function notation as \( g(x) = x^3 + 3 \), showing that the function \( g \) is a transformation of \( f(x) = x^3 \) with an added vertical displacement. This notation succinctly captures both the original function and its transformation.