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Chapter 3: Problem 40

A function and how it is to be shifted is given. Find the shifted function and then display the given function and the shifted function on the same screen of a graphing calculator. \(y=x^{3},\) down 2

Short Answer

Expert verified
The shifted function is \( y = x^3 - 2 \).

Step by step solution

01

Understand the Problem

We are given the function \( y = x^3 \) and we need to shift it down by 2 units.
02

Identify the Transformation

The operation of shifting a function down involves subtracting a constant from the function. Here we subtract 2 from the original function \( y = x^3 \).
03

Write the Shifted Function

The new function after shifting the original function down by 2 units is \( y = x^3 - 2 \). This accounts for the downward movement on the graph.
04

Display Functions on Graphing Calculator

Enter the original function \( y = x^3 \) and the shifted function \( y = x^3 - 2 \) into a graphing calculator. Ensure both functions are active or selected to plot on the same screen for comparison.

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Key Concepts

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

Graphing Calculator
A graphing calculator is a powerful tool for visualizing algebraic functions. With its ability to plot multiple functions on a single screen, it helps you easily compare changes and transformations in a function.

Here are basic steps to use one effectively:
  • Enter each function into the calculator separately. You typically do this by selecting a function slot and typing in the equation.
  • Make sure that all functions you want to see are "turned on" or selected, enabling them to display on the graph.
  • Adjust the window settings to ensure the screen captures all important points of your functions. This might include altering the x and y values to match the range of your data.
Using a graphing calculator allows you to visually see how the original function and its transformation relate, making understanding shifts easy and tangible.
Shifted Function
A shifted function occurs when a base function, like the algebraic function given as \(y = x^3\), undergoes a transformation that alters its position on a graph. In this case, we're performing a vertical shift.

Key points about shifted functions:
  • It maintains the original shape of the function, meaning the curve of \(y = x^3\) still looks like a cubic graph, just moved down.
  • Vertical shifts involve adding or subtracting a constant value to the function, shifting all its points up or down.
    • For a shift downwards, you subtract a constant from the function.
    • For an upward shift, you add a constant.
Understanding shifted functions allows you to predict and graph the position of functions after transformations easily.
Algebraic Function
An algebraic function uses algebraic expressions and is formed by combining constants, variables, and arithmetic operations. The given example, \(y = x^3\), is an algebraic function because it involves a single variable raised to a power.

Here are some features of algebraic functions:
  • They can include operations like addition, subtraction, multiplication, division, and powers.
  • These functions represent standard forms often used in algebra, calculus, and graphing.
  • Algebraic functions can be quite simple, like linear functions, or complex, involving multiple terms and variables.
Understanding these functions is crucial because they act as the foundation for more advanced mathematical problems and solutions.
Vertical Translation
A vertical translation occurs when every point of a function is moved up or down the graph by adding or subtracting a constant value from the function. This type of transformation changes the location of the graph without altering its shape.

Important characteristics of vertical translations:
  • The entire graph shifts uniformly, meaning each point on the graph moves the same distance in the vertical direction.
  • For downward translations, subtract the constant from the function's equation. For example, \(y = x^3\) becomes \(y = x^3 - 2\) when moved 2 units down.
  • For upward translations, add the constant value to the equation.
This concept helps in modifying and adapting functions for practical applications and better graph interpretations.

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Most popular questions from this chapter

In Exercises \(37-66,\) graph the indicated functions. On a taxable income of \(x\) dollars, a certain city's income \(\operatorname{tax} T\) is defined as \(T=0.02 x\) if \(0 < x \leq 20,000\) \(T=400+0.03(x-20,000)\) if \(x > 20,000 .\) Graph \(T=f(x)\) for \(0 \leq x < 100,000\)

Solve the given problems. Display the graph of \(y=c x^{4},\) with \(c=4\) and with \(c=1 / 4\) Describe the effect of the value of \(c\)

Write the equation as given by the statement. Then write the indicated function using functional notation. The electrical resistance \(R\) of a certain ammeter, in which the resistance of the coil is \(R_{c},\) is the product of 10 and \(R_{c}\) divided by the sum of 10 and \(R_{c}\)

Graph the indicated functions. Plot the graphs of \(y=2-x\) and \(y=|2-x|\) on the same coordinate system. Explain why the graphs differ.

Graph the indicated functions. A guideline of the maximum affordable monthly mortgage \(M\) on a home is \(M=0.25(I-E),\) where \(I\) is the homeowner's monthly income and \(E\) is the homeowner's monthly expenses. If E= 600 dollar, graph \(M\) as a function of \(I\) for I= 2000 dollar to I= 10,000 dollar.
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