Problem 28 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(27-38,\) graph each linear equation using the slope and y-intercept $$y=3 x+1$$

Step-by-Step Solution

Answer

The graph of the equation \(y=3x+1\) starts with a y-intercept at \(1\) and has a slope which is \(3\), which means the line increases to the right, taking one step in the x-direction and three in the y-direction.

Step 1: Identify the components of the linear equation

Given the equation \(y=3x+1\), it can be seen that this is in the standard format of linear equations \(y = mx + b\). Here, \(m\) represents the slope of the graph and \(b\) is where the line crosses the y-axis, which is the y-intercept. For this equation \(m = 3\) and \(b = 1\).

Step 2: Mark the y-intercept

Start by putting a point at the y-intercept, which is at \(1\) on the y-axis.

Step 3: Use the slope to plot the graph

The slope \(m\) is \(3\) which can be read as \(3/1\) meaning for each step to the right on the x-axis, correspondingly take three steps upwards on the y-axis. From the y-intercept point, count one unit to the right and from there count three units up, and mark that point. The line running through these two points is the graph of the equation.

Key Concepts

Slope and y-intercept

Understanding the slope and y-intercept is crucial when graphing a linear equation. The slope is a number that describes both the direction and the steepness of the line. It is often represented by the letter 'm' in the equation. If you come across a slope of 3, like in the equation \(y=3x+1\), it means that for every one unit you move to the right along the x-axis, you should move up by three units along the y-axis. A positive slope indicates the line is going upwards as you move to the right, while a negative slope means it goes downwards.

The y-intercept is where the line crosses the y-axis. It's designated by the letter 'b' in the standard linear equation format \(y=mx+b\). Here, the y-intercept is 1, which signifies that the line will cross the y-axis one unit above the origin. Remembering that the y-intercept is a point where the line will definitely pass through makes plotting easier, giving you a starting point on your graph.

Linear Equation Format

The format of a linear equation usually takes the form \(y = mx + b\), where 'm' is the slope and 'b' is the y-intercept. This is known as the slope-intercept form, and it is extremely useful for graphing because it gives you direct information about the line without needing additional calculation. In the context of our problem, the linear equation given is \(y=3x+1\). This tells us directly that the slope (m) is 3 and the y-intercept (b) is 1. The slope-intercept form is easy to work with because once those two values are identified, the line can be quickly sketched by starting at the y-intercept and following the slope.

Plotting Points

When you're graphing linear equations, plotting points accurately is vital. Start by putting a point at the y-intercept. For the equation \(y=3x+1\), place a point at (0,1) since the y-intercept is 1. Then, use the slope to determine the next point. With a slope of 3, you can think of it as the ratio \(\frac{3}{1}\), which translates to a rise of 3 units up for every 1 unit you move to the right.

From the y-intercept, move right to (1,1) and then up 3 units to (1,4), placing your next point there. You can continue this pattern to plot more points if needed. After plotting at least two points, draw a straight line through them, extending it across the graph. This line represents your linear equation. Always double-check your points to ensure they align correctly with the slope and y-intercept you calculated.