Problem 46 Find any intercepts of the graph... [FREE SOLUTION]

91影视

Precalculus with Calculus Previews

Dennis G. Zill, Jacqueline M. Dewar

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 1: Problem 46

Find any intercepts of the graph of the given equation. Determine whether the graph of the equation possesses symmetry with respect to the $x$ -axis, $y$ -axis, or origin. Do not graph. $y=x^{3}$

Short Answer

Expert verified

The x-intercept and y-intercept are both (0, 0). The graph is symmetric with respect to the origin.

Step by step solution

Find the x-intercepts

To find the x-intercepts, set y to 0 and solve for x. For the equation $y=x^3$, set $0=x^3$. The only solution is $x=0$, thus the x-intercept is the point $(0, 0)$.

Find the y-intercepts

To find the y-intercepts, set x to 0 and solve for y. Substitute $x=0$ into the equation: $y=0^3=0$. Thus, the y-intercept is also the point $(0, 0)$.

Check for symmetry with respect to the x-axis

For symmetry with respect to the x-axis, replacing $y$ with $-y$ should result in an equivalent equation. Replacing $y$ with $-y$ gives $-y = x^3$, which is not equivalent to $y = x^3$. Therefore, the equation has no symmetry with respect to the x-axis.

Check for symmetry with respect to the y-axis

For symmetry with respect to the y-axis, replacing $x$ with $-x$ should result in an equivalent equation. Substituting $-x$ results in the equation $y = (-x)^3 = -x^3$. This is not equivalent to $y = x^3$, hence the equation does not have symmetry with respect to the y-axis.

Check for symmetry with respect to the origin

For symmetry with respect to the origin, replacing $x$ with $-x$ and $y$ with $-y$ should result in an equivalent equation. Replace and verify: $-y = (-x)^3 = -x^3$, which simplifies to $y = x^3$ when both sides are multiplied by -1. This is equivalent to the original equation, showing that the graph is 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.

Intercepts

Intercepts are the points where a graph crosses the axes. Finding these points is crucial because they provide valuable information about the graph of an equation.

For the equation $ y = x^3 $:

**x-intercept:** To discover the x-intercepts, we set $ y = 0 $ and solve for $ x $. Equation results are $ 0 = x^3 $, giving us $ x = 0 $. Thus, the point $ (0, 0) $ is the x-intercept.
**y-intercept:** Similarly, to find the y-intercepts, we set $ x = 0 $ and solve for $ y $. Substituting $ x = 0 $ yields $ y = 0^3 = 0 $, resulting in the point $ (0, 0) $ as the y-intercept.

Both intercepts coincide at the origin, which is common when the function graph starts at $ (0, 0) $. Knowing the intercepts helps establish where the graph begins along each axis.

Graphing Equations

Graphing equations is like visual storytelling. It helps us see how changes in values affect the equation visually.

When graphing a function like $ y = x^3 $, each point on the graph represents an input-output pair for $ x $ and $ y $. Utilizing intercepts gives us a clear starting point. However, determining symmetry can provide more insight. Symmetry simplifies the graphing process by establishing visible patterns:

**Symmetry with respect to the x-axis:** Not present if changes in $ y $ do not yield the same equation.
**Symmetry with respect to the y-axis:** Not present when substituting $ x $ with $ -x $ alters the equation.
**Symmetry with respect to the origin:** Present when the transformation $ -x $ to $ x $ and $-y$ to $ y $ bring us back to the initial equation.

Recognizing these symmetries can make graphing faster, as it helps to predict the curve's nature and orientation without plotting every single point.

Polynomial Functions

Polynomial functions are equations like $ y = x^3 $. They are expressed in terms of variables raised to whole-number exponents.

Understanding polynomial functions involves recognizing their degree, which tells us about their behavior:

The degree of $ x^3 $ is 3, indicating it's a cubic function.
**Degree** affects the graph's shape; odd degrees like 3 produce symmetric curves around the origin, unlike even degrees.
Such functions can cross the x-axis multiple times based on their real roots.

Polynomial functions often exhibit higher complexity as degrees increase, but understanding their general form (like $ ax^n + bx^{n-1} + \ldots $) helps in comprehending their broad patterns and behaviors. Mastery over polynomial functions leads to smoother graph interpretation and easier problem solving.

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Find any intercepts of the graph of the given equation. Determine whether the graph of the equation possesses symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Do not graph. \(y=x^{3}\)

Short Answer

Step by step solution

Find the x-intercepts

Find the y-intercepts

Check for symmetry with respect to the x-axis

Check for symmetry with respect to the y-axis

Check for symmetry with respect to the origin

Key Concepts

Intercepts

Graphing Equations

Polynomial Functions

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Discrete Mathematics

Statistics

Applied Mathematics

Mechanics Maths

Decision Maths

Study anywhere. Anytime. Across all devices.