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
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 1.4.32
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.
Verified step by step guidance1
Identify the equation of the hyperbola: \( y^2 - 16x^2 = 1 \). This is a standard form of a hyperbola centered at the origin with a vertical transverse axis.
Rewrite the equation in the standard form for a hyperbola: \( \frac{y^2}{1} - \frac{x^2}{\frac{1}{16}} = 1 \). This shows that \( a^2 = 1 \) and \( b^2 = \frac{1}{16} \).
Determine the vertices and asymptotes: The vertices are at \( (0, \pm a) = (0, \pm 1) \). The equations of the asymptotes are \( y = \pm \frac{1}{4}x \).
Select a suitable viewing window for graphing: Choose a window that includes the vertices and shows the asymptotic behavior. A possible window could be \([-2, 2]\) for \(x\) and \([-2, 2]\) for \(y\).
Use graphing software to plot the upper branch of the hyperbola: Input the equation \( y = \sqrt{1 + 16x^2} \) to graph the upper branch, ensuring the graph is within the selected viewing window.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Hyperbola
A hyperbola is a type of conic section formed by the intersection of a plane with both nappes of a double cone. It consists of two disconnected curves called branches. The standard form of a hyperbola centered at the origin is x²/a² - y²/b² = 1 or y²/a² - x²/b² = 1, where a and b are real numbers that determine the shape and orientation of the hyperbola.
Graphing Functions
Graphing functions involves plotting points on a coordinate plane to visualize the behavior of a function. Key features to consider include intercepts, asymptotes, and the general shape of the graph. For a hyperbola, it's important to identify the vertices, foci, and asymptotes to accurately represent its branches.
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Graph of Sine and Cosine Function
Viewing Window
The viewing window in graphing software determines the portion of the coordinate plane visible on the screen. Selecting an appropriate viewing window is crucial for capturing the key features of a function, such as intercepts and asymptotes, and ensuring that the graph is neither too zoomed in nor too zoomed out, which can obscure important details.
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13:49Example 4: Norman Window
Related Practice
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
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
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
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