Linear Functions - Grade 8

1. What is a Linear Function?

Definition: A linear function is a function whose graph is a straight line. It has a constant rate of change.

General Form:

$ f(x) = mx + b $

where $ m $ = slope (rate of change) and $ b $ = y-intercept

Key Characteristics:

Constant rate of change: The slope m is the same everywhere
Straight line graph: No curves or bends
First-degree: The variable x has an exponent of 1
One-to-one mapping: Each input has exactly one output

Examples:

$ f(x) = 2x + 3 $ → Linear (m = 2, b = 3)
$ f(x) = -\frac{1}{2}x + 5 $ → Linear (m = -1/2, b = 5)
$ f(x) = 4 $ → Linear (m = 0, b = 4, horizontal line)
$ f(x) = x^2 + 1 $ → NOT linear (has x²)

2. Evaluate a Linear Function

Definition: To evaluate a function means to find the output value f(x) for a given input value x.

Steps to Evaluate f(x):

Start with the function equation
Substitute the given x-value into the equation
Simplify using order of operations
Write the answer as f(x) or the specific point (x, f(x))

Examples:

Example 1: If $ f(x) = 3x - 5 $, find f(4).

Step 1: $ f(x) = 3x - 5 $

Step 2: $ f(4) = 3(4) - 5 $

Step 3: $ f(4) = 12 - 5 = 7 $

Answer: f(4) = 7 or point (4, 7)

Example 2: If $ f(x) = -2x + 8 $, find f(0) and f(-3).

f(0): $ f(0) = -2(0) + 8 = 0 + 8 = 8 $

f(-3): $ f(-3) = -2(-3) + 8 = 6 + 8 = 14 $

Example 3: If $ f(x) = \frac{1}{2}x + 6 $, find f(10).

$ f(10) = \frac{1}{2}(10) + 6 = 5 + 6 = 11 $

3. Complete a Table for a Linear Function

Steps:

Identify the function equation
For each x-value in the table, substitute into the function
Calculate the corresponding y-value (or f(x))
Fill in all missing values

Example:

Complete the table for $ f(x) = 2x - 3 $:

x	f(x)	Calculation
0	-3	$ 2(0) - 3 = -3 $
1	-1	$ 2(1) - 3 = -1 $
2	1	$ 2(2) - 3 = 1 $
3	3	$ 2(3) - 3 = 3 $

4. Complete a Table and Graph a Linear Function

Steps:

Complete the table by evaluating the function for each x-value
Plot the points from the table as ordered pairs (x, f(x))
Draw a straight line through all the points
Extend the line in both directions with arrows

Tips for Graphing:

Use at least 3 points to ensure accuracy
Choose x-values that are easy to work with (like 0, 1, 2)
All points should lie on the same straight line
Label your axes and scale appropriately

Example:

For $ f(x) = x + 2 $, complete table and graph:

Points: (0, 2), (1, 3), (2, 4), (3, 5)

Plot these points and draw a line through them

5. Interpret Points on the Graph of a Linear Function

Key Concept: Every point (x, y) on the graph represents an input-output pair where y = f(x).

What Points Tell Us:

Point (a, b): When input is a, output is b
Y-intercept (0, b): Initial value when x = 0
X-intercept (a, 0): Where the function equals zero

Real-World Interpretation:

Example: A function $ C(h) = 50h + 100 $ represents the cost (in dollars) of renting equipment for h hours.

Point (0, 100): There's a $100 initial fee (even for 0 hours)
Point (2, 200): Renting for 2 hours costs $200
Point (5, 350): Renting for 5 hours costs $350
Slope = 50: Each additional hour costs $50

6. Rate of Change of a Linear Function: Tables

Definition: Rate of change is how much the output (y) changes for each unit change in input (x). For linear functions, this is the slope.

Formula:

$ \text{Rate of Change} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} $

Steps from a Table:

Choose any two rows from the table
Find the change in y-values: $ y_2 - y_1 $
Find the change in x-values: $ x_2 - x_1 $
Divide: Rate of change = $ \frac{\Delta y}{\Delta x} $

Example:

x	y
2	11
4	17
6	23

Using rows 1 and 2: $ \frac{17 - 11}{4 - 2} = \frac{6}{2} = 3 $

Using rows 2 and 3: $ \frac{23 - 17}{6 - 4} = \frac{6}{2} = 3 $

Rate of change = 3 (constant for linear functions)

7. Rate of Change of a Linear Function: Graphs

From a Graph:

The rate of change is the slope of the line. Use rise over run.

Steps:

Choose two clear points on the line
Count the vertical change (rise) between the points
Count the horizontal change (run) between the points
Calculate: $ \text{Rate of Change} = \frac{\text{rise}}{\text{run}} $

Interpreting Rate of Change:

Positive: Function is increasing (line rises from left to right)
Negative: Function is decreasing (line falls from left to right)
Zero: Function is constant (horizontal line)
Larger value: Steeper line, faster rate of change

Example:

A line passes through (1, 3) and (5, 11). Find the rate of change.

Rise: $ 11 - 3 = 8 $

Run: $ 5 - 1 = 4 $

Rate of change: $ \frac{8}{4} = 2 $

8. Interpret the Slope and Y-Intercept of a Linear Function

In Context:

Slope (m): The rate of change; how much y changes per unit of x

Units: (units of y) per (units of x)
Example: dollars per hour, miles per gallon, inches per year

Y-Intercept (b): The initial value or starting point; value of y when x = 0

Units: Same as y-axis
Example: starting amount, initial fee, beginning value

Examples:

Example 1: $ C(h) = 25h + 50 $ represents the cost C (in dollars) to rent a bike for h hours.

Slope = 25: The rental costs $25 per hour

Y-intercept = 50: There's a $50 initial rental fee

Example 2: $ d(t) = 60t + 120 $ represents distance d (in miles) after t hours of driving.

Slope = 60: The car travels 60 miles per hour (speed)

Y-intercept = 120: The starting position was 120 miles from home

Example 3: $ T(d) = -2d + 50 $ represents temperature T (°F) at depth d (feet) underground.

Slope = -2: Temperature decreases by 2°F per foot of depth

Y-intercept = 50: Surface temperature is 50°F

9. Write a Linear Function from a Table

Steps:

Find the slope (m): Use any two points: $ m = \frac{y_2 - y_1}{x_2 - x_1} $
Find the y-intercept (b):
- If x = 0 is in the table, b is the corresponding y-value
- Otherwise, substitute a point and m into y = mx + b and solve for b
Write the function: $ f(x) = mx + b $

Example:

x	y
1	7
3	13
5	19

Step 1: Find slope using (1, 7) and (3, 13):

$ m = \frac{13 - 7}{3 - 1} = \frac{6}{2} = 3 $

Step 2: Find b using point (1, 7) and m = 3:

$ 7 = 3(1) + b $ → $ 7 = 3 + b $ → $ b = 4 $

Step 3: Write function:

$ f(x) = 3x + 4 $

10. Compare Linear Functions: Tables, Graphs, and Equations

Key Concept: Compare linear functions by comparing their slopes (rate of change) and y-intercepts.

What to Compare:

Slope: Which function grows/decreases faster?
Y-intercept: Which function starts higher/lower?
Specific values: Evaluate both at the same x-value

Steps:

Find the slope and y-intercept for each function
Compare the slopes (larger slope = steeper line)
Compare the y-intercepts (starting values)
Make conclusions based on the context

Example:

Function A (equation): $ f(x) = 4x + 5 $

Slope = 4, Y-intercept = 5

Function B (graph): Line passing through (0, 3) and (2, 9)

Slope = $ \frac{9-3}{2-0} = \frac{6}{2} = 3 $, Y-intercept = 3

Comparison:

Function A has a greater slope (4 > 3) → grows faster
Function A has a higher y-intercept (5 > 3) → starts higher
At x = 0: A = 5, B = 3 → A is greater
At x = 10: A = 45, B = 33 → A is greater

11. Write Linear Functions: Word Problems

Steps:

Identify the independent variable (input, usually x)
Identify the dependent variable (output, usually y or f(x))
Find the rate of change (slope m) from the problem
Find the initial value (y-intercept b)
Write the function: $ f(x) = mx + b $

Examples:

Example 1: A gym charges a $30 membership fee plus $5 per visit. Write a function for the total cost C based on the number of visits v.

Rate of change: $5 per visit → m = 5

Initial value: $30 membership fee → b = 30

Function: $ C(v) = 5v + 30 $

Example 2: A candle is 20 cm tall and burns at a rate of 2 cm per hour. Write a function for the candle's height h after t hours.

Rate of change: -2 cm per hour (decreasing) → m = -2

Initial height: 20 cm → b = 20

Function: $ h(t) = -2t + 20 $

12. Evaluate a Linear Function: Word Problems

Steps:

Identify or write the function equation
Determine what value to substitute (given in problem)
Substitute the value into the function
Calculate the result
Interpret the answer in context with units

Examples:

Example 1: A phone plan costs $40 per month plus $0.10 per text. The function is $ C(t) = 0.10t + 40 $. Find the cost for 200 texts.

$ C(200) = 0.10(200) + 40 $

$ C(200) = 20 + 40 = 60 $

Answer: The cost for 200 texts is $60.

Example 2: A water tank has 500 gallons and drains at 15 gallons per minute. Function: $ W(t) = -15t + 500 $. How much water remains after 12 minutes?

$ W(12) = -15(12) + 500 $

$ W(12) = -180 + 500 = 320 $

Answer: After 12 minutes, 320 gallons remain.