Linear, Quadratic, and Exponential Functions - Ninth Grade Math
Overview: Three Types of Functions
Three Main Function Types:
• Linear: Constant rate of change (steady increase/decrease)
• Quadratic: Variable rate of change (parabola shape)
• Exponential: Multiplicative rate of change (rapid growth/decay)
• Linear: Constant rate of change (steady increase/decrease)
• Quadratic: Variable rate of change (parabola shape)
• Exponential: Multiplicative rate of change (rapid growth/decay)
General Forms:
Linear Function:
$$f(x) = mx + b$$
where $m$ = slope (constant rate of change), $b$ = y-intercept
Quadratic Function:
$$f(x) = ax^2 + bx + c$$
where $a \neq 0$, variable is squared
Exponential Function:
$$f(x) = ab^x$$
where $a$ = initial value, $b$ = growth/decay factor, variable is exponent
Linear Function:
$$f(x) = mx + b$$
where $m$ = slope (constant rate of change), $b$ = y-intercept
Quadratic Function:
$$f(x) = ax^2 + bx + c$$
where $a \neq 0$, variable is squared
Exponential Function:
$$f(x) = ab^x$$
where $a$ = initial value, $b$ = growth/decay factor, variable is exponent
1-2. Identify Functions from Graphs
Key Visual Characteristics
Linear Functions
Graph Shape: Straight line
Direction: Increasing, decreasing, or horizontal
Slope: Constant (same steepness throughout)
Rate of Change: Constant
Key Feature: Goes up/down by same amount for each unit of x
Direction: Increasing, decreasing, or horizontal
Slope: Constant (same steepness throughout)
Rate of Change: Constant
Key Feature: Goes up/down by same amount for each unit of x
Quadratic Functions
Graph Shape: Parabola (U-shape or ∩-shape)
Direction: Opens up or down
Vertex: Has highest or lowest point
Symmetry: Symmetric about vertical line (axis of symmetry)
Rate of Change: Variable (changes based on position)
Key Feature: Curved, increases then decreases (or vice versa)
Direction: Opens up or down
Vertex: Has highest or lowest point
Symmetry: Symmetric about vertical line (axis of symmetry)
Rate of Change: Variable (changes based on position)
Key Feature: Curved, increases then decreases (or vice versa)
Exponential Functions
Graph Shape: Curved (J-shape or decay curve)
Growth: Steep increase or decrease
Asymptote: Approaches but never touches x-axis (usually)
Rate of Change: Multiplicative (gets faster/slower)
Key Features:
• Growth ($b > 1$): Starts slow, then increases rapidly
• Decay ($0 < b < 1$): Decreases rapidly, then levels off
Growth: Steep increase or decrease
Asymptote: Approaches but never touches x-axis (usually)
Rate of Change: Multiplicative (gets faster/slower)
Key Features:
• Growth ($b > 1$): Starts slow, then increases rapidly
• Decay ($0 < b < 1$): Decreases rapidly, then levels off
Example 1: Identify function type from description
Graph A: Straight line going up
Answer: LINEAR
Graph B: U-shaped curve with lowest point at (2, -3)
Answer: QUADRATIC
Graph C: Curve that starts near zero, then shoots up rapidly
Answer: EXPONENTIAL (growth)
Graph A: Straight line going up
Answer: LINEAR
Graph B: U-shaped curve with lowest point at (2, -3)
Answer: QUADRATIC
Graph C: Curve that starts near zero, then shoots up rapidly
Answer: EXPONENTIAL (growth)
Quick Visual Test:
• Straight line? → Linear
• U or ∩ shape? → Quadratic
• One end flat, other steep? → Exponential
• Straight line? → Linear
• U or ∩ shape? → Quadratic
• One end flat, other steep? → Exponential
3-4. Identify Functions from Tables
Pattern Recognition in Tables
How to Identify from Table:
LINEAR: First differences are constant
• Calculate: $y_2 - y_1$, $y_3 - y_2$, $y_4 - y_3$, ...
• If all equal → LINEAR
QUADRATIC: Second differences are constant
• First differences vary
• Second differences (differences of differences) are constant
EXPONENTIAL: Ratios are constant
• Calculate: $\frac{y_2}{y_1}$, $\frac{y_3}{y_2}$, $\frac{y_4}{y_3}$, ...
• If all equal → EXPONENTIAL
LINEAR: First differences are constant
• Calculate: $y_2 - y_1$, $y_3 - y_2$, $y_4 - y_3$, ...
• If all equal → LINEAR
QUADRATIC: Second differences are constant
• First differences vary
• Second differences (differences of differences) are constant
EXPONENTIAL: Ratios are constant
• Calculate: $\frac{y_2}{y_1}$, $\frac{y_3}{y_2}$, $\frac{y_4}{y_3}$, ...
• If all equal → EXPONENTIAL
Linear Tables
Example 1: Identify function type
First differences:
$8 - 5 = 3$
$11 - 8 = 3$
$14 - 11 = 3$
$17 - 14 = 3$
All differences equal 3 (constant)
Answer: LINEAR with slope $m = 3$
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y | 5 | 8 | 11 | 14 | 17 |
First differences:
$8 - 5 = 3$
$11 - 8 = 3$
$14 - 11 = 3$
$17 - 14 = 3$
All differences equal 3 (constant)
Answer: LINEAR with slope $m = 3$
Quadratic Tables
Example 2: Identify function type
First differences: 1, 3, 5, 7 (not constant)
Second differences:
$3 - 1 = 2$
$5 - 3 = 2$
$7 - 5 = 2$
Second differences constant!
Answer: QUADRATIC (this is $y = x^2$)
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y | 0 | 1 | 4 | 9 | 16 |
First differences: 1, 3, 5, 7 (not constant)
Second differences:
$3 - 1 = 2$
$5 - 3 = 2$
$7 - 5 = 2$
Second differences constant!
Answer: QUADRATIC (this is $y = x^2$)
Exponential Tables
Example 3: Identify function type
Check ratios:
$\frac{6}{3} = 2$
$\frac{12}{6} = 2$
$\frac{24}{12} = 2$
$\frac{48}{24} = 2$
All ratios equal 2 (constant)
Answer: EXPONENTIAL with $b = 2$ (doubling)
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y | 3 | 6 | 12 | 24 | 48 |
Check ratios:
$\frac{6}{3} = 2$
$\frac{12}{6} = 2$
$\frac{24}{12} = 2$
$\frac{48}{24} = 2$
All ratios equal 2 (constant)
Answer: EXPONENTIAL with $b = 2$ (doubling)
Decision Process for Tables:
Step 1: Check if x-values increase by 1 (unit intervals)
Step 2: Calculate first differences
Step 3: If constant → LINEAR
Step 4: If not, calculate second differences
Step 5: If constant → QUADRATIC
Step 6: If not, check ratios
Step 7: If constant → EXPONENTIAL
Step 1: Check if x-values increase by 1 (unit intervals)
Step 2: Calculate first differences
Step 3: If constant → LINEAR
Step 4: If not, calculate second differences
Step 5: If constant → QUADRATIC
Step 6: If not, check ratios
Step 7: If constant → EXPONENTIAL
5-7. Write Functions from Tables
Writing Linear Functions
Linear: $y = mx + b$
$m$ (slope): First difference (amount added each time)
$b$ (y-intercept): Value when $x = 0$
If table doesn't start at $x = 0$, use point-slope form or substitute a point
$m$ (slope): First difference (amount added each time)
$b$ (y-intercept): Value when $x = 0$
If table doesn't start at $x = 0$, use point-slope form or substitute a point
Example 1: Write equation
First difference: $7 - 4 = 3$ → $m = 3$
Y-intercept: $b = 4$ (when $x = 0$)
Equation: $y = 3x + 4$
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y | 4 | 7 | 10 | 13 |
First difference: $7 - 4 = 3$ → $m = 3$
Y-intercept: $b = 4$ (when $x = 0$)
Equation: $y = 3x + 4$
Writing Exponential Functions
Exponential: $y = ab^x$
$a$ (initial value): Value when $x = 0$
$b$ (growth/decay factor): Common ratio (multiply by this each time)
$a$ (initial value): Value when $x = 0$
$b$ (growth/decay factor): Common ratio (multiply by this each time)
Example 2: Write equation
Initial value: $a = 5$ (when $x = 0$)
Common ratio: $\frac{10}{5} = 2$ → $b = 2$
Equation: $y = 5 \cdot 2^x$
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y | 5 | 10 | 20 | 40 |
Initial value: $a = 5$ (when $x = 0$)
Common ratio: $\frac{10}{5} = 2$ → $b = 2$
Equation: $y = 5 \cdot 2^x$
Writing Quadratic Functions
Quadratic: More complex, typically need:
• Vertex form: $y = a(x - h)^2 + k$ if vertex visible
• Three points to solve for $a$, $b$, $c$ in $y = ax^2 + bx + c$
• Vertex form: $y = a(x - h)^2 + k$ if vertex visible
• Three points to solve for $a$, $b$, $c$ in $y = ax^2 + bx + c$
Example 3: Write equation with vertex at $(0, 0)$ and point $(2, 8)$
Vertex form: $y = a(x - 0)^2 + 0 = ax^2$
Use point $(2, 8)$:
$8 = a(2)^2$
$8 = 4a$
$a = 2$
Equation: $y = 2x^2$
Vertex form: $y = a(x - 0)^2 + 0 = ax^2$
Use point $(2, 8)$:
$8 = a(2)^2$
$8 = 4a$
$a = 2$
Equation: $y = 2x^2$
8. Write Functions: Word Problems
Key Words to Identify Function Type:
LINEAR:
• Constant rate, per unit, each time, steady
• Add/subtract same amount
• "increases by $5 each hour"
EXPONENTIAL:
• Doubles, triples, halves
• Percent increase/decrease
• Multiply by same factor
• "increases by 20% each year"
LINEAR:
• Constant rate, per unit, each time, steady
• Add/subtract same amount
• "increases by $5 each hour"
EXPONENTIAL:
• Doubles, triples, halves
• Percent increase/decrease
• Multiply by same factor
• "increases by 20% each year"
Example 1: A phone costs $800 and depreciates $50 each year. Write function.
Key word: "$50 each year" → constant decrease → LINEAR
Initial value: $b = 800$
Slope: $m = -50$ (decreasing)
Equation: $V(t) = -50t + 800$
where $t$ = years
Key word: "$50 each year" → constant decrease → LINEAR
Initial value: $b = 800$
Slope: $m = -50$ (decreasing)
Equation: $V(t) = -50t + 800$
where $t$ = years
Example 2: A bacteria population of 100 doubles every hour. Write function.
Key word: "doubles" → multiply by 2 → EXPONENTIAL
Initial value: $a = 100$
Growth factor: $b = 2$
Equation: $P(t) = 100 \cdot 2^t$
where $t$ = hours
Key word: "doubles" → multiply by 2 → EXPONENTIAL
Initial value: $a = 100$
Growth factor: $b = 2$
Equation: $P(t) = 100 \cdot 2^t$
where $t$ = hours
Example 3: An investment of $1000 grows 5% per year. Write function.
Key phrase: "5% per year" → percent growth → EXPONENTIAL
Initial: $a = 1000$
Growth factor: $b = 1 + 0.05 = 1.05$
Equation: $A(t) = 1000(1.05)^t$
Key phrase: "5% per year" → percent growth → EXPONENTIAL
Initial: $a = 1000$
Growth factor: $b = 1 + 0.05 = 1.05$
Equation: $A(t) = 1000(1.05)^t$
9-10. Functions Over Unit Intervals
Unit Interval: When x increases by 1
Purpose: Analyze how function changes over equal intervals
Purpose: Analyze how function changes over equal intervals
Linear Functions Over Unit Intervals
Linear Property:
Over equal intervals (like unit intervals), linear functions change by equal amounts
If $f(x) = mx + b$, then:
$$f(x+1) - f(x) = m$$
(Constant first difference)
Over equal intervals (like unit intervals), linear functions change by equal amounts
If $f(x) = mx + b$, then:
$$f(x+1) - f(x) = m$$
(Constant first difference)
Example 1: $f(x) = 3x + 2$
$f(1) = 5$, $f(2) = 8$, $f(3) = 11$, $f(4) = 14$
Changes: +3, +3, +3 (constant)
Pattern: Linear functions add same amount each time
$f(1) = 5$, $f(2) = 8$, $f(3) = 11$, $f(4) = 14$
Changes: +3, +3, +3 (constant)
Pattern: Linear functions add same amount each time
Exponential Functions Over Unit Intervals
Exponential Property:
Over equal intervals, exponential functions change by equal ratios
If $f(x) = ab^x$, then:
$$\frac{f(x+1)}{f(x)} = b$$
(Constant ratio/multiplier)
Over equal intervals, exponential functions change by equal ratios
If $f(x) = ab^x$, then:
$$\frac{f(x+1)}{f(x)} = b$$
(Constant ratio/multiplier)
Example 2: $g(x) = 2 \cdot 3^x$
$g(0) = 2$, $g(1) = 6$, $g(2) = 18$, $g(3) = 54$
Ratios: $\frac{6}{2} = 3$, $\frac{18}{6} = 3$, $\frac{54}{18} = 3$
Pattern: Exponential functions multiply by same factor each time
$g(0) = 2$, $g(1) = 6$, $g(2) = 18$, $g(3) = 54$
Ratios: $\frac{6}{2} = 3$, $\frac{18}{6} = 3$, $\frac{54}{18} = 3$
Pattern: Exponential functions multiply by same factor each time
11. Describe Linear and Exponential Growth and Decay
Growth: Function increases as x increases
Decay: Function decreases as x increases
Decay: Function decreases as x increases
Linear Growth and Decay
Linear Growth: $y = mx + b$ where $m > 0$
• Increases by constant amount
• Slope is positive
• Example: Save $50 per week
Linear Decay: $y = mx + b$ where $m < 0$
• Decreases by constant amount
• Slope is negative
• Example: Lose $30 per day
• Increases by constant amount
• Slope is positive
• Example: Save $50 per week
Linear Decay: $y = mx + b$ where $m < 0$
• Decreases by constant amount
• Slope is negative
• Example: Lose $30 per day
Exponential Growth and Decay
Exponential Growth: $y = ab^x$ where $b > 1$
• Multiplies by factor greater than 1
• Increases rapidly
• Example: Population doubles each year ($b = 2$)
Exponential Decay: $y = ab^x$ where $0 < b < 1$
• Multiplies by factor less than 1
• Decreases rapidly at first, then slowly
• Example: Half-life, depreciation by 20% ($b = 0.8$)
• Multiplies by factor greater than 1
• Increases rapidly
• Example: Population doubles each year ($b = 2$)
Exponential Decay: $y = ab^x$ where $0 < b < 1$
• Multiplies by factor less than 1
• Decreases rapidly at first, then slowly
• Example: Half-life, depreciation by 20% ($b = 0.8$)
Example 1: Describe: A car worth $25,000 loses $2,000 per year
Type: Linear decay
Reason: Loses constant amount each year
Function: $V(t) = -2000t + 25000$
Type: Linear decay
Reason: Loses constant amount each year
Function: $V(t) = -2000t + 25000$
Example 2: Describe: A population of 500 increases by 12% each month
Type: Exponential growth
Reason: Increases by percentage (multiply by 1.12)
Function: $P(t) = 500(1.12)^t$
Type: Exponential growth
Reason: Increases by percentage (multiply by 1.12)
Function: $P(t) = 500(1.12)^t$
12-13. Compare Linear, Exponential, and Quadratic Growth
Growth Comparison: How fast each function increases over time
Growth Speed (Long Term):
Slowest → Fastest:
$$\text{Linear} < \text{Quadratic} < \text{Exponential}$$
Initially: Order may vary
Eventually: Exponential ALWAYS wins
Slowest → Fastest:
$$\text{Linear} < \text{Quadratic} < \text{Exponential}$$
Initially: Order may vary
Eventually: Exponential ALWAYS wins
Short-Term vs Long-Term
Initial Behavior (small x-values):
• Linear may start highest
• Quadratic grows faster than linear early on
• Exponential may start lowest
Long-Term Behavior (large x-values):
• Exponential eventually surpasses everything
• Quadratic eventually surpasses linear
• Linear grows slowest in long run
• Linear may start highest
• Quadratic grows faster than linear early on
• Exponential may start lowest
Long-Term Behavior (large x-values):
• Exponential eventually surpasses everything
• Quadratic eventually surpasses linear
• Linear grows slowest in long run
Example 1: Compare $f(x) = 2x$, $g(x) = x^2$, $h(x) = 2^x$
At $x = 10$: Exponential is FAR ahead!
Order: $2^{10} = 1024 > x^2 = 100 > 2x = 20$
| x | 0 | 1 | 2 | 3 | 4 | 5 | 10 |
|---|---|---|---|---|---|---|---|
| Linear: 2x | 0 | 2 | 4 | 6 | 8 | 10 | 20 |
| Quadratic: x² | 0 | 1 | 4 | 9 | 16 | 25 | 100 |
| Exponential: 2ˣ | 1 | 2 | 4 | 8 | 16 | 32 | 1024 |
At $x = 10$: Exponential is FAR ahead!
Order: $2^{10} = 1024 > x^2 = 100 > 2x = 20$
Key Insight:
Exponential growth seems slow at first but explodes later!
This is why compound interest and viral spread are so powerful.
Exponential growth seems slow at first but explodes later!
This is why compound interest and viral spread are so powerful.
Master Comparison Table
| Feature | Linear | Quadratic | Exponential |
|---|---|---|---|
| Equation | $y = mx + b$ | $y = ax^2 + bx + c$ | $y = ab^x$ |
| Variable Location | Base (standard) | Base (squared) | Exponent |
| Graph Shape | Straight line | Parabola (U-shape) | Curve (J-shape) |
| Rate of Change | Constant | Variable | Multiplicative |
| Pattern in Table | Equal differences | Equal 2nd differences | Equal ratios |
| Growth Type | Add same amount | Accelerating addition | Multiply by factor |
| Example | $y = 3x + 5$ | $y = x^2 - 4$ | $y = 2 \cdot 3^x$ |
Identification Quick Reference
| From Table | Check This | If True |
|---|---|---|
| First Differences | $y_2 - y_1$, $y_3 - y_2$, etc. | All equal → LINEAR |
| Second Differences | Difference of differences | All equal → QUADRATIC |
| Ratios | $\frac{y_2}{y_1}$, $\frac{y_3}{y_2}$, etc. | All equal → EXPONENTIAL |
Real-World Applications
| Function Type | Real-World Examples |
|---|---|
| Linear |
• Hourly wages • Constant speed travel • Simple interest • Phone bill with per-minute rate |
| Quadratic |
• Projectile motion (height vs time) • Area of squares • Braking distance • Profit models |
| Exponential |
• Population growth • Compound interest • Viral spread • Radioactive decay • Bacterial growth |
Decision Flowchart
How to Identify Function Type:
FROM GRAPH:
1. Is it a straight line? → LINEAR
2. Is it U-shaped or ∩-shaped? → QUADRATIC
3. Does it curve with one flat end? → EXPONENTIAL
FROM TABLE (with unit intervals):
1. Calculate first differences
2. All equal? → LINEAR
3. Not equal? Calculate second differences
4. All equal? → QUADRATIC
5. Not equal? Calculate ratios
6. All equal? → EXPONENTIAL
FROM WORD PROBLEM:
1. Constant rate? → LINEAR
2. Percentage/doubles/halves? → EXPONENTIAL
3. Variable is squared? → QUADRATIC
FROM EQUATION:
1. Variable in base only? → LINEAR
2. Variable squared? → QUADRATIC
3. Variable in exponent? → EXPONENTIAL
FROM GRAPH:
1. Is it a straight line? → LINEAR
2. Is it U-shaped or ∩-shaped? → QUADRATIC
3. Does it curve with one flat end? → EXPONENTIAL
FROM TABLE (with unit intervals):
1. Calculate first differences
2. All equal? → LINEAR
3. Not equal? Calculate second differences
4. All equal? → QUADRATIC
5. Not equal? Calculate ratios
6. All equal? → EXPONENTIAL
FROM WORD PROBLEM:
1. Constant rate? → LINEAR
2. Percentage/doubles/halves? → EXPONENTIAL
3. Variable is squared? → QUADRATIC
FROM EQUATION:
1. Variable in base only? → LINEAR
2. Variable squared? → QUADRATIC
3. Variable in exponent? → EXPONENTIAL
Success Tips for Comparing Functions:
✓ Linear = constant addition (equal differences)
✓ Quadratic = variable addition (equal 2nd differences)
✓ Exponential = constant multiplication (equal ratios)
✓ Graph: line vs parabola vs curve
✓ Long term: exponential always grows fastest
✓ In word problems: look for "per" vs "percent"
✓ Exponential has variable in exponent!
✓ Check where the variable appears in equation
✓ Practice all three identification methods
✓ Remember: patterns are key to identification
✓ Linear = constant addition (equal differences)
✓ Quadratic = variable addition (equal 2nd differences)
✓ Exponential = constant multiplication (equal ratios)
✓ Graph: line vs parabola vs curve
✓ Long term: exponential always grows fastest
✓ In word problems: look for "per" vs "percent"
✓ Exponential has variable in exponent!
✓ Check where the variable appears in equation
✓ Practice all three identification methods
✓ Remember: patterns are key to identification