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Chapter 1: Problem 46

In Exercises 45-56, identify any intercepts and test for symmetry. Then sketch the graph of the equation. \( y = 2x - 3 \)

Short Answer

Expert verified
The y-intercept is -3 and the x-intercept is 1.5. The graph does not show symmetry about the y-axis, x-axis or the origin.

Step by step solution

01

Finding the y-intercept

For finding the y-intercept, put x = 0 in the equation. The equation becomes \( y = 2(0) - 3 \) simplifying this gives \( y = -3 \). Thus, the y-intercept is -3.
02

Finding the x-intercept

For finding the x-intercept, put y = 0 in the equation. The equation becomes \( 0 = 2x - 3 \). Solving this for x gives \( x = 3/2 \). So, the x intercept is 3/2 or 1.5.
03

Testing for symmetry

To test for symmetry with respect to the y-axis, replace each x by -x in the equation and simplify. Upon doing this, we get \( y = 2(-x) - 3 = -2x - 3 \), which is not the same as the original equation, hence the equation is not symmetric with respect to the y-axis. To test for symmetry with respect to the x-axis, replace y by -y in the equation and simplify. We get \( -y = 2x - 3 \), which is also not the same as the original equation, hence the equation is not symmetric with respect to the x-axis. To test for symmetry with respect to the origin, replace both y and x by their negatives, simplify and see if it matches the original equation. The equation becomes \( -y = 2(-x) - 3 = -2x - 3 \). This also is not the original equation, hence the graph is not 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.

Intercepts
In linear equations like \( y = 2x - 3 \), identifying intercepts is crucial to understanding where the line crosses the axes. Intercepts are where the graph meets the x-axis or y-axis.
The x-intercept occurs where the graph crosses the x-axis, meaning
  • The y-value at this point will be zero.
  • Solve the equation by setting \( y = 0 \) and solving for \( x \).
  • For our equation: \(0 = 2x - 3\), which results in \(x = \frac{3}{2}\).
This means the x-intercept of this line is at \( (\frac{3}{2}, 0) \).
The y-intercept, on the other hand, is where the graph crosses the y-axis. Here,
  • The x-value is zero.
  • Set \( x = 0 \) in the equation and solve for \( y \).
  • Substituting gives \( y = 2(0) - 3 \) leading to \( y = -3 \).
Thus, the y-intercept is \( (0, -3) \).
Understanding intercepts helps us graph linear equations quickly and efficiently.
Symmetry
Symmetry in mathematics describes a balanced and proportionate similarity. For our equation \( y = 2x - 3 \), we check for three types of symmetry:
  • Y-axis Symmetry: To test for this, replace \( x \) with \( -x \). If the equation remains the same, it is symmetric across the y-axis. Here, \[ y = 2(-x) - 3 = -2x - 3\] This doesn't match the original equation, hence the graph isn't symmetric across the y-axis.
  • X-axis Symmetry: Change \( y \) to \( -y \). If the altered equation equals the original, it's x-axis symmetric. For our equation, \[ -y = 2x - 3 \]This result doesn't match the original, indicating no x-axis symmetry.
  • Origin Symmetry: Substitute both \( x \) and \( y \) with \( -x \) and \( -y \) respectively. The equation becomes \[ -y = 2(-x) - 3 = -2x - 3\]Once again, this diverges from the original equation, suggesting no origin symmetry.
Understanding symmetry helps in visualizing how the graph may look without detailed calculations.
Graphing
Graphing a linear equation like \( y = 2x - 3 \) involves plotting the intercepts and drawing a straight line through them. Start with the intercepts,
  • Place point \( (\frac{3}{2}, 0) \) on the x-axis.
  • Mark point \( (0, -3) \) on the y-axis.
Once the points are plotted, draw a line through these intercepts across the axis markings. Linear equations yield straight lines, hence knowing these two points is enough.
The line represents all solutions to the equation \( y = 2x - 3 \) in the coordinate plane. Graphing helps in understanding the equation's visual representation:
  • The slope (2 in this case) indicates steepness.
  • A positive slope (2) means the line rises as it moves to the right.
Visualizing equations through graphical representations is a powerful tool for better comprehension.

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Most popular questions from this chapter

In Exercises 49-58, find a mathematical model for the verbal statement. \(F\) varies directly as \(g\) and inversely as \(r^2\).

The mathematical model \(y = \frac{k}{x}\) is an example of ________ variation.

TRUE OR FALSE? In Exercises 87 and 88, decide whether the statement is true or false. Justify your answer. In the equation for kinetic energy, \(E=\frac{1}{2}mv^2\) the amount of kinetic energy \(E\) is directly proportional to the mass \(m\) of an object and the square of its velocity \(v\).

DIRECT VARIATION In Exercises 35-38, assume that is \(y\) directly proportional to \(x\). Use the given \(x\)-value and \(y\)-value to find a linear model that relates \(y\) and \(x\). \(x = 6\), \(y = 580\)

CAPSTONE Describe and correct the error. Given \(f(x) = \sqrt{x-6}\), then \(f^{-1} (x) = \frac{1}{\sqrt{x-6}}\).
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