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Ch. 1 - Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 1 - FunctionsProblem 1.2.32
Chapter 1, Problem 1.2.32

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 = −√x Right 3

Verified step by step guidance
1
Identify the transformation needed: The problem states that the graph of the equation y = -√x needs to be shifted to the right by 3 units.
Understand the effect of shifting: Shifting a graph to the right by 'h' units involves replacing 'x' with '(x - h)' in the equation. Here, h = 3.
Apply the transformation: Replace 'x' with '(x - 3)' in the original equation y = -√x to get the new equation y = -√(x - 3).
Sketch the graphs: Draw the original graph of y = -√x, which is a downward-opening half parabola starting at the origin (0,0). Then, draw the shifted graph of y = -√(x - 3), which starts at the point (3,0) and follows the same shape.
Label the graphs: Clearly label the original graph with the equation y = -√x and the shifted graph with the equation y = -√(x - 3) to distinguish between them.

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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 moving a graph horizontally or vertically without changing its shape. A horizontal shift occurs when a constant is added or subtracted from the x-variable, moving the graph left or right. For example, y = f(x - 3) shifts the graph of y = f(x) three units to the right.
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Square Root Function

The square root function, y = √x, is a basic function in calculus that produces a curve starting at the origin and increasing slowly. It is defined only for x ≥ 0, as the square root of a negative number is not real. Understanding its shape and domain is crucial for graph transformations.
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Equation Transformation

Transforming an equation involves modifying its algebraic expression to reflect changes in the graph. For a horizontal shift to the right by 3 units, replace x with (x - 3) in the equation. Thus, y = -√x becomes y = -√(x - 3), representing the shifted graph.
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Related Practice
Textbook Question

Use graphing software to graph the functions specified in Exercises 31–36.

Select a viewing window that reveals the key features of the function.


Graph two periods of the function f (x) = 3 cot (x/2) + 1.

Textbook Question

Even and Odd Functions


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


f(x) = x⁻⁵

Textbook Question

In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.


𝔂 = x²/⁵

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Textbook Question

Use graphing software to graph the functions specified in Exercises 31–36.

Select a viewing window that reveals the key features of the function.


Graph the upper branch of the hyperbola y² − 16x² = 1.

Textbook Question

In Exercises 35 and 36, find the (a) domain and (b) range.


𝔂 = { -x - 2, -2 ≤ x ≤ - 1

{ x, -1 < x ≤ 1

{ -x + 2, 1 < x ≤ 2

Textbook Question

Theory and Examples


In Exercises 69 and 70, match each equation with its graph. Do not use a graphing device, and give reasons for your answer.


a. y = x⁴

b. y = x⁷

c. y = x¹⁰


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