Graphing Linear Functions with Slope -1/2 and Y-Intercept -3

When working with linear functions, it's often useful to graph them and understand their behavior. In this article, we'll explore how to derive and graph a linear equation for a function with a specific slope and y-intercept.

Deriving the Linear Equation

The general form of a linear equation is given by:

y mx b

where m represents the slope of the line and b represents the y-intercept.

Given Values

Equation in Slope-Intercept Form

Substituting the given values into the general form:

y -1/2x - 3

This is the equation of the graph of the linear function in slope-intercept form.

Graphing the Linear Equation

To graph the linear equation, we can either use the slope-intercept form or plot points using the general form of the equation. Let's use the slope-intercept form to graph it.

Using the Slope-Intercept Form

Starting with the slope-intercept form:

y -1/2x - 3

Step 1: Find the Y-Intercept

The y-intercept is given by the value of y when x is 0. In this case, the y-intercept is -3. This means the line passes through the point (0, -3).

Step 2: Use the Slope to Find Another Point

The slope of the line is -1/2, which can be interpreted as a negative change in y for a positive change in x. If we move 2 units to the right (positive change in x), we should move 1 unit down (negative change in y) to find another point on the line.

Starting from (0, -3), moving 2 units to the right and 1 unit down gives us the point (2, -4).

Step 3: Plot the Points and Draw the Line

Plot the points (0, -3) and (2, -4) on a coordinate plane. Draw a straight line through these points to represent the graph of the linear function.

Alternative Method: Using the General Form

The general form of the linear equation is:

y mx b

Substituting the given values, we get:

y -1/2x - 3

To find another point, we can choose any value for x and solve for y.

For example, let x 4:

y -1/2(4) - 3 -2 - 3 -5

This gives us the point (4, -5).

Let x 2:

y -1/2(2) - 3 -1 - 3 -4

This gives us the point (2, -4).

P plot these points (0, -3), (4, -5), and (2, -4) on a coordinate plane and draw a straight line through them.

Key Insights

Understanding the relationship between the slope and y-intercept is crucial for graphing linear functions. By using the slope-intercept form, we can easily plot the y-intercept and use the slope to find another point, making it straightforward to graph the line.

The slope determines the direction and steepness of the line, while the y-intercept gives the starting point. By combining these elements, we can accurately represent the function on a coordinate plane.

Conclusion

Graphing a linear function with a slope of -1/2 and a y-intercept of -3 is a fundamental skill in algebra. By understanding the slope-intercept form and using it to find key points, we can confidently graph and analyze linear functions.