Identifying the slope and y-intercept from a graph is an important skill in Algebra II. Linear functions show how one thing relates to another, and their graphs are straight lines. The main goal is to learn how to get useful information from these graphs.

What is a Linear Function?

When you see the graph of a linear function, it usually looks like a straight line. Linear functions are often written in a specific way, known as slope-intercept form:

$$y = mx + b$$

In this equation:

• $m$ is the slope of the line.
• $b$ is the y-intercept. This is where the line hits the y-axis.

Understanding the Slope

The slope ($m$) shows how steep the line is and which way it goes. You can figure out the slope by looking at two points on the line. It’s calculated by the rise over the run:

$$m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$$

To find the slope from the graph, follow these steps:

1. Pick two points on the line. They should be easy to read, ideally where the line crosses grid lines.

2. Count the rise. This is the change in the y-values of the two points. Move up if the line goes up, and down if it goes down.

3. Count the run. This is the change in the x-values of the two points. Move right if the line goes to the right, and left if it goes to the left.

4. Plug these numbers into the slope formula. If the rise is positive as you go from left to right, the slope is positive. If it’s negative, then the slope is negative.

Finding the y-Intercept

The y-intercept ($b$) is where the line crosses the y-axis. To find it:

1. Look for the point on the graph where the line crosses the y-axis. This happens when $x = 0$.

2. Read the y-coordinate at this point. This y-value is your y-intercept, $b$.

Example of Finding Slope and y-Intercept

Let’s say we have a line that goes through the points (2, 3) and (4, 7).

1. To calculate the slope:
   • Rise = 7 - 3 = 4
   • Run = 4 - 2 = 2
   • So, slope $m = \frac{4}{2} = 2$.
2. To find the y-intercept, suppose the line crosses the y-axis at (0, 1). Then $b = 1$.

This means the equation of the line in slope-intercept form is:

$$y = 2x + 1$$

How to Graph the Line

Once you know the slope and y-intercept, you can draw the line:

1. Start at the y-intercept. Put a point on the y-axis where $b$ is.

2. Use the slope to find another point. From the y-intercept, use the rise over run. For a slope of 2 (which is like moving up 2 and right 1), go up 2 units and then right 1 unit.

3. Draw the line. Connect the points with a straight line that goes on in both directions.

Tips for Reading the Graph

• Make sure the axes (the x and y lines) are labeled correctly and the scales are clear.
• Look for different types of slopes—positive, negative, zero, and undefined—to understand what they mean.

Remember, practice makes perfect! The more you work with graphs, the easier it will get to understand these ideas. Solving different problems will help you get better at slope and y-intercept, making you confident in Algebra II. Knowing these parts is not just helpful for graphs, but it can also help you in everyday situations and other math classes.