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.
Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 1, Problem 1.2.34
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 = (1/2)(x + 1) + 5 Down 5, right 1
Verified step by step guidance1
Identify the original function: The given equation is \( y = \frac{1}{2}(x + 1) + 5 \). This is a linear function in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
Determine the transformations: The problem states 'Down 5, right 1'. This means we need to shift the graph down by 5 units and to the right by 1 unit.
Apply the horizontal shift: To shift the graph to the right by 1 unit, replace \( x \) with \( x - 1 \) in the equation. This gives us \( y = \frac{1}{2}((x - 1) + 1) + 5 \). Simplify this to \( y = \frac{1}{2}x + 5 \).
Apply the vertical shift: To shift the graph down by 5 units, subtract 5 from the entire equation. This results in \( y = \frac{1}{2}x + 5 - 5 \), which simplifies to \( y = \frac{1}{2}x \).
Sketch the graphs: Draw the original graph of \( y = \frac{1}{2}(x + 1) + 5 \) and the shifted graph \( y = \frac{1}{2}x \) on the same set of axes. Label each graph with its corresponding equation to clearly show the transformation.
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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 or vertically without altering its shape. Horizontal shifts are achieved by adding or subtracting a constant from the x-variable, while vertical shifts involve adding or subtracting a constant from the entire function. Understanding how these shifts affect the graph's position is crucial for accurately sketching the transformed graph.
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Graph of Sine and Cosine Function
Linear Equations
Linear equations represent straight lines and are typically expressed in the form y = mx + b, where m is the slope and b is the y-intercept. In the context of graph shifting, recognizing the components of a linear equation helps in understanding how changes to the equation affect the graph's position and orientation on the coordinate plane.
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07:17Linearization
Coordinate Plane
The coordinate plane is a two-dimensional surface defined by a horizontal axis (x-axis) and a vertical axis (y-axis). It is used to plot points, lines, and curves based on their x and y values. Familiarity with the coordinate plane is essential for graphing equations and understanding how shifts in the graph correspond to changes in the equation's parameters.
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7:42Properties of Parabolas
Related Practice
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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Textbook Question
[Technology Exercise]
You want to make an 80° angle by marking an arc on the perimeter of a 12-in.-diameter disk and drawing lines from the ends of the arc to the disk’s center. To the nearest tenth of an inch, how long should the arc be?
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 = x³/8
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Textbook Question
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = x² + 1
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