Lesson 1: Linear Functions
Estimated time: 30-40 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Define and calculate slope as rate of change (rise over run)
- Identify and interpret y-intercepts and x-intercepts
- Write linear equations in slope-intercept form (y = mx + b)
- Write linear equations in point-slope form (y - y₁ = m(x - x₁))
- Write linear equations in standard form (Ax + By = C)
- Convert between different forms of linear equations
- Recognize and interpret positive, negative, zero, and undefined slopes
What is a Linear Function?
A linear function is one of the most fundamental concepts in algebra. As the name suggests, when you graph a linear function, you get a perfectly straight line.
Linear Function: A function whose graph is a straight line. It has a constant rate of change and can be written in the form f(x) = mx + b, where m and b are constants.
Linear functions appear everywhere in real life: converting temperatures, calculating costs, determining distance traveled at constant speed, and much more. Understanding linear functions is essential for success in algebra and beyond.
Key Characteristic
The defining feature of a linear function is that it has a constant rate of change. For every unit increase in x, y changes by the same amount. This constant rate of change is called the slope.
Understanding Slope
The slope of a line measures its steepness and direction. Think of it as "how much the line rises (or falls) for every step to the right."
Slope (m): The ratio of vertical change (rise) to horizontal change (run) between any two points on a line.
m = (y₂ - y₁)/(x₂ - x₁) = rise/run
Types of Slope
Positive Slope
m > 0
Line rises from left to right
Example: y = 2x + 1 (slope = 2)
As x increases, y increases
Negative Slope
m < 0
Line falls from left to right
Example: y = -3x + 5 (slope = -3)
As x increases, y decreases
Zero Slope
m = 0
Horizontal line (flat)
Example: y = 4 (slope = 0)
y stays constant as x changes
Undefined Slope
m is undefined
Vertical line
Example: x = 3
Division by zero (run = 0)
Example 1: Calculating Slope
Find the slope of the line passing through points (2, 3) and (5, 9).
Solution:
Step 1: Identify the coordinates
(x₁, y₁) = (2, 3) and (x₂, y₂) = (5, 9)
Step 2: Apply the slope formula
m = (y₂ - y₁)/(x₂ - x₁)
m = (9 - 3)/(5 - 2)
Step 3: Simplify
m = 6/3
m = 2
Answer: The slope is 2. This means for every 1 unit increase in x, y increases by 2 units. The line rises from left to right.
Example 2: Negative Slope
Find the slope of the line passing through points (-1, 4) and (3, -2).
Solution:
m = (y₂ - y₁)/(x₂ - x₁)
m = (-2 - 4)/(3 - (-1))
m = (-6)/(4)
m = -3/2 or -1.5
Answer: The slope is -3/2. The negative slope means the line falls from left to right. For every 2 units to the right, the line drops 3 units.
Example 3: Zero Slope
Find the slope of the line passing through points (1, 5) and (7, 5).
Solution:
m = (y₂ - y₁)/(x₂ - x₁)
m = (5 - 5)/(7 - 1)
m = 0/6
m = 0
Answer: The slope is 0. This is a horizontal line where y always equals 5, regardless of x.
Intercepts
Intercepts are the points where a line crosses the axes. They provide important information about a linear function.
Y-Intercept
Definition: The point where the line crosses the y-axis
x-coordinate is always 0
Written as: (0, b)
To find: Set x = 0 and solve for y
X-Intercept
Definition: The point where the line crosses the x-axis
y-coordinate is always 0
Written as: (a, 0)
To find: Set y = 0 and solve for x
Example 4: Finding Intercepts
Find both intercepts of the line 2x + 3y = 12.
Finding the y-intercept (set x = 0):
2(0) + 3y = 12
3y = 12
y = 4
Y-intercept: (0, 4)
Finding the x-intercept (set y = 0):
2x + 3(0) = 12
2x = 12
x = 6
X-intercept: (6, 0)
Answer: Y-intercept is (0, 4) and x-intercept is (6, 0).
Slope-Intercept Form
The slope-intercept form is the most common and user-friendly way to write linear equations. It immediately shows you the slope and y-intercept.
Slope-Intercept Form
where:
m = slope (rate of change)
b = y-intercept (where line crosses y-axis)
Example 5: Identifying Slope and Y-Intercept
For the equation y = 4x - 7, identify the slope and y-intercept.
Solution:
This equation is already in slope-intercept form: y = mx + b
Comparing y = 4x - 7 to y = mx + b:
m = 4 (the coefficient of x)
b = -7 (the constant term)
Answer:
Slope = 4 (line rises; for every 1 unit right, go up 4 units)
Y-intercept = -7 (line crosses y-axis at point (0, -7))
Example 6: Writing Equation in Slope-Intercept Form
Write an equation for the line with slope -2 and y-intercept 5.
Solution:
We know: m = -2 and b = 5
Substitute into y = mx + b:
y = -2x + 5
Answer: y = -2x + 5
Example 7: Equation from Two Points
Write the equation of the line passing through points (1, 3) and (4, 12).
Solution:
Step 1: Find the slope
m = (12 - 3)/(4 - 1) = 9/3 = 3
Step 2: Use one point to find b
Using point (1, 3): 3 = 3(1) + b
3 = 3 + b
b = 0
Step 3: Write the equation
y = 3x + 0
y = 3x
Answer: y = 3x
Point-Slope Form
The point-slope form is especially useful when you know the slope and one point on the line (but don't know the y-intercept yet).
Point-Slope Form
where:
m = slope
(x₁, y₁) = a specific point on the line
Example 8: Using Point-Slope Form
Write the equation of a line with slope 5 passing through point (2, -3).
Solution:
We know: m = 5 and (x₁, y₁) = (2, -3)
Substitute into y - y₁ = m(x - x₁):
y - (-3) = 5(x - 2)
y + 3 = 5(x - 2)
Point-slope form: y + 3 = 5(x - 2)
If we want slope-intercept form, distribute and solve for y:
y + 3 = 5x - 10
y = 5x - 13
Slope-intercept form: y = 5x - 13
Example 9: Point-Slope with Two Points
Write the equation of the line through points (-1, 4) and (3, -2).
Solution:
Step 1: Find the slope
m = (-2 - 4)/(3 - (-1)) = -6/4 = -3/2
Step 2: Choose either point (let's use (-1, 4))
y - 4 = -3/2(x - (-1))
y - 4 = -3/2(x + 1)
Point-slope form: y - 4 = -3/2(x + 1)
Converting to slope-intercept form:
y - 4 = -3/2·x - 3/2
y = -3/2·x - 3/2 + 4
y = -3/2·x + 5/2
Slope-intercept form: y = -3/2·x + 5/2
Standard Form
The standard form is another way to write linear equations. It's useful for quickly finding both intercepts and is preferred in some applications.
Standard Form
where:
A, B, C are integers (whole numbers)
A should be positive (if possible)
A, B, C have no common factors other than 1
Example 10: Converting to Standard Form
Convert y = 3x - 7 to standard form.
Solution:
Step 1: Move the x-term to the left side
y = 3x - 7
-3x + y = -7
Step 2: Multiply by -1 to make A positive
3x - y = 7
Answer: 3x - y = 7
Here A = 3, B = -1, C = 7
Example 11: Standard Form to Slope-Intercept
Convert 4x + 2y = 10 to slope-intercept form.
Solution:
Solve for y:
4x + 2y = 10
2y = -4x + 10
y = -2x + 5
Answer: y = -2x + 5
Now we can see: slope = -2, y-intercept = 5
Example 12: Using Standard Form to Find Intercepts
Find both intercepts of 5x + 2y = 20.
Finding x-intercept (set y = 0):
5x + 2(0) = 20
5x = 20
x = 4
X-intercept: (4, 0)
Finding y-intercept (set x = 0):
5(0) + 2y = 20
2y = 20
y = 10
Y-intercept: (0, 10)
Converting Between Forms
Summary: Three Forms of Linear Equations
1. Slope-Intercept Form: y = mx + b
Best for: Graphing, seeing slope and y-intercept at a glance
2. Point-Slope Form: y - y₁ = m(x - x₁)
Best for: When you know slope and a point
3. Standard Form: Ax + By = C
Best for: Finding intercepts quickly, integer coefficients
Example 13: Complete Conversion
Write the line through (2, 5) with slope -3 in all three forms.
Point-Slope Form:
y - 5 = -3(x - 2)
Slope-Intercept Form:
y - 5 = -3x + 6
y = -3x + 11
Standard Form:
3x + y = 11
Check Your Understanding
Try these questions to see if you've grasped the key concepts:
1. Find the slope of the line passing through points (3, 7) and (6, 16).
m = (y₂ - y₁)/(x₂ - x₁)
m = (16 - 7)/(6 - 3)
m = 9/3
m = 3
The slope is 3, meaning the line rises 3 units for every 1 unit to the right.
2. What are the slope and y-intercept of the line y = -5x + 8?
Slope (m) = -5 (the line falls; for every 1 unit right, go down 5 units)
Y-intercept (b) = 8 (the line crosses the y-axis at (0, 8))
3. Find both intercepts of the line 3x - 4y = 24.
3x - 4(0) = 24
3x = 24
x = 8
X-intercept: (8, 0)
Y-intercept (set x = 0):
3(0) - 4y = 24
-4y = 24
y = -6
Y-intercept: (0, -6)
4. Write the equation of a line with slope 4 and y-intercept -3.
m = 4 and b = -3
y = 4x - 3
5. Write the equation of a line passing through point (1, -2) with slope 6 in point-slope form. Then convert to slope-intercept form.
y - y₁ = m(x - x₁)
y - (-2) = 6(x - 1)
y + 2 = 6(x - 1)
Converting to slope-intercept form:
y + 2 = 6x - 6
y = 6x - 8
y = 6x - 8
6. Convert the equation y = 2x + 5 to standard form.
y = 2x + 5
-2x + y = 5 (move x-term to left)
Multiply by -1 to make A positive:
2x - y = -5
Key Takeaways
- Slope measures steepness: m = (y₂ - y₁)/(x₂ - x₁) = rise/run
- Positive slope rises left to right; negative slope falls left to right
- Zero slope is horizontal; undefined slope is vertical
- Y-intercept is where the line crosses the y-axis (x = 0)
- X-intercept is where the line crosses the x-axis (y = 0)
- Slope-intercept form: y = mx + b (easiest to graph and interpret)
- Point-slope form: y - y₁ = m(x - x₁) (useful with slope and one point)
- Standard form: Ax + By = C (good for finding intercepts)
- You can convert between all three forms algebraically
Ready for More?
Next Lesson
In Lesson 2, you'll learn how to apply linear functions to real-world modeling problems. You'll interpret slope and intercepts in context and solve practical problems involving rates, costs, and more!
Start Lesson 2