Linear vs Exponential Models on the SAT: Constant Difference vs Constant Ratio

The fastest reliable distinction is:

• linear means equal input steps create a constant difference in output;
• exponential means equal input steps create a constant ratio in output.

That one idea solves many SAT questions involving:

• tables;
• word problems;
• growth and decay;
• equations;
• graphs;
• repeated percent change.

The important word is constant.

A quantity can increase quickly and still be linear. A quantity can start slowly and still be exponential.

Do not classify by how steep the graph looks. Classify by the pattern of change.

Current SAT connection: comparing linear and exponential models

College Board's current Math specifications include linear functions in Algebra, exponential functions in Advanced Math, and comparison of linear and exponential growth within Problem-Solving and Data Analysis. That makes this a cross-domain modeling skill: the representation may be an equation, table, graph, or real-world context.

Use SAT Linear Equations for constant-rate models, SAT Exponential Growth and Decay for multiplicative models, and SAT Slope From Tables when the relationship is presented as data.

Linear model

A linear function can be written as:

$$y=mx+b.$$

Here:

• $m$ is the constant rate of change;
• $b$ is the starting value when $x=0$.

If $x$ increases by 1 each step, the output changes by $m$ each step.

Example:

$$y=7x+20.$$

Outputs increase by 7 for every increase of 1 in $x$.

That is a constant difference.

Exponential model

An exponential function can be written as:

$$y=a(b)^x.$$

Here:

• $a$ is the initial value at $x=0$;
• $b$ is the multiplicative factor per equal input step.

If $x$ increases by 1, the output is multiplied by $b$.

Example:

$$y=20(1.07)^x.$$

Each step multiplies the output by:

$$1.07.$$

That means 7% growth per step.

The difference between outputs is not constant. The ratio is.

Table test: constant difference

Consider:

Differences:

$$17-12=5$$ $$22-17=5$$ $$27-22=5$$

Constant difference 5.

So the model is linear:

$$y=5x+12.$$

Table test: constant ratio

Consider:

Ratios:

$$\frac{18}{12}=1.5$$ $$\frac{27}{18}=1.5$$ $$\frac{40.5}{27}=1.5$$

Constant ratio 1.5.

So the model is exponential:

$$y=12(1.5)^x.$$

Worked example 1: identify the model from words

A gym membership account begins with a balance of $300. Each month, $25 is added.

This is linear.

Why?

The same amount is added each month:

$$+25.$$

Model:

$$B(m)=300+25m.$$

Now compare:

A savings account begins with $300 and increases by 5% each month.

This is exponential.

Each month the previous balance is multiplied by:

$$1.05.$$

Model:

$$B(m)=300(1.05)^m.$$

The difference is not “money vs percentages.”

It is add the same amount vs multiply by the same factor.

Additive language usually signals linear

Common linear phrases include:

• increases by 8 each year;
• decreases by 12 each hour;
• earns $15 per item;
• loses 3 liters per minute;
• starts at 40 and adds 6 per period.

These describe constant additive change.

Multiplicative language usually signals exponential

Common exponential phrases include:

• grows by 8% each year;
• decreases by 12% each hour;
• doubles every 5 days;
• is multiplied by 1.4 each period;
• retains 82% of its value each year.

These describe repeated multiplicative change.

Worked example 2: percent wording

A population is 50,000 and grows by 3% per year.

A common wrong model is:

$$P(t)=50000+0.03t.$$

That adds 0.03 people per year, which makes no sense.

Another wrong model is:

$$P(t)=50000+1500t.$$

That adds 3% of the original population every year. That is linear, not repeated percentage growth.

Correct exponential model:

$$P(t)=50000(1.03)^t.$$

Each year, the new population becomes 103% of the previous year's population.

Worked example 3: decay

A machine worth $40,000 loses 18% of its value each year.

It retains:

$$100\%-18\%=82\%.$$

So the decay factor is:

$$0.82.$$

Model:

$$V(t)=40000(0.82)^t.$$

Do not use:

$$0.18$$

as the factor.

That would mean the machine retains only 18% each year.

Initial value in linear models

For:

$$y=mx+b,$$

when:

$$x=0,$$

we get:

$$y=b.$$

So $b$ is the initial value.

Example:

$$y=4x+30.$$

Initial value:

$$30.$$

Rate of change:

$$4.$$

Initial value in exponential models

For:

$$y=a(b)^x,$$

when:

$$x=0,$$ $$y=a(b)^0=a.$$

So $a$ is the initial value.

Example:

$$y=80(1.2)^x.$$

Initial value:

$$80.$$

Growth factor:

$$1.2.$$

Growth rate:

$$20\%.$$

Equal input spacing matters

A constant ratio test only works directly when the $x$-values are equally spaced.

Suppose:

Ratios:

$$\frac{40}{10}=4$$

and:

$$\frac{160}{40}=4.$$

The input increases by 2 each time.

So the output multiplies by 4 every 2 units of $x$.

If you want the factor for one unit, solve:

$$b^2=4.$$

For a positive growth factor:

$$b=2.$$

So:

$$y=10(2)^x.$$

Worked example 4: unequal x-spacing trap

Suppose:

Ratios:

$$\frac{12}{6}=2$$

and:

$$\frac{48}{12}=4.$$

At first glance, the ratios are not constant.

But the input steps are different:

• 1 to 2 is one step;
• 2 to 4 is two steps.

If the per-step factor is 2, then over two steps:

$$2^2=4.$$

So this is perfectly consistent with an exponential model.

Always compare change over equal input intervals.

Graph shape is a clue, not proof

A nonhorizontal straight line is linear.

An exponential graph often curves and changes steepness.

But on a small viewing window, part of an exponential graph can look nearly straight.

Use equations, tables, or rate descriptions when available.

Linear slope stays constant

For a linear function:

$$y=mx+b,$$

the slope is always $m$.

Between any two points:

$$\frac{\Delta y}{\Delta x}=m.$$

That is why equal input increments create equal output differences.

Exponential absolute change does not stay constant

Take:

$$y=100(1.1)^x.$$

Values:

$$100,\ 110,\ 121,\ 133.1,\ldots$$

Differences:

$$10,\ 11,\ 12.1,\ldots$$

The amount added grows.

But ratios remain:

$$1.1.$$

That is the fingerprint of exponential growth.

Worked example 5: choose the equation

A bacteria culture contains 600 cells initially and doubles every 3 hours.

Which model fits?

A linear model like:

$$600+2t$$

does not represent doubling.

If $t$ is measured in 3-hour periods:

$$N(t)=600(2)^t.$$

If $h$ is measured in hours:

$$N(h)=600(2)^{h/3}.$$

The exponent must count the number of doubling periods.

Time-unit consistency

This is a common SAT modeling issue.

If a quantity changes by 4% per month, and $t$ is measured in years, you cannot simply write:

$$(1.04)^t.$$

There are 12 months in a year.

If $t$ is years, the number of monthly periods is:

$$12t.$$

So:

$$A(t)=A_0(1.04)^{12t}.$$

Always match the exponent to the change interval.

Worked example 6: linear depreciation vs percentage depreciation

Two cars both start at $30,000.

Car A loses $2,000 each year.

$$A(t)=30000-2000t.$$

Linear.

Car B loses 10% of its value each year.

$$B(t)=30000(0.90)^t.$$

Exponential.

After one year:

$$A(1)=28000$$ $$B(1)=27000.$$

After two years:

$$A(2)=26000$$ $$B(2)=24300.$$

The percentage-based decrease changes in dollar amount because it is calculated from the current value each time.

“Per” does not automatically mean linear

“$8 per ticket” usually implies linear.

But “increases by 8% per year” is exponential.

Focus on whether the repeated operation is:

• add/subtract a fixed amount;
• multiply by a fixed factor.

Worked example 7: identify from equations

Which are linear?

1. $y=7x-2$: linear.

2. $y=7(2)^x$: exponential.

3. $y=7+2x$: linear.

4. $y=7(0.8)^x$: exponential.

5. $y=x^2+7$: neither linear nor exponential.

Not every nonlinear model is exponential.

A major trap: exponential does not mean “has an exponent somewhere”

Consider:

$$y=x^2.$$

The variable is the base and the exponent is fixed.

That is a quadratic, not an exponential function.

In an exponential function such as:

$$y=2^x,$$

the variable appears in the exponent.

A second trap: linear does not mean “increases”

A decreasing line is still linear.

Example:

$$y=100-6x.$$

Constant difference:

$$-6$$

per unit.

Linear functions can increase, decrease, or be constant.

Exponential decay still has a positive base factor

A standard decay model looks like:

$$A(t)=A_0(b)^t$$

with:

$$0<b<1.$$

Example:

$$0.93$$

means retaining 93%, or losing 7%, each period.

The factor itself is not negative.

Worked example 8: infer model from successive values

A quantity has values:

$$200,\ 170,\ 144.5,\ 122.825.$$

Differences:

$$-30,\ -25.5,\ -21.675.$$

Not constant.

Ratios:

$$\frac{170}{200}=0.85$$ $$\frac{144.5}{170}=0.85$$ $$\frac{122.825}{144.5}=0.85.$$

Exponential decay with factor:

$$0.85.$$

Decay rate:

$$15\%.$$

Worked example 9: infer a linear equation from a table

Equal x-step:

$$+3.$$

Equal y-step:

$$+12.$$

Slope:

$$m=\frac{12}{3}=4.$$

Use point $(2,11)$:

$$11=4(2)+b.$$ $$b=3.$$

So:

$$\boxed{y=4x+3}.$$

Worked example 10: infer an exponential equation from a table

Constant ratio:

$$3.$$

Initial value at $x=0$:

$$5.$$

So:

$$\boxed{y=5(3)^x}.$$

Compare average rate of change

A linear function has the same average rate of change over any equal-length interval.

An exponential function generally does not.

This can be another diagnostic if the question gives interval changes rather than a full table.

Common trap 1: using percent as a decimal add-on

If a quantity grows 6% each year, do not write:

$$A=A_0+0.06t.$$

Use:

$$A=A_0(1.06)^t.$$

Common trap 2: using growth rate instead of growth factor

6% growth means:

$$r=0.06.$$

Growth factor:

$$1+r=1.06.$$

The base of the exponential model is usually:

$$1.06,$$

not:

$$0.06.$$

Common trap 3: forgetting 100% baseline

A 12% increase means the new amount is:

$$112\%$$

of the old amount.

So multiply by:

$$1.12.$$

A 12% decrease means retain:

$$88\%.$$

So multiply by:

$$0.88.$$

Common trap 4: comparing differences with unequal intervals

If x-values are not equally spaced, raw output differences are not directly comparable.

Normalize by the input interval.

For linear models, compute slope.

For exponential models, compare growth over matching intervals.

Common trap 5: assuming exponential always grows

$$y=100(0.7)^x$$

is exponential decay.

Exponential refers to the structure, not the direction.

Common trap 6: assuming any curved graph is exponential

Quadratics, radicals, rational functions, and other nonlinear functions curve too.

Use the equation or change pattern.

A practical decision tree

If you are deciding between linear and exponential:

Step 1: inspect the wording

Fixed amount per interval?

Likely linear.

Fixed percent or fixed multiplier per interval?

Likely exponential.

Step 2: inspect a table

Equal x-steps:

• constant y-difference → linear;
• constant y-ratio → exponential.

Step 3: inspect the equation

• variable to first power in $mx+b$ → linear;
• variable in exponent → exponential.

Step 4: inspect the graph

Straight line → linear.

Curved multiplicative growth/decay pattern → possibly exponential.

Confirm with other evidence if available.

Mini drill

1.

A plant grows 2.5 cm each week.

Linear.

2.

A population grows 2.5% each week.

Exponential.

3.

Outputs:

$$8,13,18,23$$

Linear because difference is 5.

4.

Outputs:

$$8,12,18,27$$

Exponential because ratio is 1.5.

5.

$$f(x)=90(0.96)^x$$

Exponential decay by:

$$4\%.$$

6.

$$g(x)=90-4x$$

Linear decrease by:

$$4$$

units per x-unit.

The deeper lesson

Linear and exponential models differ in what stays stable.

For linear models, the absolute change stays stable.

For exponential models, the relative change stays stable.

That is why repeated percentage growth compounds: each new change is based on the new amount, not the original one.

Once you see that difference, tables, formulas, graphs, and word problems become different representations of the same idea.

FAQ: linear vs exponential models on the SAT

What is the fastest way to tell linear from exponential in a table?

For equal x-steps, linear outputs have a constant difference; exponential outputs have a constant ratio.

Does a percentage increase always mean exponential?

Repeated percentage change usually does, because each period multiplies the current amount by the same factor.

Can a decreasing relationship be linear?

Yes. A constant negative additive rate is linear.

Can an exponential function decrease?

Yes. A factor between 0 and 1 creates exponential decay.

Is x² exponential because it has an exponent?

No. In an exponential function, the variable appears in the exponent, such as $2^x$. $x^2$ is quadratic.

What if the x-values in a table are not equally spaced?

Compare change over equal intervals or normalize the linear change by (Delta x). For exponential data, account for how many input steps occur between rows.

Can Desmos help compare linear and exponential models?

Yes, especially when the question involves graphs or fitted models. But tables often reveal the pattern faster by hand.

Final takeaway

Remember:

linear ⇒ constant difference

exponential ⇒ constant ratio

Then translate the context:

• add the same amount → linear;
• multiply by the same factor → exponential;
• fixed percentage change → exponential.

Do not classify a model by whether it “looks fast.” Classify it by the operation repeated over equal input intervals.

How to practice this on MastaPrep

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Related MastaPrep guides

• SAT Linear Equations
• SAT Exponential Growth and Decay
• Growth Rate vs Growth Factor
• SAT Functions
• Slope From Tables
• SAT Percentages
• Desmos for the Digital SAT

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