3.2 Linear & Exponential Functions

Slope and Rate of Change

Slope measures how fast a line rises or falls. It is the constant rate of change:

m = Δy / Δx = (y₂ − y₁) / (x₂ − x₁)

Worked example. Find the slope through (2, 5) and (6, 13):

m = (13 − 5) / (6 − 2) = 8 / 4 = 2.

Order matters only that you stay consistent — subtract the y-values and x-values in the same order. A positive slope rises; a negative slope falls; a zero slope is horizontal; an undefined slope is vertical (Δx = 0).

In a real context, slope is a unit rate with units: dollars per month, feet per second, gallons per minute. When a Regents item gives a model and asks "what does the slope represent," answer with that rate and its units, not just the number.

Three Forms of a Linear Equation

In y = mx + b, b is the y-intercept (the value when x = 0) and m is the slope.

Writing a model. A gym charges a $25 sign-up fee plus $15 per month. Cost C after m months: C = 15m + 25. Here 25 is the initial value (y-intercept) and 15 is the rate (slope). Regents items often ask you to interpret each number in context.

Point-Slope in Action

Worked example. Write a line through (4, 1) with slope −3.

• Point-slope: y − 1 = −3(x − 4)
• Distribute: y − 1 = −3x + 12
• Solve for y: y = −3x + 13

So the slope-intercept form is y = −3x + 13, with y-intercept 13. Point-slope is the fastest start when you have a point and a slope but not the intercept.

Exponential Functions: y = ab^x

An exponential function has the form y = ab^x, where a is the initial value (the y-intercept) and b is the constant multiplier (base).

Unlike linear functions that add a fixed amount, exponentials multiply by a fixed factor each step.

• Growth: b > 1. Write b = 1 + r, where r is the growth rate. A 7% increase gives b = 1.07.
• Decay: 0 < b < 1. Write b = 1 − r. A 12% decrease gives b = 0.88.

Worked example. A $20,000 car loses 15% of its value yearly. Model: V = 20000(0.85)^t, since b = 1 − 0.15 = 0.85. After 3 years: V = 20000(0.85)³ ≈ 20000(0.614) ≈ $12,283.

The initial value a is always the y-intercept: it is the amount when x = 0, before any growth or decay. Reading a, b, and the percent rate r straight out of a written situation is a high-frequency Regents skill, so practice converting "increases 7%" → b = 1.07 and "decreases 12%" → b = 0.88 instantly.

Linear vs. Exponential Growth

A classic Regents comparison: a line and an exponential start close, but exponential growth eventually overtakes any linear growth.

Key tell: in a table, a function is linear if successive y-values change by the same difference, and exponential if they change by the same ratio. Check first differences vs. ratios to classify quickly.

Average Rate of Change

For a curve, the average rate of change over [a, b] is the slope of the secant line joining the endpoints:

AROC = [f(b) − f(a)] / (b − a)

For a straight line this equals the slope everywhere; for a parabola or exponential it varies by interval.

Worked example. For f(x) = x² on [1, 4]: f(4) = 16, f(1) = 1, so AROC = (16 − 1)/(4 − 1) = 15/3 = 5. This is the average growth per unit x across that interval, not the slope at a single point.

When the function is given as a table, read the endpoint outputs directly and apply the same formula. The Regents likes to compare AROC across two different intervals to show that exponentials accelerate — the AROC on a later interval is larger than on an earlier one, evidence that the curve is steepening.