Topic 2.2 Notes – Change in Linear and Exponential Functions

Linear and Exponential Change

The whole topic is about how outputs change over equal input intervals.

• Linear change means a constant difference.
  • If $x$ goes up by 1 each time, the outputs might go $4, 7, 10, 13$.
  • You keep adding 3.
• Exponential change means a constant ratio.
  • Outputs might go $4, 8, 16, 32$.
  • You keep multiplying by 2.

That is the additive vs multiplicative split:

• Linear uses repeated addition
• Exponential uses repeated multiplication

The standard forms show that structure:

$$f(x)=mx+b$$

• $b$ is the initial value because $f(0)=b$
• $m$ is the constant rate of change

$$f(x)=ab^x$$

• $a$ is the initial value because $f(0)=a$
• $b$ is the constant multiplicative factor

A quick sequence connection:

• Arithmetic sequence $\leftrightarrow$ linear structure
• Geometric sequence $\leftrightarrow$ exponential structure

These examples show the same starting value, $4$, but different kinds of change as $x$ increases by 1.

On the left, each output increases by 3, so the sequence matches a linear function. On the right, each output doubles, so the sequence matches an exponential function.

For exponential functions, the base must satisfy:

• $b>0$
• $b\neq 1$

And for geometric sequences used in this comparison, the ratio cannot be $0$ or $1$.

Matching Sequences to Functions

When an arithmetic sequence is really a line

These are parallel forms:

$$a_n=a_0+dn \qquad\text{and}\qquad f(x)=mx+b$$

The pieces match:

• $a_0 \leftrightarrow b$ because both are outputs at input $0$
• $d \leftrightarrow m$ because both tell how much gets added per input unit

Point-based forms also match:

$$a_n=a_k+d(n-k) \qquad\text{and}\qquad f(x)=y_i+m(x-x_i)$$

If you know a term and the common difference, or a point and the slope, you can build the model.

When a geometric sequence is really an exponential

These are the matching forms:

$$g_n=g_0r^n \qquad\text{and}\qquad f(x)=ab^x$$

The pieces match:

• $g_0 \leftrightarrow a$
• $r \leftrightarrow b$

Point-based forms:

$$g_n=g_kr^{n-k} \qquad\text{and}\qquad f(x)=y_i b^{x-x_i}$$

Domains and graphs

This is where students often lose points.

• Sequences use whole-number inputs only, so their graphs are discrete points
• Functions use real-number inputs, so their graphs are continuous

A sequence and its corresponding function can agree at integer inputs and still be different because the domains differ. On a quiz or AP-style question, do not connect sequence points unless the problem is clearly extending the sequence to a function.

How to Tell Which Model Fits

Use equal-length input intervals only.

$$\text{Linear test: constant } \Delta y \qquad \text{Exponential test: constant } \frac{\text{new}}{\text{old}}$$

• If outputs change by the same amount, the model is linear.
• If outputs change by the same factor, the model is exponential.

Be careful with interval length.

• If $x$ changes by 2, slope is $\frac{\Delta y}{2}$, not just $\Delta y$.
• If the ratio over 2 input units is 9, then $b^2=9$, so $b=3$, not 9.

Two values alone do not prove whether a relationship is linear or exponential. They only determine the model once the type is known.

Building the Function from Given Information

One point and a known constant

Linear:

$$f(x)=y_i+m(x-x_i)$$

Example with point $(2,11)$ and slope $-2$:

$$f(x)=11-2(x-2)$$

Exponential:

$$f(x)=y_i b^{x-x_i}$$

Example with point $(1,12)$ and factor $2$:

$$f(x)=12\cdot 2^{x-1}$$

The known output is only the initial value when the input is $0$.

Two values for a linear function

Find slope first:

$$m=\frac{y_2-y_1}{x_2-x_1}$$

Using $(2,11)$ and $(7,1)$:

$$m=\frac{1-11}{7-2}=-2$$

So

$$f(x)=11-2(x-2)$$

Same idea for arithmetic sequences, with

$$d=\frac{a_\ell-a_k}{\ell-k}$$

Two values for an exponential function

Use the ratio equation:

$$\frac{y_2}{y_1}=b^{x_2-x_1}$$

Then solve for the base:

$$b=\left(\frac{y_2}{y_1}\right)^{1/(x_2-x_1)}$$

Using $(1,12)$ and $(4,96)$:

$$\frac{96}{12}=8=b^3 \quad\Rightarrow\quad b=2$$

So

$$f(x)=12\cdot 2^{x-1}$$

For geometric sequences, the matching formula is

$$r=\left(\frac{g_\ell}{g_k}\right)^{1/(\ell-k)}$$

Common Mistakes and Fast Checks

• In $mx+b$, $b$ is the intercept. In $ab^x$, $b$ is the base.
• A raw output difference is not the slope unless the input change is 1.
• A ratio over 2 or 3 input units is not automatically the exponential base.
• A known point $(x_i,y_i)$ gives the initial value only when $x_i=0$.
• To build a real-valued exponential model from two points, the outputs must be nonzero and have the same sign.
• Two points can fit both a line and an exponential curve.