Exponential Functions
Maya posts a short video. When it has 50 views, it starts going viral, and the number of views triples every day. That is not "add 100 views a day" like a straight line. The views get multiplied by the same number again and again. This is exponential growth, and the variable now lives in the exponent. By the end of this page you will know how many views Maya's video has after 4 days.
What is an exponential function?
Compare this with a power function like $x^3$. In $x^3$ the variable is the base. In $3^x$ the variable is the exponent. A linear function adds the same amount each step, and an exponential function multiplies by the same factor each step.
| $x$ | $0$ | $1$ | $2$ | $3$ | $4$ |
|---|---|---|---|---|---|
| $2x + 1$ (linear: add $2$) | $1$ | $3$ | $5$ | $7$ | $9$ |
| $2^x$ (exponential: multiply by $2$) | $1$ | $2$ | $4$ | $8$ | $16$ |
Drag the slider and watch two things that never change:
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Every graph $y = b^x$ passes through $(0, 1)$, because $b^0 = 1$ for any base.
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Every graph $y = b^x$ also passes through $(1, b)$.
To evaluate, remember the exponent rules: $b^0 = 1$ and $b^{-n} = \dfrac{1}{b^n}$.
Given the exponential function $f(x) = 2^x$, calculate the images for the values $x = -2$, $x = 0$ and $x = 3$.
Given the exponential function $f(x) = 2^x$, calculate the images for the values $x = -2$, $x = 0$ and $x = 3$.
Growth and decay
Try a base below $1$, such as $b = 0.5$. The graph is the mirror image of $y = 2^x$ across the $y$-axis. That is because $\left(\tfrac{1}{2}\right)^x = 2^{-x}$. In the same way, $y = \left(\tfrac{1}{3}\right)^x$ is the mirror image of $y = 3^x$.
To read a graph, look from left to right. If the curve rises, the base is greater than $1$. If it falls, the base is between $0$ and $1$.
Indicate, reasoning your answer, which of the following exponential functions are increasing and which are decreasing: a) $f(x) = 1,2^x$ b) $g(x) = 0,7^x$ c)…
Indicate, reasoning your answer, which of the following exponential functions are increasing and which are decreasing:
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a) $f(x) = 1,2^x$
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b) $g(x) = 0,7^x$
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c) $h(x) = \left(\frac{4}{3}\right)^x$
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d) $i(x) = \left(\frac{2}{5}\right)^x$
Observe graphs A and B represented in the following figure. Match each of them with its corresponding analytical expression: $y = 3^x$ or…
Observe graphs A and B represented in the following figure. Match each of them with its corresponding analytical expression: $y = 3^x$ or $y = \left(\frac{1}{3}\right)^x$.
Domain, range, y-intercept, and asymptote
Look at the graph. On the left the curve gets closer and closer to the $x$-axis but never touches it. That line is a horizontal asymptote: a line the graph approaches but never reaches.
Determine the domain, the range and the point of intersection with the y-axis of the exponential function $f(x) = 0,5^x$.
Determine the domain, the range and the point of intersection with the y-axis of the exponential function $f(x) = 0,5^x$.
Solving equations by matching bases
Exponential functions never repeat an output: they are always rising or always falling. So if two powers with the same base are equal, their exponents must be equal.
The plan: write both sides as powers of the same base, then set the exponents equal and solve the resulting equation. Useful facts: $81 = 3^4$, $\dfrac{1}{8} = 2^{-3}$, $125 = 5^3$.
Find the value of $x$ in the following elementary exponential equalities: a) $2^x = 64$ b) $3^x = \frac{1}{9}$
Find the value of $x$ in the following elementary exponential equalities:
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a) $2^x = 64$
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b) $3^x = \frac{1}{9}$
Solve the following exponential equations by equating the bases on both sides of the equality: a) $5^{2x-1} = 125$ b) $2^{x+3} = \frac{1}{4}$
Solve the following exponential equations by equating the bases on both sides of the equality:
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a) $5^{2x-1} = 125$
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b) $2^{x+3} = \frac{1}{4}$
Equations with a common factor
Sometimes the equation has a sum of powers with the same base but different exponents, like $2^x + 2^{x+2} = 40$. You cannot match bases right away. First rewrite every term using the property $b^{x+n} = b^x \cdot b^n$, then factor out the common power.
If there are three terms, such as $2^{x-1} + 2^{x-2} + 2^{x-3}$, factor out the one with the smallest exponent, $2^{x-3}$, and rewrite the other terms as $2^{x-3}$ times a number.
Solve the following exponential equation by first taking out a common factor: $3^x + 3^{x+1} = 36$
Solve the following exponential equation by first taking out a common factor:
$3^x + 3^{x+1} = 36$
Solve the following exponential equation by applying the properties of exponents and taking out a common factor: $2^{x-1} + 2^{x-2} + 2^{x-3} = 56$
Solve the following exponential equation by applying the properties of exponents and taking out a common factor:
$2^{x-1} + 2^{x-2} + 2^{x-3} = 56$
Finding the equation from information
To write $f(x) = a \cdot b^x$ you need the two numbers $a$ and $b$. Use the information you are given.
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The y-intercept (value at $x = 0$) is $a$.
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Substitute the other point into the formula and solve for $b$.
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In a table with $x$ going up by $1$, divide each output by the previous one. That ratio is $b$.
The graph of an exponential function of the type $f(x) = a^x$ passes through the point $(3, 8)$. Calculate the value of the base $a$.
The graph of an exponential function of the type $f(x) = a^x$ passes through the point $(3, 8)$. Calculate the value of the base $a$.
The evolution of a certain insect plague follows an exponential model of the form $N(t) = k \cdot a^t$, where $N(t)$ is the number of insects and $t$ is the time in…
The evolution of a certain insect plague follows an exponential model of the form $N(t) = k \cdot a^t$, where $N(t)$ is the number of insects and $t$ is the time in weeks. Given that at the initial instant ($t=0$) there were $5$ insects and that at $2$ weeks there were $45$ insects, calculate the values of the constants $k$ and $a$. Then, determine how many insects there will be at $4$ weeks.
Shifting the graph
Adding or subtracting numbers in $f(x) = b^{x-h} + k$ moves the graph:
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$x - h$ in the exponent moves the graph right $h$ units (left if $h$ is negative).
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$+k$ at the end moves the graph up $k$ units (down if $k$ is negative).
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The horizontal asymptote moves with it: it becomes the line $y = k$.
(In the widget, the slider $p$ is the shift $h$ and the slider $q$ is the shift $k$.) To graph a shifted exponential, build a table of values first. Then plot the points and draw the asymptote.
To find an x-intercept, always set $f(x) = 0$, isolate the power, and solve the exponential equation. If the asymptote is above the $x$-axis ($k > 0$ and $a > 0$), the graph never reaches $y = 0$, so there is no x-intercept.
Plot the function $f(x) = 3^{x-2} + 1$ graphically by first drawing up a table of values. Indicate the equation of its horizontal asymptote and calculate the coordinates…
Plot the function $f(x) = 3^{x-2} + 1$ graphically by first drawing up a table of values. Indicate the equation of its horizontal asymptote and calculate the coordinates of its point of intersection with the y-axis.
The following graph corresponds to an exponential function of the form $f(x) = a^x + b$. Observe its characteristic elements, such as the horizontal asymptote and the…
The following graph corresponds to an exponential function of the form $f(x) = a^x + b$. Observe its characteristic elements, such as the horizontal asymptote and the points it passes through, and determine the values of the parameters $a$ and $b$.
Find, analytically, the exact coordinates of the points of intersection with the x-axis and the y-axis of the exponential function $f(x) = 2^{x+1} - 8$.
Find, analytically, the exact coordinates of the points of intersection with the x-axis and the y-axis of the exponential function $f(x) = 2^{x+1} - 8$.
Modeling growth and decay
A real-life quantity that is multiplied by the same factor in equal time periods follows $$y = a \cdot b^t$$ where $a$ is the starting amount and $b$ is the factor for one time unit.
If the problem says the quantity doubles, triples, or halves every $T$ time units, the exponent counts how many periods have passed. For doubling or tripling use $y = a \cdot 2^{t/T}$ or $y = a \cdot 3^{t/T}$. For decay by half every $T$ units (a half-life) use $$y = a \cdot \left(\tfrac{1}{2}\right)^{t/T}$$ Why $t/T$? After $t$ years, $t/T$ periods of length $T$ have gone by, and each one multiplies the amount by $\tfrac{1}{2}$.
A culture of bacteria is studied in a laboratory. Initially there are $100$ bacteria and it is observed that their number triples every hour. Write the function that…
A culture of bacteria is studied in a laboratory. Initially there are $100$ bacteria and it is observed that their number triples every hour. Write the function that relates the number of bacteria to the time elapsed in hours, and calculate how many bacteria there will be after $3$ hours.
A sample of a radioactive isotope has an initial mass of $200$ ounces. It is known that its mass halves every $10$ years. Write the analytical expression of the function…
A sample of a radioactive isotope has an initial mass of $200$ ounces. It is known that its mass halves every $10$ years. Write the analytical expression of the function that gives the remaining mass as a function of time (in years) and calculate exactly how many ounces will remain after $30$ years.
Pharmacokinetics studies how a medicine is eliminated from the human body. A patient is administered an initial dose of $50$ mg of a drug. The amount in the blood…
Pharmacokinetics studies how a medicine is eliminated from the human body. A patient is administered an initial dose of $50$ mg of a drug. The amount in the blood decreases exponentially so that every $6$ hours the amount is halved.
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a) Write the function $C(t)$ that expresses the milligrams of the drug in the blood in terms of the time $t$ measured in hours.
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b) What amount of the drug will remain in the patient's blood after a full day ($24$ hours)?
Percent change and compound interest
A percent increase or decrease is a growth or decay factor in disguise.
A family deposits $\$2000$ into a bank account that offers a compound interest of $4 \%$ per year. What capital will they have after $2…
A family deposits $\$2000$ into a bank account that offers a compound interest of $4 \%$ per year. What capital will they have after $2$ years if they do not withdraw any money from the account?
The price of a newly released high-end cell phone is $\$800$. Due to technological obsolescence, its value depreciates by $20 \%…
The price of a newly released high-end cell phone is $\$800$. Due to technological obsolescence, its value depreciates by $20 \%$ each year. Write the function that models the price of the cell phone in terms of the years elapsed and calculate what its value will be after $4$ years.
An investor wishes to obtain a final capital of $\$5000$ in $5$ years' time. If their bank offers them a deposit with a compound interest of $3 \%…
An investor wishes to obtain a final capital of $\$5000$ in $5$ years' time. If their bank offers them a deposit with a compound interest of $3 \%$ per year, what initial capital must they invest today to achieve their goal? (Round the result to the nearest cent).
Simple vs. compound interest
With simple interest, you earn interest only on the original principal: $A = P(1 + rt)$. That is a linear function: the same number of dollars is added every year. With compound interest, the interest earns interest, so the function is exponential.
If interest is added $n$ times per year, the annual rate $r$ is split into $n$ equal parts and there are $nt$ periods: $$A = P\left(1 + \frac{r}{n}\right)^{nt}$$ For example, monthly compounding means $n = 12$, and quarterly means $n = 4$.
Both lines above start at $\$500$ and use an $8\%$ rate. The straight line is simple interest. The curve is compound interest. The curve starts a little lower in slope but takes over.
Two brothers receive an inheritance of $\$10000$ each. The first decides to invest his money at a simple interest of $5 \%…
Two brothers receive an inheritance of $\$10000$ each. The first decides to invest his money at a simple interest of $5 \%$ per annum, while the second invests it at a compound interest of $4 \%$ per annum. Write the functions that describe the capital of each brother over time. Who will have more money after $10$ years? And after $20$ years?
A bank offers an investment product with a nominal annual interest rate of $6 \%$. a) Write the function that models the final capital if $\$3000…
A bank offers an investment product with a nominal annual interest rate of $6 \%$.
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a) Write the function that models the final capital if $\$3000$ is invested and the interest is paid annually (annual compound interest).
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b) Write the function if the interest is paid monthly (monthly compound interest).
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c) Calculate the exact difference in capital obtained between both methods after $5$ years.
When the bases do not match: guess and check
Not every exponential equation can be written with the same base. For example, $1.05^t = 1.5$ has no whole-number answer. At this level, you can solve it by guess and check with a calculator. Try values of $t$, see whether you are too high or too low, and narrow in. In Algebra 2 you will learn logarithms, which solve $b^t = c$ directly.
When a model has an added constant, such as a cooling cup of tea $T(t) = 20 + 70 \cdot 0.8^t$, first isolate the power. Subtract the constant, then divide by the coefficient. The constant $20$ is also the horizontal asymptote: as $t$ gets large, $0.8^t$ approaches $0$, so $T$ approaches $20$.
The population of a city A is currently $100000$ inhabitants and grows at a constant rate of $4 \%$ per year. The population of a city B is $150000$ inhabitants and…
The population of a city A is currently $100000$ inhabitants and grows at a constant rate of $4 \%$ per year. The population of a city B is $150000$ inhabitants and decreases at a rate of $2 \%$ per year. Write the population functions for both cities. Check, using a calculator by trial and improvement (or by applying logarithms), in exactly which year city A will exceed city B in population.
According to Newton's law of cooling, the temperature of a cup of tea in a room is modelled by the function $T(t) = 20 + 70 \cdot 0,8^t$, where $T(t)$ is the temperature…
According to Newton's law of cooling, the temperature of a cup of tea in a room is modelled by the function $T(t) = 20 + 70 \cdot 0,8^t$, where $T(t)$ is the temperature in degrees Fahrenheit and $t$ is the time in minutes elapsed since it was served.
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a) What was the initial temperature of the tea?
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b) What is the ambient temperature of the room? (Reason what happens when the time $t$ becomes very large).
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c) Set up the equation to calculate in which minute the temperature of the tea will be exactly $55$ °F and solve it.
Absolute value of an exponential function
The absolute value $|u|$ keeps non-negative numbers as they are and changes negative numbers to their opposite. To write $f(x) = |2^x - 4|$ as a piecewise function, find where the inside expression changes sign.
Given the absolute value function $f(x) = |3^x - 9|$: a) Write it as a piecewise function, first calculating at which point the inner expression changes sign. b)…
Given the absolute value function $f(x) = |3^x - 9|$:
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a) Write it as a piecewise function, first calculating at which point the inner expression changes sign.
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b) Determine its points of intersection with the coordinate axes.
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c) Indicate in which intervals the function is increasing and in which it is decreasing.
Back to the start: Maya's video starts at $50$ views and triples every day, so $V(t) = 50 \cdot 3^t$. After $4$ days it has $50 \cdot 3^4 = 50 \cdot 81 = 4{,}050$ views.