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Chapter 1: Problem 41

In Problems \(41-66\), 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. \(y=-3 x\)

Short Answer

Expert verified
The intercept is at (0,0), and the graph is symmetric with respect to the origin.

Step by step solution

01

Finding the x-intercept

An x-intercept is a point where the graph intersects the x-axis. At this point, the value of y is 0. To find the x-intercept, set y = 0 and solve the equation.\[0 = -3x\]Solving for x gives:\[x = 0\]So, the x-intercept is at the point (0,0).
02

Finding the y-intercept

A y-intercept is a point where the graph intersects the y-axis. At this point, the value of x is 0. To find the y-intercept, set x = 0 and use the equation.\[y=-3(0)\]This simplifies to:\[y = 0\]So, the y-intercept is at the point (0,0).
03

Testing for symmetry with respect to the x-axis

For a graph to be symmetric with respect to the x-axis, replacing y with -y in the equation should yield an equivalent equation. Start with the equation:\[y = -3x\]Replacing y with -y gives:\[-y = -3x\]Multiplying the whole equation by -1 results in:\[y = 3x\]Since this is not equivalent to the original equation, the graph is not symmetric with respect to the x-axis.
04

Testing for symmetry with respect to the y-axis

For a graph to be symmetric with respect to the y-axis, replacing x with -x in the equation should yield an equivalent equation. Start with the equation:\[y = -3x\]Replacing x with -x gives:\[y = -3(-x)\]This simplifies to:\[y = 3x\]Since this is not equivalent to the original equation, the graph is not symmetric with respect to the y-axis.
05

Testing for symmetry with respect to the origin

For a graph to be symmetric with respect to the origin, replacing both x with -x and y with -y should yield an equivalent equation. Start with the equation:\[y = -3x\]Replace x with -x and y with -y:\[-y = -3(-x)\]This simplifies to:\[-y = 3x\]Multiplying the whole equation by -1 yields:\[y = -3x\]Since this results in the original equation, the graph is symmetric with respect to the origin.

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Key Concepts

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

Understanding the X-Intercept
The x-intercept of a graph is where it crosses the x-axis. At this point, the value of y is 0. Let's see how that works: if you set y to 0 in the equation, you're solving for the point where it hits the x-axis. For the equation \( y = -3x \), substituting y with 0 gives us:
  • \[ 0 = -3x \]
  • Solving this, we find that \( x = 0 \)
This means that the x-intercept is at the point (0, 0). It's important to understand that an x-intercept gives us useful information about where our graph touches the x-axis. This can often tell us more about the graph's direction or stability at that point.
Understanding x-intercepts can help in graph analysis and predict where other intercepts might be based on the graph's behavior.
Exploring the Y-Intercept
The y-intercept is the point where the graph crosses the y-axis. Here, the value of x is 0. By setting x to 0, we can find this precise location on the graph. Looking at the equation \( y = -3x \) and substituting x with 0:
  • \[ y = -3(0) \]
  • y simplifies to 0
Thus, the y-intercept is also at the point (0, 0). Similar to the x-intercept, the y-intercept provides insights into the graph's position relative to the axes.
This intercept is particularly useful for finding initial starting points when sketching a graph as it shows where the graph starts from the y-axis.
Investigating Graph Symmetry
Graph symmetry helps us understand balance in a graph's appearance. There are three main types of symmetry to consider: with respect to the x-axis, y-axis, and origin.
  • X-Axis Symmetry: Replacing y with -y in the original equation does not result in the same equation (\( y = 3x \)). Hence, there's no symmetry about the x-axis.
  • Y-Axis Symmetry: Replacing x with -x gives us \( y = 3x \), which also is not the same as the original. Thus, the graph is not symmetric about the y-axis.
  • Origin Symmetry: Replacing both x and y with their negatives, \(-y = 3x\), can be rearranged to return to the original equation, \( y = -3x \). This confirms symmetry about the origin.
This particular graph shows origin symmetry, meaning you can rotate it 180 degrees around the origin, and it looks unchanged.
Understanding symmetry can simplify graphing and better predict graph behavior around key axes changes.

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Most popular questions from this chapter

Find an equation for the upper half of the circle \(x^{2}+(y-3)^{2}=4\). Repeat for the right half of the circle.

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. \(y=(x-2)^{2}(x+2)^{2}\)

Explain why there are no points \(P(x, x)\) that are a distance \(\sqrt{10}\) from the point (3,-5)

Use factorization to simplify the given expression in part (a). Then, if instructed, find the indicated limit in part \((b)\). (a) \(\frac{x^{5}+2 x^{4}+x^{3}}{x^{4}-2 x^{2}+1}\) (b) \(\lim _{x \rightarrow-1} \frac{x^{5}+2 x^{4}+x^{3}}{x^{4}-2 x^{2}+1}\)

Use factorization to simplify the given expression in part (a). Then, if instructed, find the indicated limit in part \((b)\). (a) \(\frac{x^{4}-5 x^{3}+4 x-20}{x^{4}-5 x^{3}+x-5}\) (b) \(\lim _{x \rightarrow 5} \frac{x^{4}-5 x^{3}+4 x-20}{x^{4}-5 x^{3}+x-5}\)
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