Nonlinear Functions - Grade 8

1. What is a Nonlinear Function?

Definition: A nonlinear function is a function whose graph is NOT a straight line. It can be any curve, parabola, exponential curve, or other non-straight shape.

Key Characteristic:

A nonlinear function does NOT have a constant rate of change (slope varies).

Examples of Nonlinear Functions:

Quadratic: \( f(x) = x^2 \), \( f(x) = x^2 + 3x - 5 \)
Cubic: \( f(x) = x^3 \), \( f(x) = 2x^3 - x \)
Exponential: \( f(x) = 2^x \), \( f(x) = 3^x + 1 \)
Absolute Value: \( f(x) = |x| \), \( f(x) = |x - 2| + 1 \)
Reciprocal: \( f(x) = \frac{1}{x} \)
Square Root: \( f(x) = \sqrt{x} \)

What Makes It Nonlinear:

Variables with exponents other than 1 (like \( x^2, x^3 \))
Variables in denominators (like \( \frac{1}{x} \))
Variables under roots (like \( \sqrt{x} \))
Variables as exponents (like \( 2^x \))
Products of variables (like \( xy \))

2. Linear vs. Nonlinear Functions

Feature	Linear Function	Nonlinear Function
Graph	Straight line	Curved or not a straight line
Rate of Change	Constant (same slope)	NOT constant (slope varies)
Equation Form	\( f(x) = mx + b \)	Any other form
Variable Exponent	Always 1 (or 0)	Can be 2, 3, fractions, etc.
Example	\( y = 3x + 2 \)	\( y = x^2 + 1 \)

3. Identify Linear and Nonlinear Functions: Graphs and Equations

From Graphs:

Linear: The graph is a straight line

Use a ruler or straight edge - if it touches all points, it's linear

Nonlinear: The graph is curved, wavy, or any shape other than a straight line

Examples: parabolas (U-shape), exponential curves, circles, hyperbolas

From Equations:

How to Identify:

Look at the highest exponent on the variable
Check if variables are multiplied together
Check if variables are in denominators, under roots, or as exponents

Examples:

Equation	Linear?	Reason
\( y = 5x - 3 \)	Yes ✓	Form \( y = mx + b \)
\( y = x^2 + 2 \)	No ✗	Has \( x^2 \) (exponent 2)
\( y = 7 \)	Yes ✓	Horizontal line (m=0)
\( y = 2^x \)	No ✗	Exponential (x is exponent)
\( y = -\frac{1}{2}x + 4 \)	Yes ✓	Form \( y = mx + b \)
\( y = \|x\| \)	No ✗	Absolute value (V-shape)
\( y = \frac{1}{x} \)	No ✗	Variable in denominator
\( y = x^3 - 2x \)	No ✗	Has \( x^3 \) (cubic)
\( y = \sqrt{x} \)	No ✗	Variable under square root

4. Identify Linear and Nonlinear Functions: Tables

Method: Check Rate of Change

Linear: Rate of change (slope) is constant for all pairs of consecutive points

Nonlinear: Rate of change varies; NOT constant

Steps:

Find the change in x between consecutive rows (usually constant in tables)
Find the change in y between consecutive rows
Calculate \( \frac{\Delta y}{\Delta x} \) for each consecutive pair
If all ratios are the SAME → Linear
If ratios are DIFFERENT → Nonlinear

Example 1: Linear Function

x	y	Δy/Δx
1	5	—
2	8	3/1 = 3
3	11	3/1 = 3
4	14	3/1 = 3

✓ All ratios = 3 (constant) → This IS LINEAR

Example 2: Nonlinear Function

x	y	Δy/Δx
0	0	—
1	1	1/1 = 1
2	4	3/1 = 3
3	9	5/1 = 5

✗ Ratios are different (1, 3, 5) → This IS NONLINEAR (it's \( y = x^2 \))

5. Is (x, y) a Solution to the Nonlinear Equation?

Definition: A point (x, y) is a solution to an equation if substituting these values makes the equation true.

Steps:

Substitute the x-value into the equation
Substitute the y-value into the equation
Simplify both sides carefully (follow order of operations)
If both sides are equal → It IS a solution ✓
If both sides are NOT equal → It is NOT a solution ✗

Examples:

Example 1: Is (3, 10) a solution to \( y = x^2 + 1 \)?

Substitute x = 3 and y = 10:

\( 10 = (3)^2 + 1 \)

\( 10 = 9 + 1 \)

\( 10 = 10 \) ✓

Yes, (3, 10) IS a solution.

Example 2: Is (2, 5) a solution to \( y = x^2 \)?

Substitute x = 2 and y = 5:

\( 5 = (2)^2 \)

\( 5 = 4 \) ✗

No, (2, 5) is NOT a solution.

Example 3: Is (4, 2) a solution to \( y = \sqrt{x} \)?

Substitute x = 4 and y = 2:

\( 2 = \sqrt{4} \)

\( 2 = 2 \) ✓

Yes, (4, 2) IS a solution.

Example 4: Is (-1, 3) a solution to \( y = 2^x + 2 \)?

Substitute x = -1 and y = 3:

\( 3 = 2^{-1} + 2 \)

\( 3 = \frac{1}{2} + 2 = 2.5 \) ✗

No, (-1, 3) is NOT a solution.

6. Evaluate a Nonlinear Function

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

Steps:

Start with the function equation
Substitute the given x-value
Apply exponents, roots, or other operations first
Follow order of operations (PEMDAS)
Simplify to get the final answer

Examples:

Example 1: If \( f(x) = x^2 - 3x + 5 \), find f(4).

\( f(4) = (4)^2 - 3(4) + 5 \)

\( f(4) = 16 - 12 + 5 \)

\( f(4) = 9 \)

Answer: f(4) = 9

Example 2: If \( g(x) = 2x^3 + 1 \), find g(-2).

\( g(-2) = 2(-2)^3 + 1 \)

\( g(-2) = 2(-8) + 1 \)

\( g(-2) = -16 + 1 = -15 \)

Example 3: If \( h(x) = |x - 5| \), find h(3) and h(8).

h(3): \( h(3) = |3 - 5| = |-2| = 2 \)

h(8): \( h(8) = |8 - 5| = |3| = 3 \)

Example 4: If \( f(x) = \sqrt{x + 9} \), find f(0) and f(16).

f(0): \( f(0) = \sqrt{0 + 9} = \sqrt{9} = 3 \)

f(16): \( f(16) = \sqrt{16 + 9} = \sqrt{25} = 5 \)

Example 5: If \( f(x) = 3^x \), find f(0), f(2), and f(-1).

f(0): \( f(0) = 3^0 = 1 \)

f(2): \( f(2) = 3^2 = 9 \)

f(-1): \( f(-1) = 3^{-1} = \frac{1}{3} \)

Example 6: If \( f(x) = \frac{12}{x} \), find f(3) and f(-4).

f(3): \( f(3) = \frac{12}{3} = 4 \)

f(-4): \( f(-4) = \frac{12}{-4} = -3 \)

7. Common Types of Nonlinear Functions

Type	General Form	Example	Graph Shape
Quadratic	\( f(x) = ax^2 + bx + c \)	\( f(x) = x^2 - 4 \)	Parabola (U-shape)
Cubic	\( f(x) = ax^3 + ... \)	\( f(x) = x^3 \)	S-shaped curve
Exponential	\( f(x) = a^x \)	\( f(x) = 2^x \)	Rapid growth curve
Absolute Value	\( f(x) = \|x - h\| + k \)	\( f(x) = \|x\| \)	V-shape
Square Root	\( f(x) = \sqrt{x} \)	\( f(x) = \sqrt{x + 2} \)	Half-parabola (sideways)
Reciprocal	\( f(x) = \frac{a}{x} \)	\( f(x) = \frac{1}{x} \)	Hyperbola (two curves)

Quick Reference: Nonlinear Functions

Definition:

A nonlinear function is any function that is NOT of the form \( f(x) = mx + b \)

Key Characteristics:

Graph: NOT a straight line (curved, V-shaped, etc.)
Rate of change: NOT constant (varies)
Exponents: Can have x², x³, etc.
Operations: Can have x in denominators, roots, or as exponents

Identification:

From graph: Any non-straight line
From equation: Look for exponents ≠ 1, variables in denominator/root/exponent
From table: Rate of change is NOT constant