Graph of the Linear Function

Oct 13, 2025 by thesweetdesigns.com 114 views

Step 1: Identify the Slope

The first step in graphing a linear function is to identify the slope. The slope is a measure of how steep the line is and can be calculated using the equation:

Slope = (change in y) / (change in x)

In the given equation, $y - 2 = -\frac{2}{3}(x + 1)$ , we can see that the slope is $-\frac{2}{3}$ . This means that for every 3 units we move to the right, the line will move down by 2 units.

Step 2: Complete the Table

To graph the linear function, we need to create a table with x and y values. We can start by choosing some x values and then calculating the corresponding y values using the equation.

x	y
-1	2
0	0
1	-2
2	-4
3	-6

Discussion

Now that we have completed the table, we can see that the line passes through the points (-1, 2), (0, 0), (1, -2), (2, -4), and (3, -6). We can use these points to graph the line.

Step 3: Plot the Points

To graph the line, we need to plot the points we have calculated. We can start by plotting the point (-1, 2) and then draw a line through the other points.

Step 4: Draw the Line

Once we have plotted the points, we can draw a line through them. The line should be straight and pass through all the points we have calculated.

Step 5: Add the y-Intercept

The y-intercept is the point where the line intersects the y-axis. In this case, the y-intercept is (0, 2). We can add this point to the graph and draw a line through it.

Step 6: Add the x-Intercept

The x-intercept is the point where the line intersects the x-axis. In this case, the x-intercept is (-1, 0). We can add this point to the graph and draw a line through it.

Step 7: Add the Slope

The slope is the measure of how steep the line is. In this case, the slope is $-\frac{2}{3}$ . We can add this to the graph and draw a line through it.

Step 8: Add the Equation

The equation of the line is $y - 2 = -\frac{2}{3}(x + 1)$ . We can add this to the graph and draw a line through it.

Conclusion

Graphing a linear function involves identifying the slope, completing a table, plotting the points, drawing the line, adding the y-intercept, adding the x-intercept, adding the slope, and adding the equation. By following these steps, we can graph the linear function whose equation is $y - 2 = -\frac{2}{3}(x + 1)$ .

Graph of the Linear Function

Here is the graph of the linear function:

  +---------------------------------------+
  |                                            |
  |  y = -2/3(x + 1) + 2                  |
  |                                            |
  +---------------------------------------+
  |  -1 | 2  |
  |  0  | 0  |
  |  1  | -2 |
  |  2  | -4 |
  |  3  | -6 |
  +---------------------------------------+

Key Takeaways

The slope of the line is $-\frac{2}{3}$ .
The y-intercept is (0, 2).
The x-intercept is (-1, 0).
The equation of the line is $y - 2 = -\frac{2}{3}(x + 1)$ .

Real-World Applications

Graphing linear functions has many real-world applications, including:

Modeling population growth
Predicting stock prices
Calculating interest rates
Determining the cost of goods

Conclusion

$Graphing linear functions is an important skill in mathematics and has many real-world applications. By following the steps outlined in this article, we can graph the linear function whose equation is $y - 2 = -\frac{2}{3}(x + 1)$.<br/> **Graphing Linear Functions: A Q&A Guide**$

Q: What is a linear function?

A: A linear function is a function that can be written in the form y = mx + b, where m is the slope and b is the y-intercept.

Q: What is the slope of a linear function?

A: The slope of a linear function is a measure of how steep the line is. It can be calculated using the equation:

Slope = (change in y) / (change in x)

Q: How do I graph a linear function?

A: To graph a linear function, you need to follow these steps:

Identify the slope.
Complete a table with x and y values.
Plot the points on a coordinate plane.
Draw a line through the points.
Add the y-intercept and x-intercept.
Add the slope and equation.

Q: What is the y-intercept of a linear function?

A: The y-intercept of a linear function is the point where the line intersects the y-axis. It is the value of y when x is equal to 0.

Q: What is the x-intercept of a linear function?

A: The x-intercept of a linear function is the point where the line intersects the x-axis. It is the value of x when y is equal to 0.

Q: How do I find the equation of a linear function?

A: To find the equation of a linear function, you need to use the slope and y-intercept. The equation is in the form:

y = mx + b

where m is the slope and b is the y-intercept.

Q: What are some real-world applications of graphing linear functions?

A: Graphing linear functions has many real-world applications, including:

Modeling population growth
Predicting stock prices
Calculating interest rates
Determining the cost of goods

Q: How do I use graphing linear functions in real-world situations?

A: To use graphing linear functions in real-world situations, you need to:

Identify the problem and the variables involved.
Determine the slope and y-intercept of the linear function.
Graph the linear function using the slope and y-intercept.
Use the graph to make predictions or determine the cost of goods.

Q: What are some common mistakes to avoid when graphing linear functions?

A: Some common mistakes to avoid when graphing linear functions include:

Not identifying the slope and y-intercept correctly.
Not completing the table with x and y values.
Not plotting the points correctly on the coordinate plane.
Not drawing the line through the points correctly.

Q: How do I check my work when graphing linear functions?

A: To check your work when graphing linear functions, you need to:

Verify that the slope and y-intercept are correct.
Check that the points are plotted correctly on the coordinate plane.
Verify that the line is drawn through the points correctly.
Check that the equation of the linear function is correct.

Conclusion

Graphing linear functions is an important skill in mathematics and has many real-world applications. By following the steps outlined in this article and avoiding common mistakes, you can graph linear functions with confidence.