Problem 36 - College Algebra

Question

For each of the following exercises, construct a table and graph the equation by plotting at least three points. $$y=-3 x+1$$

Step-by-Step Solution

Answer

Plot points (-1,4), (0,1), and (1,-2) and connect them with a line to graph y=-3x+1.

Step 1: Select Values for x

Choose at least three different values for \(x\). For simplicity, let's use \(x = -1\), \(x = 0\), and \(x = 1\).

Step 2: Calculate Corresponding y-values

Use the equation \(y = -3x + 1\) to determine the value of \(y\) for each chosen \(x\). - If \(x = -1\), then \(y = -3(-1) + 1 = 3 + 1 = 4\). - If \(x = 0\), then \(y = -3(0) + 1 = 0 + 1 = 1\). - If \(x = 1\), then \(y = -3(1) + 1 = -3 + 1 = -2\).

Step 3: Construct a Table of Values

Compile the \(x\) and \(y\) pairs into a table:\[\begin{array}{|c|c|}\hlinex & y \\hline-1 & 4 \0 & 1 \1 & -2 \\hline\end{array}\]

Step 4: Plot the Points on a Graph

On a coordinate plane, plot the points \((-1, 4)\), \((0, 1)\), and \((1, -2)\).

Step 5: Draw the Line

Using a ruler, draw a straight line through the plotted points. This line represents the graph of the equation \(y = -3x + 1\).

Key Concepts

Graphing

Graphing a linear equation is a fundamental skill in mathematics that helps us visualize the relationship between two variables. To graph an equation like \( y = -3x + 1 \), you begin by selecting a few values for \( x \) and then calculate the corresponding \( y \) values using the equation.

Here's a simple way to get started:

• Choose at least three different values for \( x \).
• Substitute these \( x \) values into the equation to find the corresponding \( y \) values.
• These \( (x, y) \) pairs are your points to plot on the graph.

Once the points are plotted, use a ruler to draw a straight line through them. This line visually represents the solution to the linear equation.

Remember, linear equations result in straight lines, and every point on this line is a solution to the equation. Each time you graph, you're basically sketching the line that satisfies the equation.

Coordinate Plane

The coordinate plane is an essential tool for graphing equations. It consists of two number lines that intersect at a right angle. The horizontal line is called the \( x \)-axis, and the vertical line is called the \( y \)-axis. These axes divide the plane into four sections or quadrants.

Here's what you should know:

• The point where the \( x \)-axis and \( y \)-axis intersect is called the origin, labeled as \( (0, 0) \).
• Each point on the coordinate plane is identified by an ordered pair \( (x, y) \), which tells you its position relative to the origin.
• The first number in the pair is the \( x \)-coordinate, which tells you how far left or right the point is.
• The second number is the \( y \)-coordinate, which indicates how far up or down the point is.

When plotting points for graphing, always start from the origin and move horizontally according to the \( x \)-coordinate, then vertically according to the \( y \)-coordinate.

Slope-Intercept Form

The slope-intercept form is a method of expressing linear equations. It's given by the formula \( y = mx + b \), where:

• \( m \) represents the slope of the line.
• \( b \) is the \( y \)-intercept, which is the point where the line crosses the \( y \)-axis.

Understanding the slope is crucial, as it defines the steepness and direction of the line. If \( m \) is positive, the line rises as it moves from left to right. If \( m \) is negative, the line falls. In our example, the equation \( y = -3x + 1 \) tells us:

• The slope \( m = -3 \) indicates the line decreases 3 units vertically for every 1 unit it moves horizontally.
• The \( y \)-intercept \( b = 1 \) means the line crosses the \( y \)-axis at \( y = 1 \).

Using the slope-intercept form makes it easy to quickly graph these equations, especially when combined with a few calculated points as described earlier.