Linear functions

Linear Functions

A linear function is a function that can be expressed in the form f(x) = mx + c, where:

• m is the slope (gradient) of the line.
• c is the y-intercept, the point where the line crosses the y-axis.

The graph of a linear function is a straight line. Understanding linear functions is essential as they form the foundation for more complex functions.

Identifying Linear Functions

To identify a linear function, check if it can be written in the form f(x) = mx + c. For example:

• f(x) = 2x + 3 is a linear function.
• f(x) = -x + 5 is also a linear function.
• f(x) = x^2 + 2 is not a linear function because it contains x^2.

Graphing Linear Functions

To graph a linear function, you need two key points. You can find these points by substituting values for x into the function to calculate the corresponding y values.

Example 1: Graphing f(x) = 2x + 1

1. Choose values for x. Let’s use x = -1 and x = 2.

2. Calculate the corresponding y values:

• For x = -1: f(-1) = 2(-1) + 1 = -2 + 1 = -1. So, the point is (-1, -1).
• For x = 2: f(2) = 2(2) + 1 = 4 + 1 = 5. So, the point is (2, 5).

3. Plot the points (-1, -1) and (2, 5) on a Cartesian plane.

4. Draw a straight line through the points. This line represents the function f(x) = 2x + 1.

The Slope of a Linear Function

The slope (m) of a linear function represents the rate of change of y with respect to x. It indicates how steep the line is. A positive slope means the line rises from left to right, while a negative slope means it falls from left to right.

Example 2: Finding the slope

For the function f(x) = 3x - 4, the slope is 3. This means that for every unit increase in x, y increases by 3 units.

Finding the Y-Intercept

The y-intercept (c) is found by setting x = 0 in the function. This gives you the point where the line crosses the y-axis.

Example 3: Finding the y-intercept

For the function f(x) = -2x + 6:

1. Set x = 0: f(0) = -2(0) + 6 = 6.
2. The y-intercept is (0, 6).

Writing Linear Functions from Two Points

Sometimes, you may need to write the equation of a linear function given two points. To do this, follow these steps:

1. Identify the two points, say A(x_1, y_1) and B(x_2, y_2).
2. Calculate the slope using m = (y_2 - y_1) / (x_2 - x_1).
3. Use one of the points and the slope to find the equation in the form y - y_1 = m(x - x_1).

Example 4: Finding the equation from two points

Given the points A(1, 2) and B(3, 4):

1. Calculate the slope: m = (4 - 2) / (3 - 1) = 2 / 2 = 1.
2. Using point A(1, 2), plug into the point-slope form: y - 2 = 1(x - 1).
3. Simplify: y - 2 = x - 1 → y = x + 1.
4. The equation of the line is f(x) = x + 1.

Parallel and Perpendicular Lines

Lines can be classified as parallel or perpendicular based on their slopes:

• Parallel lines have the same slope. For example, the lines f(x) = 2x + 1 and g(x) = 2x - 3 are parallel because both have a slope of 2.
• Perpendicular lines have slopes that are negative reciprocals of each other. For example, if one line has a slope of m, the other line will have a slope of -1/m.

Example 5: Finding perpendicular lines

If a line has a slope of 3, a line perpendicular to it will have a slope of -1/3.

Common Mistakes

Summary

• A linear function is in the form f(x) = mx + c.
• The slope (m) indicates how steep the line is.
• The y-intercept (c) is where the line crosses the y-axis.
• Two points can be used to find the equation of a linear function.
• Parallel lines have the same slope; perpendicular lines have slopes that are negative reciprocals.

Check your understanding

1. Identify if the function f(x) = 4x - 7 is linear.
2. Find the slope of the line passing through the points (2, 3) and (4, 7).
3. Write the equation of a line with a slope of -2 that passes through the point (1, 5).
4. Determine whether the lines f(x) = 3x + 2 and g(x) = -1/3x - 5 are parallel, perpendicular, or neither.