Exponential functions
An account that goes viral triples its followers every day. A radioactive sample loses half of its mass after a fixed period of time. In both cases, the quantity does not change by always adding the same amount (as in a straight line), but by always multiplying by the same factor. This is exponential growth or decay, and the variable is no longer in the base but in the exponent.
The exponential function
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All the functions $y=a^x$ pass through $(0,1)$, because $a^0=1$ always.
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They also pass through $(1,a)$.
Given the exponential function $f(x) = 2^x$, calculate the outputs for the values $x = -2$, $x = 0$ and $x = 3$.
Given the exponential function $f(x) = 2^x$, calculate the outputs for the values $x = -2$, $x = 0$ and $x = 3$.
The graph of an exponential function of the form $f(x) = a^x$ passes through the point $(3, 8)$. Calculate the value of the base $a$.
The graph of an exponential function of the form $f(x) = a^x$ passes through the point $(3, 8)$. Calculate the value of the base $a$.
Growth if $a>1$, decay if $0<a<1$
State, giving reasons for 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)…
State, giving reasons for 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) $i(x) = \left(\frac{2}{5}\right)^x$
Look at graphs A and B shown in the following figure. Match each of them with its corresponding algebraic expression: $y = 3^x$ or $y = \left(\frac{1}{3}\right)^x$.
Look at graphs A and B shown in the following figure. Match each of them with its corresponding algebraic expression: $y = 3^x$ or $y = \left(\frac{1}{3}\right)^x$.
Find the value of $x$ in the following basic exponential equations: a) $2^x = 64$ b) $3^x = \frac{1}{9}$
Find the value of $x$ in the following basic exponential equations:
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a) $2^x = 64$
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b) $3^x = \frac{1}{9}$
The horizontal asymptote and translations
In $g(x)=a^{x-p}+q$, the exponent $x-p$ shifts the curve $p$ units horizontally, and the final $+q$ shifts it $q$ units vertically, taking the asymptote with it: the new asymptote is $y=q$.
Determine the domain, the range and the y-intercept of the exponential function $f(x) = 0,5^x$.
Determine the domain, the range and the y-intercept of the exponential function $f(x) = 0,5^x$.
Plot the graph of the function $f(x) = 3^{x-2} + 1$ by first making a table of values. State the equation of its horizontal asymptote and calculate the coordinates of…
Plot the graph of the function $f(x) = 3^{x-2} + 1$ by first making a table of values. State the equation of its horizontal asymptote and calculate the coordinates of its y-intercept.
You can do it on paper and send a photo with your answer.
Find algebraically the exact coordinates of the x-intercepts and y-intercepts of the exponential function $f(x) = 2^{x+1} - 8$.
Find algebraically the exact coordinates of the x-intercepts and y-intercepts of the exponential function $f(x) = 2^{x+1} - 8$.
Basic exponential equations
When both sides can be written with the same base, it is enough to equate the exponents. If several terms with the same base but different exponents appear, take out a common factor.
Solve the following exponential equations by making the bases equal on both sides of the equation: a) $5^{2x-1} = 125$ b) $2^{x+3} = \frac{1}{4}$
Solve the following exponential equations by making the bases equal on both sides of the equation:
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a) $5^{2x-1} = 125$
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b) $2^{x+3} = \frac{1}{4}$
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 laws of indices and taking out a common factor: $2^{x-1} + 2^{x-2} + 2^{x-3} = 56$
Solve the following exponential equation by applying the laws of indices and taking out a common factor:
$2^{x-1} + 2^{x-2} + 2^{x-3} = 56$
Compound interest and decay
A family deposits $2000$ € into a bank account offering a compound interest rate of $4 \%$ per annum. What capital will they have after $2$ years if they do not withdraw…
A family deposits $2000$ € into a bank account offering a compound interest rate of $4 \%$ per annum. What capital will they have after $2$ years if they do not withdraw any money from the account?
The number of followers of an account that has just gone viral triples every day. If it initially has $100$ followers, write the function relating the number of…
The number of followers of an account that has just gone viral triples every day. If it initially has $100$ followers, write the function relating the number of followers to the time elapsed in days, and calculate how many followers there will be after $3$ days.
An investor wishes to obtain a final capital of $5000$ € in $5$ years' time. If their bank offers a deposit with a compound interest rate of $3 \%$ per annum, what…
An investor wishes to obtain a final capital of $5000$ € in $5$ years' time. If their bank offers a deposit with a compound interest rate of $3 \%$ per annum, what initial capital must they invest today to achieve their goal? (Round the answer to the nearest cent).
A sample of a radioactive isotope has an initial mass of $200$ grams. It is known that its mass halves every $10$ years. Write the algebraic expression for the function…
A sample of a radioactive isotope has an initial mass of $200$ grams. It is known that its mass halves every $10$ years. Write the algebraic expression for the function giving the remaining mass in terms of time (in years) and calculate how many grams will remain after exactly $30$ years.
The price of a high-end mobile phone newly released on the market is $800$ €. Due to technological obsolescence, its value depreciates by $20 \%$ each year. Write the…
The price of a high-end mobile phone newly released on the market is $800$ €. Due to technological obsolescence, its value depreciates by $20 \%$ each year. Write the function modelling the price of the phone in terms of the years elapsed and calculate what its value will be after $4$ years.
The growth of a certain insect pest 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.…
The growth of a certain insect pest 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. Knowing that initially ($t=0$) there were $5$ insects and that after $2$ weeks there were $45$ insects, calculate the values of the constants $k$ and $a$. Then, determine how many insects there will be after $4$ weeks.
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 Celsius and $t$ is the time in minutes elapsed since it was poured.
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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 as the time $t$ becomes very large).
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c) Set up the equation to calculate at which minute the temperature of the tea will be exactly $55$ ºC and solve it.
Pharmacokinetics studies how a medication is eliminated from the human body. A patient is administered an initial dose of $50$ mg of a drug. The amount in the…
Pharmacokinetics studies how a medication is eliminated from the human body. A patient is administered an initial dose of $50$ mg of a drug. The amount in the bloodstream decreases exponentially such that every $6$ hours the amount halves.
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a) Write the function $C(t)$ expressing the milligrams of drug in the bloodstream in terms of time $t$ measured in hours.
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b) What amount of drug will remain in the patient's bloodstream after a full day ($24$ hours)?
Back to the beginning: If the account that goes viral starts with $50$ followers and triples them every day, the function is $S(t) = 50 \cdot 3^t$, so after $4$ days it will have $50 \cdot 3^4 = 50 \cdot 81 = 4050$ followers.