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Chapter 8: Problem 4

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 horizontally to the right 2 units.

Short Answer

Expert verified
Based on the step-by-step solution provided, 1. The original function is f(x) = x^3. 2. The function needs to be shifted horizontally to the right by 2 units. 3. The new function g(x) representing the horizontal shift is g(x) = (x-2)^3. 4. The graph of g(x) has the same general shape as f(x) but is shifted 2 units to the right, intersecting the x-axis at (2, 0).

Step by step solution

01

Identify the given function and transformation

The given function is f(x) = x^3. The graph of this function needs to be shifted horizontally to the right by 2 units. To perform the horizontal shift, we need to adjust the value inside the function related to x.
02

Apply the horizontal shift

Shifting the function to the right by 2 units means replacing x in the given function f(x) with (x-2). So, the function g(x) becomes: g(x) = [(x-2)]^3 Now, we need to sketch the graph of g(x).
03

Sketch the original graph of f(x)

First, sketch the original graph of the function f(x) = x^3, which is a cubic function. The graph passes through the origin (0, 0), and it goes from the bottom left quadrant to the top right quadrant, crossing the x-axis and y-axis at the origin.
04

Sketch the horizontally shifted graph of g(x)

To sketch the graph of g(x) = (x-2)^3, take the graph of f(x) = x^3, and shift it 2 units horizontally to the right. This means the entire curve will move to the right, and the point of intersection with the x-axis will now be at (2, 0) instead of the origin.
05

Verify the result

The graph of g(x) = (x-2)^3 is the horizontal shift of the original graph f(x) = x^3 to the right by 2 units. The point of intersection with the x-axis is now at (2, 0) instead of the origin, and the general shape of the graph remains the same as that of f(x). Thus, the formula for g(x) representing the horizontal shift of f(x) is given by: g(x) = (x-2)^3

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cubic Functions
A cubic function is a type of polynomial function where the highest exponent of the variable is three. The standard form of a cubic function is written as \( f(x) = ax^3 + bx^2 + cx + d \). In many cases, the simpler form \( f(x) = x^3 \) is used, which is what we have in the problem at hand.
Cubic functions exhibit particular characteristics that set them apart from linear and quadratic functions:
  • They have at most three real roots.
  • The graph of a cubic function is an S-shaped curve.
  • It can have one or two inflection points, depending on the coefficients.
  • The steepness of the curve is dictated by the leading coefficient.
The graph of \( f(x) = x^3 \) specifically passes through the origin and displays symmetry about the origin, known as point symmetry. Understanding these traits makes it easier to predict how alterations will affect the graph.
Graph Transformations
Graph transformations involve shifting, stretching, or reflecting the graph of a function in the coordinate plane. This exercise focused specifically on horizontal transformations鈥攁 common type of graph alteration.
Horizontal transformations modify the function's input. When you add or subtract a value directly inside the function, it shifts the graph left or right. In the function \( f(x) = x^3 \), modifying it to \( g(x) = (x-2)^3 \) shifts the graph 2 units to the right.
  • Adding to the inside of the function shifts the graph left: \( f(x) = (x + n)^3 \)
  • Subtracting from the inside of the function shifts the graph right: \( f(x) = (x - n)^3 \)
It鈥檚 crucial to adjust components inside the function accurately to predict the graph's new position. Visualizing these shifts helps in understanding how particular modifications affect function graphs.
Function Notation
Function notation serves to easily define and work with functions in a concise manner. At its core, a function typically pairs each input value from a set with exactly one output value. We denote functions using symbols, often \( f(x) \), where "\( f \)" names the function, and "\( x \)" represents the variable input.
In our example, \( g(x) = (x-2)^3 \) indicates that the function \( g \) is derived from \( f(x) = x^3 \) by implementing a transformation.
  • Horizontal shifts involve modifying the argument of the function \( g(x) = f(x-n) \).
  • Function notation provides flexibility, allowing various transformations to be enacted and represented clearly.
Using function notation correctly is fundamental for expressing transformations and defining the behavior of different mathematical functions.

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Most popular questions from this chapter

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