Problem 46 - Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.

Key Concepts

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.