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Linear Function
Modeling

Learn to use linear functions to model real-world situations involving distance, cost, and constant rates of change.

Pre-Algebra

What Is a Linear Function?

A linear function is a relationship between two quantities where one changes at a constant rate compared to the other. Its graph is always a straight line.

Equation Form
\( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept
Constant Rate
Every time \( x \) increases by 1, \( y \) changes by exactly \( m \)

The Building Blocks: Slope and Y-Intercept

Slope \( m \)
The rate of change: how much \( y \) increases (or decreases) for each unit increase in \( x \). Calculated as \( m = \frac{\text{rise}}{\text{run}} \).
Y-Intercept \( b \)
The starting value: the value of \( y \) when \( x = 0 \). It is where the line crosses the y-axis.

Visualizing \( y = 2x + 3 \)

x y 0 1 2 3 4 5 0 1 2 3 4 5 6 (0, 3) (1, 5) (2, 7) +2 +1 \( b = 3 \)

The Core Formula

$$ y = mx + b $$
The slope-intercept form of a linear equation
\(y\)
Output (dependent variable): the quantity you want to find: total cost, distance traveled, amount earned
\(m\)
Slope (rate of change): how fast \( y \) changes per unit of \( x \): dollars per hour, miles per minute
\(x\)
Input (independent variable): the quantity you control or measure: hours, items, minutes
\(b\)
Y-intercept (starting value): the value of \( y \) when \( x = 0 \): base fee, initial position

Finding Slope from Two Points

$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
The slope formula: change in \( y \) divided by change in \( x \)
x y (1, 2) (4, 8) rise = 6 run = 3
\( m = \frac{6}{3} = 2 \)

Linear vs. Non-Linear

Linear Non-Linear
Graph Shape Straight line Curve (parabola, exponential, etc.)
Rate of Change Constant Changes
Equation Example \( y = 3x + 5 \) \( y = x^2 + 1 \)
Table Pattern Equal differences in \( y \) Unequal differences in \( y \)
Real-World Example Driving at constant speed Ball thrown in the air

How to Build a Linear Model

1
Identify the variables. What is the input \( x \) and what is the output \( y \)?
2
Find the rate of change \( m \). Look for words like "per", "each", or "every": that is your slope.
3
Find the starting value \( b \). What is the value when \( x = 0 \)? Look for a flat fee, initial amount, or starting point.
4
Write the equation as \( y = mx + b \) and use it to solve the problem.
5
Check your answer. Substitute a known pair back into the equation to verify it works.

Example 1: Taxi Cost

A taxi charges a $3 base fee plus $2 per mile. Write a linear function for the total cost \( C \) after \( d \) miles, and find the cost of a 7-mile ride.

1
Variables: \( d = \) miles driven, \( C = \) total cost in dollars
2
Slope: \( m = 2 \) (cost per mile)
3
Y-intercept: \( b = 3 \) (base fee)
4
Equation: \( C = 2d + 3 \)

Example 2: Distance Traveled

A hiker starts 4 miles from the trailhead and walks at a steady 3 miles per hour. Write a function for the hiker's distance \( d \) from the trailhead after \( t \) hours.

t (hours) d (miles) 0 1 2 3 4 0 2 4 6 8 10 (0, 4) (1, 7) (2, 10) \( b = 4 \)

Example 3: Savings Account

Maya has $50 in savings and deposits $15 each week. After how many weeks will she have at least $200?

1
Variables: \( w = \) weeks, \( S = \) total savings ($)
2
Equation: \( S = 15w + 50 \) (\( \text{slope} = 15 \), \( \text{y-intercept} = 50 \))
3
Set up inequality: \( 15w + 50 \geq 200 \)
4
Solve: \( 15w \geq 150 \), so \( w \geq 10 \)

Interpreting Slope in Context

The slope \( m \) always tells you the rate at which the output changes. Pay attention to units to interpret it correctly.

Positive Slope
\( m > 0 \): output increases as input increases. Example: earning $12/hour means each hour adds $12.
Negative Slope
\( m < 0 \): output decreases as input increases. Example: a candle burns 2 cm/hour, losing height over time.
Zero Slope
\( m = 0 \): output stays constant. A flat line: no change no matter the input.

Interpreting the Y-Intercept

The y-intercept \( b \) is the output value before any input happens. In context, it is the starting condition.

Cost Model
\( C = 2x + 3 \): the $3 is the base fee before any miles are driven.
Distance Model
\( d = 3t + 4 \): the 4 miles is the starting distance from the trailhead.
Savings Model
\( S = 15w + 50 \): the $50 is the initial amount already in the account.

Example: Draining a Pool

A pool has 1200 gallons of water. A pump removes 50 gallons per minute. Write a function for the water remaining \( W \) after \( t \) minutes.

t (min) W (gal) 0 6 12 18 24 0 300 600 900 1200 (0, 1200) (24, 0)
\( m = -50 \) gal/min

Common Mistakes to Avoid

✕ Confusing Rate (m) and Fee (b)
Confusing the rate and the fixed amount. For a $3/mile rate (\(m=3\)) and a $15 fee (\(b=15\)), the correct equation is \( y = 3x + 15 \), not \( y = 15x + 3 \).
✕ Forgetting Units
If \( x \) is in hours, the slope must be "per hour". Mismatched units (minutes vs. hours) give wrong answers.
✕ Ignoring Negative Slopes
When a quantity decreases, the slope is negative. Draining, cooling, spending: all use \( m < 0 \).

Summary: Linear Models at a Glance

Scenario Slope \( m \) Intercept \( b \) Equation
Taxi ride $2/mile $3 base fee \( C = 2x + 3 \)
Hiking 3 mi/hr 4 mi start \( d = 3t + 4 \)
Savings $15/week $50 initial \( S = 15w + 50 \)
Pool drain -50 gal/min 1200 gal \( W = -50t + 1200 \)

Question 1 of 5

A gym charges a $25 monthly membership fee plus $5 per class. Which equation models the total monthly cost \( C \) for attending \( n \) classes?

Question 2 of 5

A candle is 30 cm tall and burns at a rate of 2 cm per hour. What is the height \( h \) after \( t \) hours?

Question 3 of 5

The equation \( y = 8x + 20 \) models the cost of a phone plan. What does the 8 represent?

Question 4 of 5

A car rental costs $40/day plus a one-time $15 insurance fee. How much does a 5-day rental cost?

Question 5 of 5

A table shows: when \( x = 2 \), \( y = 11 \) and when \( x = 5 \), \( y = 23 \). What is the slope of the linear function?