Problem 58 - Precalculus Mathematics for Calculus

Question

Make a table of values and sketch the graph of the equation. Find the x- and y-intercepts and test for symmetry. $$y=3 x+3$$

Step-by-Step Solution

Answer

The x-intercept is (-1,0) and the y-intercept is (0,3); the graph is not symmetric.

Step 1: Create a Table of Values

To begin, we need to calculate some values for our graph. Choose several values for \( x \), then calculate the corresponding \( y \) using the equation \( y = 3x + 3 \). Common choices for \( x \) are -2, -1, 0, 1, and 2.| x | y ||---|---|| -2 | 3(-2)+3 = -3 || -1 | 3(-1)+3 = 0 || 0 | 3(0)+3 = 3 || 1 | 3(1)+3 = 6 || 2 | 3(2)+3 = 9 |

Step 2: Sketch the Graph

Now, using the table of values, plot these points on a coordinate plane. Connect the dots to form a straight line. This line represents the graph of the equation \( y = 3x + 3 \).

Step 3: Find the x-intercept

The x-intercept occurs where \( y = 0 \). Set \( y = 0 \) and solve for \( x \):\[ 0 = 3x + 3 \]\[ 3x = -3 \]\[ x = -1 \]Thus, the x-intercept is \( (-1, 0) \).

Step 4: Find the y-intercept

The y-intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into the equation:\[ y = 3(0) + 3 \]\[ y = 3 \]Thus, the y-intercept is \( (0, 3) \).

Step 5: Test for Symmetry

We will check for symmetry about the x-axis, y-axis, and origin.- For x-axis symmetry: Replace \( y \) with \( -y \). Our equation becomes: \( -y = 3x + 3 \), which is not equivalent to the original equation. - For y-axis symmetry: Replace \( x \) with \( -x \). Our equation becomes: \( y = 3(-x) + 3 = -3x + 3 \), which is not equivalent to the original equation. - For origin symmetry: Replace \( y \) with \( -y \) and \( x \) with \( -x \). Our equation becomes: \( -y = -3x + 3 \), which is not equivalent to the original equation.Therefore, the graph is not symmetric about the x-axis, y-axis, or origin.

Key Concepts

Understanding the X-Intercept

The x-intercept of a graph is the point where the line crosses the x-axis. At this point, the value of y is zero. To find the x-intercept for the equation \( y = 3x + 3 \), we set \( y \) to 0 and solve for \( x \).
Here's how it works:

• Set \( y = 0 \) in the equation: \( 0 = 3x + 3 \).
• Subtract 3 from both sides to get: \( 3x = -3 \).
• Divide by 3 to solve for \( x \): \( x = -1 \).

This gives us the x-intercept as the point \((-1, 0)\). It means if you look on the graph along the x-axis, the line will pass through \( x = -1 \).
By understanding this concept, you can visualize how the graph interacts with the axes, offering clues about its behavior.

Grasping the Y-Intercept

The y-intercept of a graph is the point where the line crosses the y-axis. Here, \( x = 0 \). For the equation \( y = 3x + 3 \), finding the y-intercept is straightforward because you only need to substitute \( x = 0 \) into the equation.
Here's the process:

• Substitute \( x = 0 \) into the equation: \( y = 3(0) + 3 \).
• Simplify to find \( y = 3 \).

This means the y-intercept is at the point \((0, 3)\). From a graphing perspective, this is the location where the line will touch or cross the y-axis.
Being familiar with y-intercepts helps you quickly place one of the fundamental points of the graph, making graphing easier and reducing errors.

Exploring Symmetry

Symmetry in graphing refers to how certain visual aspects of a graph mirror each other. To test for symmetry, we assess how the graph behaves in relation to the x-axis, the y-axis, and the origin.
Here's how:

• X-axis Symmetry: Replace \( y \) with \( -y \). This gives us \( -y = 3x + 3 \). Since this is not the original equation, there's no symmetry about the x-axis.
• Y-axis Symmetry: Replace \( x \) with \( -x \). The equation becomes \( y = -3x + 3 \), which differs from the original, indicating no y-axis symmetry.
• Origin Symmetry: Swap both \( x \) with \( -x \) and \( y \) with \( -y \). The equation transforms to \( -y = -3x + 3 \), which does not match the original equation, thus no symmetry around the origin exists.

Understanding these symmetry checks is crucial, as symmetrical graphs can simplify plotting and provide insight into the structure and patterns of the graph.

Creating a Table of Values

A table of values is a helpful tool for plotting points on a graph effectively. This table pairs selected \( x \) values with their corresponding \( y \) values from the equation \( y = 3x + 3 \).
Follow these steps to see how it works:

• Choose several values for \( x \), for instance, -2, -1, 0, 1, 2.
• Use the equation to find the corresponding \( y \) for each \( x \). For example, when \( x = -2 \), \( y = 3(-2) + 3 = -3 \).
• Continue this for the selected \( x \) values.

The resulting pairs \((-2, -3), (-1, 0), (0, 3), (1, 6), (2, 9)\) help you easily plot the graph by marking these points and drawing a line through them.
Creating a table of values gives you a tangible way to understand the relationship between \( x \) and \( y \), and helps visualize the function as a line on the graph.