Discover the general form of linear functions, understand slope and y-intercept, and learn how to graph linear functions through real-world examples.
Understanding the standard form and its components
y = mx + b
Slope-Intercept Form: y = mx + b
Most common form, shows slope and y-intercept directly
Point-Slope Form: y - y₁ = m(x - x₁)
Useful when you know a point and the slope
Standard Form: Ax + By = C
General form, useful for systems of equations
Let's explore linear functions using mobile phone plan pricing
Pricing: $20 base fee + $0.10 per minute
Monthly cost = Base fee + (Rate × Minutes used)
Linear Function: C(m) = 0.10m + 20
• Slope (m = 0.10): Cost increases by $0.10 per minute
• Y-intercept (b = 20): Base cost of $20 even with 0 minutes
• Domain: m ≥ 0 (can't use negative minutes)
Pricing: $40 base fee + $0.05 per minute
Higher base cost but lower per-minute rate
Linear Function: C(m) = 0.05m + 40
• Slope (m = 0.05): Cost increases by $0.05 per minute
• Y-intercept (b = 40): Base cost of $40
• Break-even point: When both plans cost the same
Deep dive into the meaning and calculation of these key components
Definition: Rate of change, steepness of the line
How much y changes for every 1-unit increase in x
Formula: m = (y₂ - y₁)/(x₂ - x₁)
Rise over run, or change in y divided by change in x
Types of Slope:
• Positive: Line rises from left to right
• Negative: Line falls from left to right
• Zero: Horizontal line (no change)
• Undefined: Vertical line (infinite steepness)
Definition: Point where line crosses the y-axis
The value of y when x = 0
Finding Y-intercept:
Set x = 0 in the equation and solve for y
Real-world Meaning:
• Starting value: Initial amount before any changes
• Fixed cost: Base fee or setup cost
• Baseline: Reference point for the function
Step-by-step process for creating accurate linear function graphs
Plot the point (0, b) on the y-axis
From the y-intercept, move according to the slope (rise/run)
Connect the points with a straight line extending in both directions
Step 1: Y-intercept = 3
Plot point (0, 3)
Step 2: Slope = 2 = 2/1
From (0, 3), go up 2, right 1 to get (1, 5)
Step 3: Draw the line
Connect (0, 3) and (1, 5) with a straight line
Explore more complex aspects of linear functions
Parallel Lines: Same slope, different y-intercepts
Example: y = 2x + 3 and y = 2x - 1 are parallel
Perpendicular Lines: Slopes are negative reciprocals
Example: y = 2x + 1 and y = -½x + 3 are perpendicular
Rule: If slope₁ × slope₂ = -1, lines are perpendicular
2 × (-½) = -1 ✓
Formula: y - y₁ = m(x - x₁)
Where (x₁, y₁) is a point on the line and m is the slope
Example: Line through (2, 3) with slope 4
y - 3 = 4(x - 2) → y - 3 = 4x - 8 → y = 4x - 5
How changes in the equation affect the graph
Up: y = 2x + 3 (moves up 3 units)
Adding to b shifts the line up
Down: y = 2x - 2 (moves down 2 units)
Subtracting from b shifts the line down
Steeper: y = 4x + 1 (slope = 4)
Larger slope = steeper line
Flatter: y = 0.5x + 1 (slope = 0.5)
Smaller slope = flatter line
Positive: y = 2x + 1 (slope = 2)
Line rises from left to right
Negative: y = -2x + 1 (slope = -2)
Line falls from left to right
Apply your knowledge of linear function properties
Given: f(x) = -3x + 7
Identify the slope and y-intercept. What does the negative slope tell us?
Solution:
• Slope (m): -3 (line falls from left to right)
• Y-intercept (b): 7 (line crosses y-axis at (0, 7))
• Meaning: y decreases by 3 for every 1-unit increase in x
Given: Plan A: C(m) = 0.10m + 20, Plan B: C(m) = 0.05m + 40
At how many minutes do both plans cost the same?
Solution:
Set equations equal: 0.10m + 20 = 0.05m + 40
0.10m - 0.05m = 40 - 20
0.05m = 20
m = 400 minutes
Answer: Both plans cost $60 at 400 minutes