Linear Functions In A Coordinate Planes

Unit: Functions

Chapter: Linear Functions in a Coordinated Plane

Reference: – What is a Linear Function, Slope (Rate of Change), y-intercept, Slope-Intercept Form (y = mx + b), Graphing Linear Functions, Horizontal and Vertical Lines, Standard Form (Ax + By = C), Point-Slope Form, Finding Slope from Two Points, Interpreting Slope and y-intercept in Real Life, Comparing Linear Functions, Solved Examples, Odd-One-Out Problems, Common Mistakes

After studying this chapter, you should be able to understand:

• What is a Linear Function and How to Identify It
• Slope as Rate of Change
• Graphing Linear Functions Using Slope and y-intercept
• Writing Equations in Slope-Intercept Form
• Understanding Standard Form and Point-Slope Form
• Interpreting Linear Functions in Real-World Contexts

Introduction to Linear Functions in a Coordinate Plane

Definition

A linear function is a function whose graph is a straight line. It can be written in the form f(x) = mx + b, where m is the slope (rate of change) and b is the y-intercept (where the line crosses the y-axis).

In a coordinate plane, every linear function represents a straight line. The line shows all the ordered pairs (x, y) that satisfy the equation.

When we study linear functions in a coordinate plane, we essentially ask:

"How does the line look on the graph? How steep is it? Where does it cross the axes?"

Once we understand these features, we can graph any linear function quickly and interpret what it means in real life.

Importance of Linear Functions

• Models constant rate situations (speed, cost per item, temperature change)
• Foundation for understanding more complex functions
• Used in economics (supply and demand), physics (motion), and business (profit)
• Helps make predictions based on patterns
• Essential for understanding data trends and scatter plots

Example

The function f(x) = 2x + 1 is linear.
Its graph is a straight line with slope 2 (goes up 2 units for every 1 unit right) and crosses the y-axis at (0, 1).
So, if we have f(x) = x², that is NOT linear (the graph is a curve).

Subtopics

1. Slope (Rate of Change)

The slope of a line measures how steep the line is. It tells us how much y changes when x increases by 1. Slope is often called the "rate of change."

Formula for Slope:
Given two points (x₁, y₁) and (x₂, y₂) on a line, the slope m is:

m = (y₂ – y₁) / (x₂ – x₁)

Ways to Think About Slope:

Slope = rise / run
Rise is the vertical change (how many units up or down)
Run is the horizontal change (how many units right)

Examples of Slope:

If m = 2, the line goes up 2 units for every 1 unit right (steep upward)

If m = 1/2, the line goes up 1 unit for every 2 units right (gentle upward)

If m = 0, the line is horizontal (flat, no rise)

If m = -3, the line goes down 3 units for every 1 unit right (steep downward)

If m = -1/4, the line goes down 1 unit for every 4 units right (gentle downward)

Special Slopes:

Positive slope (m > 0) – line rises from left to right

Negative slope (m < 0) – line falls from left to right

Zero slope (m = 0) – horizontal line

Undefined slope – vertical line (the run is zero, so we cannot divide by zero)

2. y-intercept

The y-intercept is the point where the line crosses the y-axis. At this point, x = 0. The y-intercept is written as (0, b), where b is the y-coordinate.

Example: In f(x) = 3x + 5, the y-intercept is 5, so the line crosses the y-axis at (0, 5).

Why the y-intercept matters: It tells us the starting value or initial amount in real-world problems. For example, if a taxi charge 2 flat fee, the y-intercept (2) is the flat fee.

3. Slope-Intercept Form (y = mx + b)

This is the most useful form for graphing linear functions quickly.

General Form: y = mx + b
m = slope
b = y-intercept

Example 1: y = 2x – 3
Slope m = 2, y-intercept b = -3 (crosses y-axis at (0, -3))

Example 2: y = -4x + 7
Slope m = -4, y-intercept b = 7

Example 3: y = x + 2 (here m = 1, b = 2)

Example 4: y = 5 (here m = 0, b = 5 – horizontal line)

How to Graph a Line Using Slope-Intercept Form:

Step 1: Plot the y-intercept (0, b) on the y-axis.

Step 2: Use the slope m = rise/run to find another point. From the y-intercept, go up (if positive rise) or down (if negative rise) and then right.

Step 3: Draw a straight line through the two points.

Example – Graph y = 2x + 1

Step 1: y-intercept is 1 → plot (0, 1)

Step 2: Slope is 2 = 2/1, so rise = 2, run = 1. From (0, 1), go up 2 and right 1 → reach (1, 3)

Step 3: Draw the line through (0, 1) and (1, 3)

Example – Graph y = -1/2 x + 3

Step 1: y-intercept is 3 → plot (0, 3)

Step 2: Slope is -1/2 = -1/2, so rise = -1, run = 2. From (0, 3), go down 1 and right 2 → reach (2, 2)

Step 3: Draw the line through (0, 3) and (2, 2)

4. Horizontal and Vertical Lines

Horizontal Lines:
Equation: y = c (where c is a constant)
Slope m = 0
y-intercept = c
Graph is a flat line crossing the y-axis at (0, c)

Example: y = 4 is a horizontal line through (0, 4). Every point on this line has y = 4.

Vertical Lines:
Equation: x = c (where c is a constant)
Slope is undefined
No y-intercept (unless c = 0, then it's the y-axis itself)
Graph is a straight up-and-down line crossing the x-axis at (c, 0)

Example: x = -2 is a vertical line through (-2, 0). Every point on this line has x = -2.

Important: A vertical line is NOT a function because one input (x = c) gives infinitely many outputs. But we still study it in the coordinate plane.

5. Standard Form (Ax + By = C)

Another way to write linear equations is standard form: Ax + By = C, where A, B, and C are integers, and A is usually positive.

Example: 3x + 2y = 6 is in standard form.

To graph from standard form:
Method 1 – Convert to slope-intercept form by solving for y.
Method 2 – Find the x-intercept (set y = 0) and y-intercept (set x = 0), then plot.

Example – Graph 2x + y = 4

Method 1: Solve for y: y = -2x + 4 → slope = -2, y-intercept = 4

Method 2: Find intercepts –
x-intercept: set y=0 → 2x = 4 → x = 2 → (2, 0)
y-intercept: set x=0 → y = 4 → (0, 4)
Plot (2,0) and (0,4) and draw the line.

6. Point-Slope Form

When you know the slope and one point on the line, use point-slope form: y – y₁ = m(x – x₁)

Example: A line has slope 3 and passes through (2, 5).
Equation: y – 5 = 3(x – 2)
Simplify: y – 5 = 3x – 6 → y = 3x – 1

When to use point-slope form: When you are given a slope and a point, or two points (find slope first, then use one point).

7. Finding Slope from Two Points

Given two points (x₁, y₁) and (x₂, y₂), use the slope formula: m = (y₂ – y₁) / (x₂ – x₁)

Example 1: Find slope between (2, 3) and (5, 9)
m = (9 – 3) / (5 – 2) = 6 / 3 = 2

Example 2: Find slope between (4, 7) and (4, 10)
m = (10 – 7) / (4 – 4) = 3 / 0 = undefined (vertical line)

Example 3: Find slope between (2, 5) and (6, 5)
m = (5 – 5) / (6 – 2) = 0 / 4 = 0 (horizontal line)

8. Interpreting Slope and y-intercept in Real Life

In real-world situations, the slope represents the rate of change, and the y-intercept represents the starting value.

Example 1 – Car Rental:
A car rental company charges 50 one-time fee.
Equation: C = 30d + 50, where C is total cost and d is number of days.
Slope = 30 (cost increases by $30 per day)
y-intercept = 50 (the initial fee before any days)

Example 2 – Water Tank:
A water tank has 200 gallons and is draining at 5 gallons per minute.
Equation: W = -5t + 200, where W is water in gallons and t is time in minutes.
Slope = -5 (water decreases by 5 gallons each minute)
y-intercept = 200 (starting amount of water)

Example 3 – Earnings:
A worker earns 20 bonus.
Equation: E = 15h + 20, where E is earnings and h is hours worked.
Slope = 15 (earns $15 per hour)
y-intercept = 20 (bonus earned even with zero hours)

Solved Examples

Example 1: Find the slope and y-intercept of y = -3x + 7.

Solution: The equation is in slope-intercept form y = mx + b. Here m = -3 and b = 7.

Answer: Slope = -3, y-intercept = 7

Example 2: Graph the function f(x) = (1/2) – 2.

Solution:
y-intercept is -2 → plot (0, -2)
Slope is 1/2, so rise = 1, run = 2. From (0, -2), go up 1 and right 2 → reach (2, -1)
Draw the line through (0, -2) and (2, -1)

Answer: [Graph description – line rising gently from left to right crossing y-axis at -2]

Example 3: Find the slope between the points (-3, 2) and (4, -5).

Solution:
m = (y₂ – y₁) / (x₂ – x₁) = (-5 – 2) / (4 – (-3)) = (-7) / (7) = -1

Answer: Slope = -1

Example 4: Write the equation of a line with slope 4 that passes through (1, 3).

Solution: Use point-slope form: y – y₁ = m(x – x₁)
y – 3 = 4(x – 1)
y – 3 = 4x – 4
y = 4x – 1

Answer: y = 4x – 1

Example 5: Graph the equation 2x – y = 3.

Solution: Convert to slope-intercept form:
y = 2x – 3
y-intercept = -3 → plot (0, -3)
Slope = 2 = 2/1 → from (0, -3), go up 2 and right 1 → reach (1, -1)
Draw the line through (0, -3) and (1, -1)

Answer: [Graph description – line rising steeply crossing y-axis at -3]

Example 6 – Odd One Out (Slope Values):

Examine the five slopes described below. Exactly one is different from the others in a significant way. Identify it.

Solution:

Slope = 0 → horizontal line (A)
Slope = -2 → negative, steep downward (B)
Slope = 5 → positive, steep upward (C)
Slope = 1/3 → positive, gentle upward (D)
Slope = undefined → vertical line (E)

Three reasons why E is the odd one out:

(A) Slope undefined means the line is vertical, which is NOT a function. All other slopes (0, -2, 5, 1/3) represent functions.
(B) A vertical line cannot be written in slope-intercept form y = mx + b. All others can.
(C) The run for a vertical line is zero, making the slope formula undefined due to division by zero. All other slopes have non-zero run.

Conclusion: E (slope undefined) is the odd one out.

Example 7 – Odd One Out (Linear vs non-linear):

Examine the five equations below. Exactly one is NOT a linear function. Identify it.

Solution:

P: y = 2x – 5 → linear (form y = mx + b)
Q: y = -x + 3 → linear (m = -1, b = 3)
R: y = 4 → linear (horizontal line, m = 0, b = 4)
S: y = x² + 1 → NOT linear (x is squared, graph is a parabola)
T: y = (1/2)x → linear (m = 1/2, b = 0)

Three reasons why S is the odd one out:

(A) S has x raised to the power 2, making it a quadratic function, not linear. All others have x to the first power only.
(B) The graph of S is a parabola (U-shaped curve), while the graphs of P, Q, R, T are straight lines.
(C) Linear functions have a constant rate of change (slope). S has a changing rate of change.

Conclusion: S (y = x² + 1) is the odd one out.

Example 8 – Odd One Out (Graph Interpretation):

Examine the five-line descriptions below. Exactly one represents a line that is NOT a function. Identify it.

Solution:

A: Horizontal line through (0,3) → equation y = 3 → function ✓

B: Slope 0 → horizontal line → function ✓

C: Vertical line through (-2,0) → equation x = -2 → NOT a function (fails vertical line test) ✗

D: y = 4x → linear function ✓

E: Points (1,2) and (3,2) → horizontal line y = 2 → function ✓

Three reasons why C is the odd one out:

(A) A vertical line fails the vertical line test (a vertical line drawn at x = -2 touches the line at every point). All others pass the vertical line test.
(B) C cannot be written as y = mx + b because the slope is undefined. All others can.
(C) In C, one input (x = -2) produces infinitely many outputs (any y-value), violating the definition of a function. All others have one output for each input.

Conclusion: C (vertical line through (-2, 0)) is the odd one out.

Common Mistakes to Avoid

Mistake 1 – Switching rise and run in slope
Slope = rise/run, not run/rise.
Correct understanding: rise is vertical change (y), run is horizontal change (x).

Mistake 2 – Forgetting the negative sign in slope
A negative slope means the line goes DOWN as you move right.
Correct understanding: Negative rise (go down) with positive run gives negative slope.

Mistake 3 – Thinking y = 3 is not a function
A horizontal line is a function because each x has exactly one y (y = 3 for all x).
Correct understanding: Only vertical lines are NOT functions.

Mistake 4 – Confusing y-intercept with x-intercept
y-intercept is where x = 0; x-intercept is where y = 0.
Correct understanding: y-intercept is (0, b); x-intercept is (a, 0).

Mistake 5 – Plotting slope incorrectly from y-intercept
If slope is 2, from the y-intercept go up 2 AND right 1. Do NOT go up 2 and right 0.
Correct understanding: Slope 2 = 2/1, so run must be 1.

Mistake 6 – Believing all straight lines are functions
Vertical lines are straight but are NOT functions.
Correct understanding: A line is a function if it passes the vertical line test.

Quick Reference Summary

Linear Function: f(x) = mx + b (graph is a straight line)

Slope (m): m = (y₂ – y₁) / (x₂ – x₁) = rise/run

Positive slope: Line rises from left to right

Negative slope: Line falls from left to right

Zero slope: Horizontal line (y = b)

Undefined slope: Vertical line (x = c) – NOT a function

y-intercept (b): Point where line crosses y-axis (0, b)

Slope-Intercept Form: y = mx + b

Standard Form: Ax + By = C

Point-Slope Form: y – y₁ = m(x – x₁)

To Graph a Line: Plot y-intercept, use slope to find second point, draw line