Linear Functions - Ninth Grade Math
Introduction to Linear Functions
Linear Function: A function whose graph is a straight line
Standard Form: $f(x) = mx + b$ or $y = mx + b$
Degree: 1 (highest power of $x$ is 1)
Key Property: Constant rate of change (slope)
Standard Form: $f(x) = mx + b$ or $y = mx + b$
Degree: 1 (highest power of $x$ is 1)
Key Property: Constant rate of change (slope)
General Form of Linear Function:
$$f(x) = mx + b$$
or
$$y = mx + b$$
where:
• $m$ = slope (rate of change)
• $b$ = y-intercept (starting value)
• $x$ = independent variable (input)
• $y$ or $f(x)$ = dependent variable (output)
$$f(x) = mx + b$$
or
$$y = mx + b$$
where:
• $m$ = slope (rate of change)
• $b$ = y-intercept (starting value)
• $x$ = independent variable (input)
• $y$ or $f(x)$ = dependent variable (output)
Characteristics of Linear Functions:
• Graph is a straight line
• Constant rate of change (slope)
• Degree is exactly 1
• No exponents higher than 1
• No variables in denominators
• No variables inside radicals
• Domain: All real numbers (unless restricted)
• Range: All real numbers (unless horizontal line)
• Graph is a straight line
• Constant rate of change (slope)
• Degree is exactly 1
• No exponents higher than 1
• No variables in denominators
• No variables inside radicals
• Domain: All real numbers (unless restricted)
• Range: All real numbers (unless horizontal line)
1-2. Identify Linear Functions
1. From Graphs and Equations
From Graphs:
A graph represents a linear function if:
• It forms a STRAIGHT line
• Passes the vertical line test (each x has one y)
• Has constant slope throughout
NOT Linear if:
• Curved (parabola, circle, etc.)
• Multiple pieces or breaks
• Vertical line
A graph represents a linear function if:
• It forms a STRAIGHT line
• Passes the vertical line test (each x has one y)
• Has constant slope throughout
NOT Linear if:
• Curved (parabola, circle, etc.)
• Multiple pieces or breaks
• Vertical line
From Equations:
An equation represents a linear function if it can be written as $y = mx + b$
Linear Examples:
• $y = 3x + 2$ ✓
• $y = -x + 5$ ✓
• $y = 7$ (horizontal line) ✓
• $2x + 3y = 6$ (can be solved for y) ✓
NOT Linear:
• $y = x^2 + 1$ ✗ (has $x^2$)
• $y = \frac{1}{x}$ ✗ (x in denominator)
• $y = \sqrt{x}$ ✗ (x under radical)
• $xy = 5$ ✗ (variables multiplied)
• $x = 3$ ✗ (vertical line, not a function)
An equation represents a linear function if it can be written as $y = mx + b$
Linear Examples:
• $y = 3x + 2$ ✓
• $y = -x + 5$ ✓
• $y = 7$ (horizontal line) ✓
• $2x + 3y = 6$ (can be solved for y) ✓
NOT Linear:
• $y = x^2 + 1$ ✗ (has $x^2$)
• $y = \frac{1}{x}$ ✗ (x in denominator)
• $y = \sqrt{x}$ ✗ (x under radical)
• $xy = 5$ ✗ (variables multiplied)
• $x = 3$ ✗ (vertical line, not a function)
Example 1: Is $y = 4x - 7$ linear?
Yes! It's in form $y = mx + b$ with $m = 4$, $b = -7$
Yes! It's in form $y = mx + b$ with $m = 4$, $b = -7$
Example 2: Is $3x + 2y = 12$ linear?
Solve for y: $2y = -3x + 12$ → $y = -\frac{3}{2}x + 6$
Yes! It's linear with slope $-\frac{3}{2}$
Solve for y: $2y = -3x + 12$ → $y = -\frac{3}{2}x + 6$
Yes! It's linear with slope $-\frac{3}{2}$
Example 3: Is $y = 2x^2 + 5$ linear?
No! It has $x^2$, so it's a quadratic function
No! It has $x^2$, so it's a quadratic function
2. From Tables
To Check if Table Shows Linear Function:
Step 1: Check if x-values increase by a constant amount
Step 2: Calculate rate of change between consecutive points
Step 3: If rate of change is CONSTANT → Linear ✓
Step 4: If rate of change varies → NOT Linear ✗
Step 1: Check if x-values increase by a constant amount
Step 2: Calculate rate of change between consecutive points
Step 3: If rate of change is CONSTANT → Linear ✓
Step 4: If rate of change varies → NOT Linear ✗
Constant Rate of Change Test:
$$\frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \text{constant}$$
$$\frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \text{constant}$$
Example 1: Linear Table
Check rate of change:
$\frac{8-5}{1-0} = 3$, $\frac{11-8}{2-1} = 3$, $\frac{14-11}{3-2} = 3$
Rate is constant = 3
This IS a linear function ✓
Equation: $y = 3x + 5$
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y | 5 | 8 | 11 | 14 |
Check rate of change:
$\frac{8-5}{1-0} = 3$, $\frac{11-8}{2-1} = 3$, $\frac{14-11}{3-2} = 3$
Rate is constant = 3
This IS a linear function ✓
Equation: $y = 3x + 5$
Example 2: NOT Linear Table
Check rate of change:
$\frac{1-0}{1-0} = 1$, $\frac{4-1}{2-1} = 3$, $\frac{9-4}{3-2} = 5$
Rate changes: 1, 3, 5
This is NOT linear ✗ (actually $y = x^2$)
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y | 0 | 1 | 4 | 9 |
Check rate of change:
$\frac{1-0}{1-0} = 1$, $\frac{4-1}{2-1} = 3$, $\frac{9-4}{3-2} = 5$
Rate changes: 1, 3, 5
This is NOT linear ✗ (actually $y = x^2$)
3. Complete a Table and Graph a Linear Function
Steps to Complete Table from Equation:
Step 1: Identify the equation $y = mx + b$
Step 2: For each x-value in table, substitute into equation
Step 3: Calculate corresponding y-value
Step 4: Fill in the table
Step 5: Graph by plotting points and drawing a line
Step 1: Identify the equation $y = mx + b$
Step 2: For each x-value in table, substitute into equation
Step 3: Calculate corresponding y-value
Step 4: Fill in the table
Step 5: Graph by plotting points and drawing a line
Example: Complete table and graph for $y = 2x - 3$
Calculate:
$x = -1$: $y = 2(-1) - 3 = -5$
$x = 0$: $y = 2(0) - 3 = -3$
$x = 1$: $y = 2(1) - 3 = -1$
$x = 2$: $y = 2(2) - 3 = 1$
Completed Table:
To Graph: Plot points $(-1,-5)$, $(0,-3)$, $(1,-1)$, $(2,1)$ and draw straight line through them
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y | ? | ? | ? | ? |
Calculate:
$x = -1$: $y = 2(-1) - 3 = -5$
$x = 0$: $y = 2(0) - 3 = -3$
$x = 1$: $y = 2(1) - 3 = -1$
$x = 2$: $y = 2(2) - 3 = 1$
Completed Table:
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y | -5 | -3 | -1 | 1 |
To Graph: Plot points $(-1,-5)$, $(0,-3)$, $(1,-1)$, $(2,1)$ and draw straight line through them
4. Evaluate a Linear Function from its Graph: Word Problems
Evaluating from Graph: Finding output values by reading the graph
Common Context: Distance vs. time, cost vs. quantity, temperature vs. time
Common Context: Distance vs. time, cost vs. quantity, temperature vs. time
Steps to Evaluate:
Step 1: Identify what x and y represent
Step 2: Find the given x-value on horizontal axis
Step 3: Draw vertical line to graph
Step 4: Read y-value at intersection
Step 5: Interpret in context
Step 1: Identify what x and y represent
Step 2: Find the given x-value on horizontal axis
Step 3: Draw vertical line to graph
Step 4: Read y-value at intersection
Step 5: Interpret in context
Example: A graph shows distance (miles) vs. time (hours). The line passes through $(0, 0)$ and $(2, 120)$.
Question: How far has the car traveled after 1.5 hours?
Solution:
• Find equation: slope = $\frac{120-0}{2-0} = 60$ mph
• Equation: $d = 60t$
• At $t = 1.5$: $d = 60(1.5) = 90$ miles
Answer: The car has traveled 90 miles after 1.5 hours
Question: How far has the car traveled after 1.5 hours?
Solution:
• Find equation: slope = $\frac{120-0}{2-0} = 60$ mph
• Equation: $d = 60t$
• At $t = 1.5$: $d = 60(1.5) = 90$ miles
Answer: The car has traveled 90 miles after 1.5 hours
5-6. Rate of Change of a Linear Function
Rate of Change = Slope
For linear functions, rate of change is ALWAYS constant
$$\text{Rate of Change} = m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\Delta y}{\Delta x}$$
For linear functions, rate of change is ALWAYS constant
$$\text{Rate of Change} = m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\Delta y}{\Delta x}$$
5. From Tables
Example: Find rate of change
Calculate:
$\text{Rate} = \frac{65-10}{1-0} = \frac{55}{1} = 55$ miles/hour
Verify constant:
$\frac{120-65}{2-1} = 55$ ✓
$\frac{175-120}{3-2} = 55$ ✓
Answer: Rate of change = 55 mph (constant)
| Time (h) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Distance (mi) | 10 | 65 | 120 | 175 |
Calculate:
$\text{Rate} = \frac{65-10}{1-0} = \frac{55}{1} = 55$ miles/hour
Verify constant:
$\frac{120-65}{2-1} = 55$ ✓
$\frac{175-120}{3-2} = 55$ ✓
Answer: Rate of change = 55 mph (constant)
6. From Graphs
Example: Find rate of change from graph passing through $(1, 3)$ and $(4, 9)$
Use slope formula:
$m = \frac{9-3}{4-1} = \frac{6}{3} = 2$
Answer: Rate of change = 2 units per unit
Use slope formula:
$m = \frac{9-3}{4-1} = \frac{6}{3} = 2$
Answer: Rate of change = 2 units per unit
Key Understanding:
• For linear functions, rate of change = slope = m
• It's CONSTANT throughout the function
• Positive rate → increasing function
• Negative rate → decreasing function
• Zero rate → horizontal line (constant function)
• For linear functions, rate of change = slope = m
• It's CONSTANT throughout the function
• Positive rate → increasing function
• Negative rate → decreasing function
• Zero rate → horizontal line (constant function)
7. Interpret the Slope and Y-Intercept
Real-World Meaning:
Slope (m): Rate of change, "per" statements
Y-intercept (b): Starting value, initial amount
Slope (m): Rate of change, "per" statements
Y-intercept (b): Starting value, initial amount
Interpretation Guide:
Slope Represents:
• How much y changes for each unit increase in x
• Rate (speed, cost per item, growth rate)
• Words like "per," "each," "every"
Y-Intercept Represents:
• Value of y when x = 0
• Starting amount, initial value
• Fixed cost, base fee
• Words like "starts at," "initial," "base"
Slope Represents:
• How much y changes for each unit increase in x
• Rate (speed, cost per item, growth rate)
• Words like "per," "each," "every"
Y-Intercept Represents:
• Value of y when x = 0
• Starting amount, initial value
• Fixed cost, base fee
• Words like "starts at," "initial," "base"
Example 1: $C(n) = 15n + 25$ models cost of party ($) for $n$ guests
Slope = 15:
• Each additional guest costs $15
• Rate: $15 per guest
Y-intercept = 25:
• Starting cost (when 0 guests) is $25
• Fixed/base fee: $25
Slope = 15:
• Each additional guest costs $15
• Rate: $15 per guest
Y-intercept = 25:
• Starting cost (when 0 guests) is $25
• Fixed/base fee: $25
Example 2: $T(h) = -5h + 80$ models temperature (°F) after $h$ hours
Slope = -5:
• Temperature decreases 5°F per hour
• Rate of cooling: 5°F/hour
Y-intercept = 80:
• Initial temperature is 80°F
• Starting value: 80°F
Slope = -5:
• Temperature decreases 5°F per hour
• Rate of cooling: 5°F/hour
Y-intercept = 80:
• Initial temperature is 80°F
• Starting value: 80°F
Example 3: $d(t) = 60t$ models distance (miles) after $t$ hours
Slope = 60: Speed is 60 mph
Y-intercept = 0: Starting position is 0 miles (at origin)
Slope = 60: Speed is 60 mph
Y-intercept = 0: Starting position is 0 miles (at origin)
8. Write a Linear Function: Word Problems
Steps to Write Linear Function from Context:
Step 1: Identify what x and y represent
Step 2: Find the rate of change (slope m)
• Look for "per," "each," "every"
Step 3: Find the y-intercept (b)
• Starting value, initial amount, when x=0
Step 4: Write equation: $y = mx + b$
Step 5: Check with given information
Step 1: Identify what x and y represent
Step 2: Find the rate of change (slope m)
• Look for "per," "each," "every"
Step 3: Find the y-intercept (b)
• Starting value, initial amount, when x=0
Step 4: Write equation: $y = mx + b$
Step 5: Check with given information
Example 1: A gym charges $30 per month plus a $50 joining fee. Write function for total cost.
Identify:
• $x$ = number of months
• $y$ = total cost
• Rate: $30 per month → $m = 30$
• Initial fee: $50 → $b = 50$
Function: $C(x) = 30x + 50$
Identify:
• $x$ = number of months
• $y$ = total cost
• Rate: $30 per month → $m = 30$
• Initial fee: $50 → $b = 50$
Function: $C(x) = 30x + 50$
Example 2: A water tank starts with 500 gallons and drains at 25 gallons per hour.
Identify:
• $x$ = hours
• $y$ = gallons remaining
• Rate: -25 gallons/hour → $m = -25$ (negative because draining)
• Initial: 500 gallons → $b = 500$
Function: $W(x) = -25x + 500$
Identify:
• $x$ = hours
• $y$ = gallons remaining
• Rate: -25 gallons/hour → $m = -25$ (negative because draining)
• Initial: 500 gallons → $b = 500$
Function: $W(x) = -25x + 500$
Example 3: A plant is 4 inches tall and grows 1.5 inches per week.
Function: $H(w) = 1.5w + 4$
where $w$ = weeks, $H$ = height in inches
Function: $H(w) = 1.5w + 4$
where $w$ = weeks, $H$ = height in inches
9-10. Domain and Range of Linear Functions
General Domain and Range:
For linear function $f(x) = mx + b$ (where $m \neq 0$):
• Domain: All real numbers, $(-\infty, \infty)$ or $\mathbb{R}$
• Range: All real numbers, $(-\infty, \infty)$ or $\mathbb{R}$
Special Case (Horizontal Line):
For $f(x) = b$ (where $m = 0$):
• Domain: $(-\infty, \infty)$
• Range: $\{b\}$ (single value)
For linear function $f(x) = mx + b$ (where $m \neq 0$):
• Domain: All real numbers, $(-\infty, \infty)$ or $\mathbb{R}$
• Range: All real numbers, $(-\infty, \infty)$ or $\mathbb{R}$
Special Case (Horizontal Line):
For $f(x) = b$ (where $m = 0$):
• Domain: $(-\infty, \infty)$
• Range: $\{b\}$ (single value)
9. From Graphs
Reading from Graph:
• Domain: All x-values covered by the line (look left-right)
• Range: All y-values covered by the line (look up-down)
• If line extends forever → use infinity symbols
• If endpoints exist → use those as boundaries
• Domain: All x-values covered by the line (look left-right)
• Range: All y-values covered by the line (look up-down)
• If line extends forever → use infinity symbols
• If endpoints exist → use those as boundaries
Example 1: Line extends infinitely in both directions
Domain: $(-\infty, \infty)$ or all real numbers
Range: $(-\infty, \infty)$ or all real numbers
Domain: $(-\infty, \infty)$ or all real numbers
Range: $(-\infty, \infty)$ or all real numbers
Example 2: Line segment from $(1, 3)$ to $(5, 11)$
Domain: $[1, 5]$ (x from 1 to 5, inclusive)
Range: $[3, 11]$ (y from 3 to 11, inclusive)
Domain: $[1, 5]$ (x from 1 to 5, inclusive)
Range: $[3, 11]$ (y from 3 to 11, inclusive)
Example 3: Horizontal line $y = 4$ extending infinitely
Domain: $(-\infty, \infty)$
Range: $\{4\}$ (only y = 4)
Domain: $(-\infty, \infty)$
Range: $\{4\}$ (only y = 4)
10. From Word Problems
Context Restrictions:
In real-world problems, domain and range may be restricted by:
• Physical constraints (can't be negative)
• Time limits
• Capacity limits
• Practical considerations
In real-world problems, domain and range may be restricted by:
• Physical constraints (can't be negative)
• Time limits
• Capacity limits
• Practical considerations
Example 1: $C(n) = 15n + 25$ for party cost with $n$ guests, max capacity 50
Domain: $[0, 50]$ or $\{0, 1, 2, 3, ..., 50\}$
• Can't have negative guests
• Maximum 50 guests
Range: $[25, 775]$
• Minimum: $C(0) = 25$
• Maximum: $C(50) = 15(50) + 25 = 775$
Domain: $[0, 50]$ or $\{0, 1, 2, 3, ..., 50\}$
• Can't have negative guests
• Maximum 50 guests
Range: $[25, 775]$
• Minimum: $C(0) = 25$
• Maximum: $C(50) = 15(50) + 25 = 775$
Example 2: $d(t) = 60t$ for distance in miles after $t$ hours, trip is 4 hours
Domain: $[0, 4]$ (time from 0 to 4 hours)
Range: $[0, 240]$ (distance from 0 to 240 miles)
Domain: $[0, 4]$ (time from 0 to 4 hours)
Range: $[0, 240]$ (distance from 0 to 240 miles)
11-12. Compare Linear Functions
Comparing Functions: Determining which has greater/lesser slope or y-intercept
Key Comparisons:
• Which is increasing/decreasing faster?
• Which starts higher?
• Which will be greater at a specific x-value?
Key Comparisons:
• Which is increasing/decreasing faster?
• Which starts higher?
• Which will be greater at a specific x-value?
11. Graphs and Equations
Comparison Strategies:
Comparing Slopes:
• Steeper line (visually) → larger absolute value of slope
• Line rising faster → greater positive slope
• Line falling faster → more negative slope
Comparing Y-Intercepts:
• Where line crosses y-axis
• Higher crossing point → greater y-intercept
Comparing Slopes:
• Steeper line (visually) → larger absolute value of slope
• Line rising faster → greater positive slope
• Line falling faster → more negative slope
Comparing Y-Intercepts:
• Where line crosses y-axis
• Higher crossing point → greater y-intercept
Example: Compare $f(x) = 3x + 2$ with graph of $g(x)$ passing through $(0, 5)$ and $(2, 9)$
Function f:
• Slope: $m = 3$
• Y-intercept: $b = 2$
Function g from graph:
• Slope: $m = \frac{9-5}{2-0} = \frac{4}{2} = 2$
• Y-intercept: $b = 5$ (crosses at $(0,5)$)
Comparison:
• $f$ has greater slope (3 > 2) → increases faster
• $g$ has greater y-intercept (5 > 2) → starts higher
• Function g: $g(x) = 2x + 5$
Function f:
• Slope: $m = 3$
• Y-intercept: $b = 2$
Function g from graph:
• Slope: $m = \frac{9-5}{2-0} = \frac{4}{2} = 2$
• Y-intercept: $b = 5$ (crosses at $(0,5)$)
Comparison:
• $f$ has greater slope (3 > 2) → increases faster
• $g$ has greater y-intercept (5 > 2) → starts higher
• Function g: $g(x) = 2x + 5$
12. Tables, Graphs, and Equations
Example: Compare three functions
Function A: $f(x) = 4x + 1$
Function B (Table):
Slope: $\frac{6-3}{1-0} = 3$, Y-int: 3
Equation: $g(x) = 3x + 3$
Function C (Graph): Passes through $(0, 2)$ and $(1, 7)$
Slope: $\frac{7-2}{1-0} = 5$, Y-int: 2
Equation: $h(x) = 5x + 2$
Comparison:
• Greatest slope: Function C (5)
• Greatest y-intercept: Function B (3)
• At $x = 3$:
$f(3) = 13$, $g(3) = 12$, $h(3) = 17$
Function C is greatest at $x = 3$
Function A: $f(x) = 4x + 1$
Function B (Table):
| x | 0 | 1 | 2 |
|---|---|---|---|
| y | 3 | 6 | 9 |
Equation: $g(x) = 3x + 3$
Function C (Graph): Passes through $(0, 2)$ and $(1, 7)$
Slope: $\frac{7-2}{1-0} = 5$, Y-int: 2
Equation: $h(x) = 5x + 2$
Comparison:
• Greatest slope: Function C (5)
• Greatest y-intercept: Function B (3)
• At $x = 3$:
$f(3) = 13$, $g(3) = 12$, $h(3) = 17$
Function C is greatest at $x = 3$
13. Transformations of Linear Functions
Transformation: A change in position, shape, or size of a function's graph
Parent Function: $f(x) = x$ (the basic linear function)
Parent Function: $f(x) = x$ (the basic linear function)
Types of Transformations:
1. Vertical Translation (Up/Down):
$$g(x) = f(x) + k$$
• $k > 0$ → shift UP k units
• $k < 0$ → shift DOWN |k| units
2. Horizontal Translation (Left/Right):
$$g(x) = f(x - h)$$
• $h > 0$ → shift RIGHT h units
• $h < 0$ → shift LEFT |h| units
3. Vertical Stretch/Compression:
$$g(x) = a \cdot f(x)$$
• $|a| > 1$ → vertical stretch (steeper)
• $0 < |a| < 1$ → vertical compression (flatter)
4. Reflection:
• $g(x) = -f(x)$ → reflect over x-axis
• $g(x) = f(-x)$ → reflect over y-axis
1. Vertical Translation (Up/Down):
$$g(x) = f(x) + k$$
• $k > 0$ → shift UP k units
• $k < 0$ → shift DOWN |k| units
2. Horizontal Translation (Left/Right):
$$g(x) = f(x - h)$$
• $h > 0$ → shift RIGHT h units
• $h < 0$ → shift LEFT |h| units
3. Vertical Stretch/Compression:
$$g(x) = a \cdot f(x)$$
• $|a| > 1$ → vertical stretch (steeper)
• $0 < |a| < 1$ → vertical compression (flatter)
4. Reflection:
• $g(x) = -f(x)$ → reflect over x-axis
• $g(x) = f(-x)$ → reflect over y-axis
Vertical Translations
Example 1: Transform $f(x) = x$ by adding 3
$g(x) = f(x) + 3 = x + 3$
Effect: Shifts entire graph UP 3 units
• New y-intercept: 3 (was 0)
• Slope remains: 1
$g(x) = f(x) + 3 = x + 3$
Effect: Shifts entire graph UP 3 units
• New y-intercept: 3 (was 0)
• Slope remains: 1
Example 2: Transform $f(x) = 2x$ by subtracting 5
$g(x) = 2x - 5$
Effect: Shifts entire graph DOWN 5 units
• New y-intercept: -5 (was 0)
• Slope remains: 2
$g(x) = 2x - 5$
Effect: Shifts entire graph DOWN 5 units
• New y-intercept: -5 (was 0)
• Slope remains: 2
Horizontal Translations
Example 3: Transform $f(x) = x$ by $f(x - 2)$
$g(x) = f(x - 2) = (x - 2) = x - 2$
Effect: Shifts entire graph RIGHT 2 units
• Points move right
• Note: Opposite sign! $(x - 2)$ moves RIGHT
$g(x) = f(x - 2) = (x - 2) = x - 2$
Effect: Shifts entire graph RIGHT 2 units
• Points move right
• Note: Opposite sign! $(x - 2)$ moves RIGHT
Example 4: Transform $f(x) = x$ by $f(x + 3)$
$g(x) = f(x + 3) = (x + 3) = x + 3$
Effect: Shifts entire graph LEFT 3 units
• $(x + 3)$ moves LEFT
$g(x) = f(x + 3) = (x + 3) = x + 3$
Effect: Shifts entire graph LEFT 3 units
• $(x + 3)$ moves LEFT
Vertical Stretch and Compression
Example 5: Transform $f(x) = x$ by multiplying by 3
$g(x) = 3f(x) = 3x$
Effect: Vertical STRETCH by factor of 3
• Graph becomes steeper
• Slope changes from 1 to 3
$g(x) = 3f(x) = 3x$
Effect: Vertical STRETCH by factor of 3
• Graph becomes steeper
• Slope changes from 1 to 3
Example 6: Transform $f(x) = x$ by multiplying by $\frac{1}{2}$
$g(x) = \frac{1}{2}f(x) = \frac{1}{2}x$
Effect: Vertical COMPRESSION by factor of $\frac{1}{2}$
• Graph becomes flatter
• Slope changes from 1 to $\frac{1}{2}$
$g(x) = \frac{1}{2}f(x) = \frac{1}{2}x$
Effect: Vertical COMPRESSION by factor of $\frac{1}{2}$
• Graph becomes flatter
• Slope changes from 1 to $\frac{1}{2}$
Reflections
Example 7: Transform $f(x) = 2x + 3$ by $-f(x)$
$g(x) = -f(x) = -(2x + 3) = -2x - 3$
Effect: Reflects over x-axis (flips upside down)
• Positive slopes become negative
• Line slopes opposite direction
$g(x) = -f(x) = -(2x + 3) = -2x - 3$
Effect: Reflects over x-axis (flips upside down)
• Positive slopes become negative
• Line slopes opposite direction
Example 8: Transform $f(x) = x$ by $f(-x)$
$g(x) = f(-x) = -x$
Effect: Reflects over y-axis
• Left and right sides swap
$g(x) = f(-x) = -x$
Effect: Reflects over y-axis
• Left and right sides swap
Combined Transformations
Example 9: $g(x) = 2(x - 3) + 5$ starting from $f(x) = x$
Transformations in order:
1. Horizontal shift RIGHT 3: $f(x-3) = x - 3$
2. Vertical stretch by 2: $2(x-3) = 2x - 6$
3. Vertical shift UP 5: $2x - 6 + 5 = 2x - 1$
Final: $g(x) = 2x - 1$
• Slope: 2 (stretched)
• Y-intercept: -1
Transformations in order:
1. Horizontal shift RIGHT 3: $f(x-3) = x - 3$
2. Vertical stretch by 2: $2(x-3) = 2x - 6$
3. Vertical shift UP 5: $2x - 6 + 5 = 2x - 1$
Final: $g(x) = 2x - 1$
• Slope: 2 (stretched)
• Y-intercept: -1
Transformation Summary:
• Adding/subtracting OUTSIDE function → vertical shift
• Adding/subtracting INSIDE function → horizontal shift (opposite sign)
• Multiplying OUTSIDE → vertical stretch/compression
• Negative sign OUTSIDE → reflect over x-axis
• Negative sign INSIDE → reflect over y-axis
• Adding/subtracting OUTSIDE function → vertical shift
• Adding/subtracting INSIDE function → horizontal shift (opposite sign)
• Multiplying OUTSIDE → vertical stretch/compression
• Negative sign OUTSIDE → reflect over x-axis
• Negative sign INSIDE → reflect over y-axis
Quick Reference Guide
Linear Function Standard Form:
$$f(x) = mx + b$$
• $m$ = slope (rate of change)
• $b$ = y-intercept (starting value)
$$f(x) = mx + b$$
• $m$ = slope (rate of change)
• $b$ = y-intercept (starting value)
Identifying Linear Functions:
✓ Linear:
• Degree = 1
• Constant rate of change
• Straight line graph
• Form: $y = mx + b$
✗ NOT Linear:
• Exponents > 1
• Variables in denominator
• Variables under radicals
• Variables multiplied together
✓ Linear:
• Degree = 1
• Constant rate of change
• Straight line graph
• Form: $y = mx + b$
✗ NOT Linear:
• Exponents > 1
• Variables in denominator
• Variables under radicals
• Variables multiplied together
Domain and Range:
• General: Domain = Range = $(-\infty, \infty)$
• Horizontal line: Domain = $(-\infty, \infty)$, Range = $\{b\}$
• Restricted context: Use practical limits
• General: Domain = Range = $(-\infty, \infty)$
• Horizontal line: Domain = $(-\infty, \infty)$, Range = $\{b\}$
• Restricted context: Use practical limits
| Concept | Formula/Rule | Interpretation |
|---|---|---|
| Slope (m) | $m = \frac{y_2-y_1}{x_2-x_1}$ | Rate of change, "per" statements |
| Y-intercept (b) | Value when $x = 0$ | Starting value, initial amount |
| Rate of Change | Always equals slope | Constant for linear functions |
| Domain | All possible x-values | Usually $(-\infty, \infty)$ |
| Range | All possible y-values | Usually $(-\infty, \infty)$ |
Transformation Quick Reference
| Transformation | Notation | Effect | Example |
|---|---|---|---|
| Vertical shift up | $f(x) + k$ | Move up k units | $x + 3$ |
| Vertical shift down | $f(x) - k$ | Move down k units | $x - 2$ |
| Horizontal shift right | $f(x - h)$ | Move right h units | $(x - 4)$ |
| Horizontal shift left | $f(x + h)$ | Move left h units | $(x + 3)$ |
| Vertical stretch | $a \cdot f(x)$, $|a| > 1$ | Steeper, slope multiplied | $3x$ |
| Vertical compression | $a \cdot f(x)$, $0 < |a| < 1$ | Flatter, slope multiplied | $\frac{1}{2}x$ |
| Reflect over x-axis | $-f(x)$ | Flip vertically | $-x$ |
| Reflect over y-axis | $f(-x)$ | Flip horizontally | $-x$ |
Success Tips for Linear Functions:
✓ Always check for constant rate of change
✓ Slope = rate of change (always constant for linear)
✓ Y-intercept = starting value in word problems
✓ Graph is always a straight line
✓ Domain and range usually all real numbers
✓ Compare functions by looking at slope and y-intercept
✓ For transformations: outside affects y, inside affects x
✓ Practice identifying linear vs. nonlinear functions
✓ Real-world context often restricts domain and range
✓ Always check for constant rate of change
✓ Slope = rate of change (always constant for linear)
✓ Y-intercept = starting value in word problems
✓ Graph is always a straight line
✓ Domain and range usually all real numbers
✓ Compare functions by looking at slope and y-intercept
✓ For transformations: outside affects y, inside affects x
✓ Practice identifying linear vs. nonlinear functions
✓ Real-world context often restricts domain and range