Ch. 1 - Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 1 - Functions

Problem 1.2.29

Chapter 1, Problem 1.2.29

Shifting Graphs

Exercises 27–36 tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation.

y = x³ Left 1, down 1

Verified step by step guidance

Identify the original function: The given function is \( y = x^3 \).

Determine the horizontal shift: A shift to the left by 1 unit means replacing \( x \) with \( x + 1 \) in the function. This gives us \( y = (x + 1)^3 \).

Determine the vertical shift: A shift down by 1 unit means subtracting 1 from the entire function. This modifies the equation to \( y = (x + 1)^3 - 1 \).

Write the equation for the shifted graph: The new equation after applying both shifts is \( y = (x + 1)^3 - 1 \).

Sketch the graphs: Draw the original graph of \( y = x^3 \) and the shifted graph \( y = (x + 1)^3 - 1 \) on the same set of axes, labeling each graph with its respective equation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Graph Shifting

Graph shifting involves translating a graph horizontally and vertically on the coordinate plane. A horizontal shift moves the graph left or right, while a vertical shift moves it up or down. For the equation y = x³, shifting left by 1 unit and down by 1 unit results in the new equation y = (x + 1)³ - 1.

Horizontal Shift

A horizontal shift changes the x-values of a function. If a graph is shifted left by 'a' units, the transformation is represented by replacing x with (x + a) in the equation. For y = x³, shifting left by 1 unit modifies the equation to y = (x + 1)³, effectively moving the graph one unit to the left.

Vertical Shift

A vertical shift affects the y-values of a function. Shifting a graph down by 'b' units involves subtracting 'b' from the entire function. In the case of y = x³, shifting down by 1 unit results in the equation y = x³ - 1, which lowers the graph by one unit on the y-axis.

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Related Practice

Textbook Question

Even and Odd Functions

In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.

g(x) = x/(x² − 1)

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Textbook Question

Increasing and Decreasing Functions

Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

y = −x³

280

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Textbook Question

Using the Addition Formulas

Use the addition formulas to derive the identities in Exercises 31–36.

sin (A − B) = sin A cos B − cos A sin B

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Textbook Question

Algebraic Combinations

In Exercises 1 and 2, find the domains of f, g, f + g, and f ⋅ g.

f(x) = √(x + 1), g(x) = √(x − 1)

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Textbook Question

Increasing and Decreasing Functions

Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

y = (−x)²/³

188

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Textbook Question

[Technology Exercise]

a. Graph the functions f(x) = x/2 and g(x) = 1 + (4/x) together to identify the values of x for which

x/2 > 1 + 4/x

b. Confirm your findings in part (a) algebraically.

177

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