/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 20 Determine whether the graph of t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 20

Determine whether the graph of the equation is symmetric with respect to the \(x\) -axis, \(y\) -axis, origin, or none of these. $$ y=x^{2}+6 x+1 $$

Short Answer

Expert verified
The graph is symmetric with respect to none of these.

Step by step solution

01

Identify the Equation

The given equation is \( y = x^2 + 6x + 1 \).
02

Test for Symmetry with Respect to the x-axis

To test for symmetry with respect to the x-axis, replace \( y \) with \( -y \) in the equation and see if the resulting equation is equivalent to the original equation. Substitute \( -y \) for \( y \): en\( -y = x^2 + 6x + 1 \)Since this equation is not equivalent to the original, the graph is not symmetric with respect to the x-axis.
03

Test for Symmetry with Respect to the y-axis

To test for symmetry with respect to the y-axis, replace \( x \) with \( -x \) in the equation and see if the resulting equation is equivalent to the original equation. Substitute \( -x \) for \( x \): \( y = (-x)^2 + 6(-x) + 1 \)Simplify: \( y = x^2 - 6x + 1 \)Since this equation is not equivalent to the original, the graph is not symmetric with respect to the y-axis.
04

Test for Symmetry with Respect to the Origin

To test for symmetry with respect to the origin, replace \( x \) with \( -x \) and \( y \) with \( -y \) in the equation and see if the resulting equation is equivalent to the original equation. Substitute \( -x \) for \( x \) and \( -y \) for \( y \): \( -y = (-x)^2 + 6(-x) + 1 \)Simplify: \( -y = x^2 - 6x + 1 \)Since this equation is not equivalent to the original, the graph is not symmetric with respect to the origin.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

x-axis Symmetry Test
Graphs displaying symmetry about the x-axis appear mirrored above and below the x-axis. To determine if an equation's graph is symmetric with respect to the x-axis, replace every instance of y with -y in the equation.
Let's take the given equation, which is
\(y = x^2 + 6x + 1\)
Replace \(y\) with \(-y\)
\(-y = x^2 + 6x + 1\)
We need the modified equation to match the original equation for it to exhibit x-axis symmetry. Since
\(-y = x^2 + 6x + 1 eq y = x^2 + 6x + 1\)
This equation does not mirror itself about the x-axis, hence it lacks x-axis symmetry.
y-axis Symmetry Test
For a graph to show symmetry about the y-axis, it should look the same on the left and right side of the y-axis. To test for y-axis symmetry in an equation, replace every x with -x and check if the resulting equation is the same as the original.
Taking our equation
\(y = x^2 + 6x + 1\)
replace x with -x:
\(y = (-x)^2 + 6(-x) + 1\)
Simplify this to get:
\(y = x^2 - 6x + 1\)
Since
\(y = x^2 - 6x + 1 eq y = x^2 + 6x + 1\)
The equation does not reflect over the y-axis either, so it is not symmetric with respect to the y-axis.
Origin Symmetry Test
Graph symmetry with respect to the origin implies that rotating the graph 180 degrees around the origin produces an indistinguishable graph. To confirm this, replace both x with -x and y with -y in the given equation. If the equation remains unchanged, symmetry with respect to the origin is present.
Using \(y = x^2 + 6x + 1\)
substitute x with -x and y with -y:
\(-y = (-x)^2 + 6(-x) + 1\)
Simplify:
\(-y = x^2 - 6x + 1\)
We now compare the transformed equation to the original:
\(-y = x^2 - 6x + 1 eq y = x^2 + 6x + 1\)
Since they are not identical, the graph of the given equation does not have origin symmetry.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

An artist makes jewelry from polished stones. The rent for her studio, Internet service, and phone come to \(\$ 640\) per month. She also estimates that it costs \(\$ 3.50\) in supplies to make one necklace. At art shows and online, she sells the necklaces for \(\$ 25\) each. a. Write a linear cost function that represents the \(\operatorname{cost} C(x)\) to produce \(x\) necklaces during a one-month period. b. Write a linear revenue function to represent the revenue \(R(x)\) for selling \(x\) necklaces. c. Evaluate \((R-C)(x)\) and interpret its meaning in the context of this problem. d. Determine the profit if the artist sells 212 necklaces during a one-month period.

Refer to the functions \(r, p,\) and \(q .\) Evaluate the function and write the domain in interval notation. \(r(x)=-3 x \quad p(x)=x^{2}+3 x \quad q(x)=\sqrt{1-x}\) $$(p \cdot q)(x)$$

Suppose that the average rate of change of a continuous function between any two points to the left of \(x=a\) is positive, and the average rate of change of the function between any two points to the right of \(x=a\) is negative. Does the function have a relative minimum or maximum at \(a\) ?

The amount of \(\mathrm{CO}_{2}\) emitted per year \(A(x)\) (in tons) for a vehicle that burns \(x\) miles per gallon of gas, can be approximated by \(A(x)=0.0092 x^{2}-0.805 x+21.9 .\) (Source: U.S. Department of Energy, http://energy.gov) a. Determine the difference quotient. \(\frac{A(x+h)-A(x)}{h}\) b. Evaluate the difference quotient on the interval \([20,25],\) and interpret its meaning in the context of this problem. c. Evaluate the difference quotient on the interval \([35,40],\) and interpret its meaning in the context of this problem.

Given \(k(x)=-3 x+1\) and \(m(x)=\frac{1}{x},\) a. Find \((k \circ m)(x)\). b. Find \((m \circ k)(x)\). c. Is \((k \circ m)(x)=(m \circ k)(x) ?\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.