Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 1 - Functions

Problem 83

Chapter 1, Problem 83

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.
$x^{\frac{2}{3}} + y^{\frac{2}{3}} = 1$

Verified step by step guidance

Step 1: Understand the types of symmetry. A graph is symmetric about the x-axis if replacing y with -y yields an equivalent equation. It is symmetric about the y-axis if replacing x with -x yields an equivalent equation. It is symmetric about the origin if replacing both x with -x and y with -y yields an equivalent equation.

Step 2: Check for symmetry about the x-axis. Replace y with -y in the equation: $x^{\frac{2}{3}} + (-y)^{\frac{2}{3}} = 1$. Since $(-y)^{\frac{2}{3}} = y^{\frac{2}{3}}$, the equation remains the same, indicating symmetry about the x-axis.

Step 3: Check for symmetry about the y-axis. Replace x with -x in the equation: $(-x)^{\frac{2}{3}} + y^{\frac{2}{3}} = 1$. Since $(-x)^{\frac{2}{3}} = x^{\frac{2}{3}}$, the equation remains the same, indicating symmetry about the y-axis.

Step 4: Check for symmetry about the origin. Replace both x with -x and y with -y in the equation: $(-x)^{\frac{2}{3}} + (-y)^{\frac{2}{3}} = 1$. Since both terms remain unchanged, the equation is symmetric about the origin.

Step 5: Verify by graphing. Graph the equation $x^{\frac{2}{3}} + y^{\frac{2}{3}} = 1$ to visually confirm the symmetries about the x-axis, y-axis, and the origin.

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Key Concepts

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

Symmetry in Graphs

Symmetry in graphs refers to the property where a graph remains unchanged under certain transformations. A graph is symmetric about the x-axis if replacing y with -y yields the same equation, symmetric about the y-axis if replacing x with -x does, and symmetric about the origin if replacing both x and y with their negatives results in the same equation. Understanding these transformations is crucial for analyzing the symmetry of functions.

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize the relationship between variables. This process helps in identifying key features of the function, such as intercepts, slopes, and symmetry. By graphing the given equation, one can visually confirm the symmetry properties and better understand the behavior of the function across different quadrants.

Implicit Functions

Implicit functions are defined by equations that relate variables without explicitly solving for one variable in terms of another. The equation x^(2/3) + y^(2/3) = 1 is an example of an implicit function, where both x and y are intertwined. Analyzing such equations often requires techniques like implicit differentiation or algebraic manipulation to explore their properties, including symmetry.

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Textbook Question

Composition of polynomials

Let ƒ be an nth-degree polynomial and let g be an mth-degree polynomial.

What is the degree of the following polynomials?

ƒ ⋅ f

Textbook Question

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.

$ƒ (x) = x ∣ x ∣$

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Textbook Question

Prove the following identities.

$\frac{\sin\theta}{1+\cos\theta}\)=\(\frac{1-\cos\theta}{\sin\theta}$

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Textbook Question

Parabola properties Consider the general quadratic function ƒ(x) = ax² + bx + c , with a ≠ 0.

a. Find the coordinates of the vertex of the graph of the parabola y= ƒ(x) in terms of a, b, and c.

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Textbook Question

A culture of bacteria has a population of $150$ cells when it is first observed. The population doubles every $12~$\text{hr}$$ , which means its population is governed by the function $p$\left$(t$\right$)=150$\cdot{2^{\frac{t}{12}$}}$ , where $t$ is the number of hours after the first observation.

How long does it take the population to reach $10,000$ ?

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Textbook Question

Composition of polynomials

Let ƒ be an nth-degree polynomial and let g be an mth-degree polynomial.

What is the degree of the following polynomials?

ƒ o g

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Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.x23+y23=1x^{\(\frac\)23}+y^{\(\frac\)23}=1

Verified video answer for a similar problem:

Key Concepts

Symmetry in Graphs

Graphing Functions

Implicit Functions

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.
$x^{\frac{2}{3}} + y^{\frac{2}{3}} = 1$