Lesson 6 of 8Exponential Functions

Unit 2 · Lesson 6 of 8

Finding an Exponential Equation

Work backwards from information to the rule: find $a$ and $b$ in $y = a \cdot b^{x}$ from two points or a table (even with $x$ -steps of 2 or 5), handle a known asymptote with $y = a \cdot b^{x} + c$ , interpret the parameters, and fit real data with exponential regression.

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Key things to remember

1From the $y$ -intercept and one point: $a$ is the $y$ -intercept, then solve $a \cdot b^{x_{1}} = y_{1}$ for $b$ (positive root only).
2From two points: $\frac{y _{2}}{y _{1}} = b^{x_{2} - x_{1}}$ cancels $a$ . Find $b$ first, then $a = \frac{y _{1}}{b ^{x_{1}}}$ .
3Table with step $h$ and ratio $r$ : $b = r^{1/ h}$ , or write the model exactly as $y = a \cdot r^{x / h}$ .
4Known asymptote $y = c$ : subtract $c$ from every $y$ -value, find $a$ and $b$ from $y - c = a \cdot b^{x}$ . The $y$ -intercept is then $a + c$ .
5 $a$ is the starting value and $b$ the factor per unit; $b - 1$ is the percent change per unit ( $b = 0.956$ means about 4.4% decrease).
6Exponential regression fits $y = a \cdot b^{x}$ to all the data; check the residual pattern and keep predictions close to the data's range.

So far you have started with an exponential equation and worked out its graph, its asymptote and its values. Real problems usually run the other way. A biologist has a few population counts, an engineer has temperature readings from a cooling part, a buyer knows what a car cost new and what it is worth now. They need the equation that produced those numbers. The good news is that $y = a \cdot b^{x}$ has only two unknowns, so two pieces of information are enough to pin it down. The skill is choosing the order: find $b$ from how the values multiply, then find $a$ . If the graph levels off at some value other than zero, you also need the asymptote $y = c$ from Transformations of Exponential Functions, and you subtract it off first.

Fig 1From (1, 10) and (4, 80): three unit steps in x multiply y by b three times, so b cubed is 80 divided by 10, which is 8, and b is 2. One step back to x = 0 divides by 2 and gives a = 5.

Case 1: the y-intercept and one more point

The $y$ -intercept hands you $a$ immediately, because $a \cdot b^{0} = a \cdot 1 = a$ . Substitute the second point and solve for $b$ . If a model passes through $(0, 3)$ and $(4, 48)$ , then $a = 3$ and $3 b^{4} = 48$ , so $b^{4} = 16$ and $b = 416 = 2$ . The equation $b^{4} = 16$ is also solved by $b = - 2$ , but an exponential base must be positive, so reject it. The model is $y = 3 (2)^{x}$ . Roots that are not whole numbers are fine too: through $(0, 500)$ and $(3, 864)$ , $b^{3} = \frac{864}{500} = 1.728$ , and $3 1.728 = 1.2$ (a calculator does this as $1.72 8^{1/3}$ ).

Case 2: two points, neither on the y-axis

When neither point has $x = 0$ , write both equations and divide them. The $a$ 's cancel, leaving one equation in $b$ alone. Through $(2, 36)$ and $(5, 972)$ : $36 = a b^{2}$ and $972 = a b^{5}$ . Dividing, $\frac{972}{36} = b^{5 - 2}$ , so $b^{3} = 27$ and $b = 3$ . Then substitute back into either equation: $36 = a (3)^{2} = 9 a$ , so $a = 4$ , and the model is $y = 4 (3)^{x}$ . Check with the other point: $4 (3)^{5} = 4 (243) = 972$ ✓. The ratio of the $y$ -values is $b$ raised to the gap in $x$ , which is exactly the staircase in the figure.

\frac{y _{2}}{y _{1}} = \frac{a b ^{x_{2}}}{a b ^{x_{1}}} = b^{x_{2} - x_{1}}

Case 3: a table with a constant ratio

A table is just many points, so the same ideas apply, with two traps. First, $a$ is the $y$ -value at $x = 0$ , which may not be the first row: if $x = 2, 4, 6$ gives $y = 12, 48, 192$ , the ratio 4 spans a step of 2, so $b^{2} = 4$ , $b = 2$ , and $a = \frac{12}{2 ^{2}} = 3$ , giving $y = 3 (2)^{x}$ . Second, when the $x$ -step is $h$ , the ratio is $b^{h}$ , not $b$ . The table below has a step of 5 and a ratio of 1.5, so $b = 1. 5^{1/5} \approx 1.0845$ . You can also keep the ratio and put the step in the exponent, $y = 200 (1.5)^{x /5}$ , which is exact and shows the meaning (×1.5 every 5 units). Do not round $b$ too early: $1.0845$ rounded to $1.08$ looks harmless, but it compounds, and $200 (1.08)^{20} \approx 932$ while the true value at $x = 20$ is $1012.5$ . Keep four or more decimals in $b$ , or use the exact form $a \cdot r^{x / h}$ , where $r$ is the table's ratio and $h$ its step.

x	y	Ratio over each step of 5
0	200
5	300	×1.5
10	450	×1.5
15	675	×1.5
20	1012.5	×1.5

Ratio 1.5 per 5 units: y = 200(1.5) to the power x/5, or y ≈ 200(1.0845) to the x per single unit.

Case 4: a graph with a shifted asymptote

If the graph levels off at $y = c$ instead of $y = 0$ , the model is $y = a \cdot b^{x} + c$ , and the ratio test fails on the raw $y$ -values. The fix is to subtract $c$ from every $y$ -value first: $y - c = a \cdot b^{x}$ is an ordinary exponential, so Cases 1 to 3 apply to the shifted values. Suppose a graph has asymptote $y = - 4$ and passes through $(0, 2)$ and $(2, 50)$ . Shift up by 4: the points become $(0, 6)$ and $(2, 54)$ . So $a = 6$ , $6 b^{2} = 54$ , $b^{2} = 9$ , $b = 3$ , and the model is $y = 6 (3)^{x} - 4$ . Notice $a$ is not the $y$ -intercept here: the intercept is $a + c = 6 + (- 4) = 2$ . Cooling problems work exactly this way, with room temperature as $c$ .

Parameter	Meaning	Example: $V = 32000 (0.8)^{t /5}$
$a$	The value at $x = 0$ (when $c = 0$ ): the starting amount.	The car cost 32 000 dollars new.
$b$	The factor per one unit of $x$ . $b - 1$ is the percent change per unit.	Per year $b = 0. 8^{1/5} \approx 0.956$ : about 4.4% lost each year.
$r$ in $r^{x / h}$	The factor per $h$ units of $x$ .	It keeps 80% of its value every 5 years.
$c$	The level the quantity approaches: room temperature, a minimum value.	0 here: the model says the value approaches 0 dollars.

What each parameter means in context, using a car that keeps 80% of its value every 5 years.

Modelling real data with exponential regression

Real data never fit perfectly, and using just two of the points throws the rest away. A graphing tool fits a curve to all of them. In Desmos type the data in a table and enter $y_{1} \sim a \cdot b^{x_{1}}$ ; on a graphing calculator use ExpReg. Take a town's population, with $t$ in years since 2010: $8200, 8870, 9590, 10380, 11220, 12150$ at $t = 0, 2, 4, 6, 8, 10$ . The ratios over each 2-year step are all close to 1.08, so an exponential is sensible. Regression gives $a \approx 8200$ and $b \approx 1.040$ : the town grows about 4% per year, and the model predicts about 14 800 people for 2025 ( $t = 15$ ).

Plot the data and check the shape and the ratios first. Regression will fit an exponential to anything, including data that are really linear.
If the data level off at a non-zero value (cooling toward room temperature), subtract that value from every $y$ first. Regression for $y = a \cdot b^{x}$ assumes the asymptote is $y = 0$ .
Run the regression and read off $a$ and $b$ . Interpret both in context: starting amount, and factor (or percent change) per unit of $x$ .
Judge the fit: the curve should pass close to every point, with points above and below it scattered rather than all on one side. $r^{2}$ near 1 helps but does not prove the model right.
State a reasonable domain. Predictions slightly beyond the data are fairly safe; predictions far beyond are not. The town model gives roughly 280 000 people by 2100, which no one should believe, because real growth slows as land, housing and jobs run out.

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

A fish population was 500 when first counted and 864 three years later. Assuming exponential growth, find the model, interpret

b

, and predict the population after 6 years.

Example 2

An exponential function

y = a \cdot b^{x}

passes through

(- 1, 40)

and

(2, 5)

. Find its equation.

Example 3

A car's value is recorded every 5 years: 32 000, 25 600, 20 480 and 16 384 dollars at

t = 0, 5, 10, 15

. Find an exponential model, the yearly percent loss, and the value at

t = 8

Example 4

A metal part comes out of an oven at 180 °C into a 20 °C room. After 10 minutes it is 100 °C. Model its temperature, and find when it reaches 40 °C.

Where marks get lost

Taking $a$ to be the first $y$ -value in a table that does not start at $x = 0$ . If the first row is $x = 2$ , step back to $x = 0$ by dividing by $b^{2}$ .
Using a table's ratio as $b$ when the $x$ -step is 2 or 5. A ratio of 1.5 per 5 units means $b = 1. 5^{1/5} \approx 1.0845$ , not 1.5.
Forgetting to subtract the asymptote. For a drink cooling toward 21 °C, the ratios of the raw temperatures are not constant; the ratios of (temperature − 21) are.
Keeping a negative root: $b^{2} = 9$ gives $b = 3$ only, since an exponential base must be positive.
Rounding $b$ early, or trusting a regression model far outside the range of the data (a town of 8000 that "reaches 280 000 by 2100").

Flashcards 8

a

from the

y

-intercept

Two points, neither at x = 0

Table ratio

r

over an

x

-step of

h

Known asymptote y = c

+ 4 more

Practice

A town's population model

P = 8200 (1.04)^{t}

was fitted to data from 2010 (

t = 0

) to 2020. Which prediction from it deserves the most trust?

A quantity is 2000 at

t = 0

and 1458 at

t = 3

, and decays exponentially. By what percent does it decrease per unit of time?

An exponential function

y = a \cdot b^{x}

passes through

(1, 15)

and

(3, 135)

. Find

a

An exponential function

y = a \cdot b^{x}

passes through

(2, 20)

and

(4, 5)

. Find

a

A table of an exponential function has

x = 0, 2, 4, 6

and

y = 7, 63, 567, 5103

. Find

b

A table of an exponential function has

x = 0, 5, 10

and

y = 400, 600, 900

. Find

b

y = a \cdot b^{x}

, to 4 decimal places.

An exponential function passes through

(0, 3)

and

(1, 6)

. Write its equation in the form

y = a (b)^{x}

A graph has horizontal asymptote

y = 5

and passes through

(0, 8)

and

(1, 17)

. Which equation matches it?

A drink cools in a 20 °C room. It is 84 °C at t = 0 and 68 °C at t = 1 minute. Assuming exponential cooling toward room temperature, what is its temperature at t = 3 minutes?

°C

Coffee cools in a 21 °C room. Before running exponential regression on the readings, why should you subtract 21 from every temperature?

A drink's temperature is modelled by

T = 60 (0.9)^{t} + 22

, with

t

in minutes. Match each part to its meaning.

0.9

Order the steps to find

y = a \cdot b^{x}

through

(2, 36)

and

(5, 972)

1Check: $4 (3)^{5} = 972$
2Substitute back: $36 = 9 a$ , so $a = 4$
3Take the cube root: $b = 3$
4Write $36 = a b^{2}$ and $972 = a b^{5}$
5Divide to cancel $a$ : $b^{3} = 27$

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