Grades Algebra I Functions — IntroductionComparing Linear, Quadratic & Exponential Functions

Comparing Linear, Quadratic & Exponential Functions

📈 Functions — Introduction

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Comparing Linear, Quadratic & Exponential Functions

Identify and compare the long-run behavior of three fundamental function families.

Linear, quadratic, and exponential functions each grow at fundamentally different rates. Linear growth adds a constant amount per step. Quadratic growth adds an increasing amount (second differences are constant). Exponential growth multiplies by a constant factor each step.

Linear f(x) = mx + b

Constant rate of changeGraph: straight lineFirst differences constant

Quadratic f(x) = ax² + bx + c

Rate of change increasesGraph: parabolaSecond differences constant

Exponential f(x) = abˣ

Rate of change proportional to valueGraph: curved, never zeroRatio of consecutive terms constant

For large x, exponential functions eventually outgrow both linear and quadratic functions.

\text{Exponential growth: } f(x) = P_0 \cdot r^x \quad (r > 1)

\text{As } x \to \infty: \quad mx + b \ll ax^2 \ll b^x \text{ for } b > 1

📊Recognizing Growth from a Table

A table with x = 0,1,2,3 and f(x) = 2, 6, 18, 54 shows each output is multiplied by 3. This is exponential growth: f(x) = 2 · 3ˣ.

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Remember This!

Check first differences for linear behavior, second differences for quadratic, and ratios of consecutive terms for exponential. Only one of these will be constant.

✏️ Try It!

A function has the values f(0)=5, f(1)=8, f(2)=11, f(3)=14. What type of function is this?

Practice with CalcVerse

📈Line Grapher (y = mx + b)

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