Consider the exponential function \(f(x)=6(1.3)^x\). State the initial value, the factor for an increase of \(1\) in \(x\), and whether the function shows growth or decay.

An exponential quantity starts at \(40\) and doubles each hour. Write a function \(Q(t)\) for the quantity after \(t\) hours.

The table represents an exponential function. <table><tr><th>\(x\)</th><td>\(0\)</td><td>\(1\)</td><td>\(2\)</td><td>\(3\)</td></tr><tr><th>\(f(x)\)</th><td>\(7\)</td><td>\(14\)</td><td>\(28\)</td><td>\(\square\)</td></tr></table> Find the missing output.

Let \(f(x)=20(0.75)^x\). Find \(f(1)\).

Two number patterns start at \(5\). - Pattern A: \(5,10,20,40\) - Pattern B: \(5,10,15,20\) Which pattern shows exponential growth? Give its step-by-step growth factor and explain your choice.

This table follows an exponential pattern. Fill in the missing output and explain how you found it. <table> <tr><th>\(x\)</th><th>\(-1\)</th><th>\(0\)</th><th>\(1\)</th><th>\(2\)</th></tr> <tr><th>\(f(x)\)</th><td>\(48\)</td><td>\(24\)</td><td>\(12\)</td><td>\(\square\)</td></tr> </table>

a) What is \(f(0)\)? b) By what factor does the output change when \(x\) increases by \(1\)?

Which curve shows exponential growth and which shows exponential decay? Explain how you know.

Rowan sees the pattern \(40,60,90,135\) and says, “It goes up by \(20\) each step.” Explain the mistake and give the correct rule for moving to the next term.

A bacterial culture starts with \(1200\) bacteria. After \(5\) hours, it has \(3000\) bacteria. Assume the population grows exponentially. Write a function \(N(t)\) for the population after \(t\) hours. Use an exact expression; do not round the growth factor.

Consider three population models, with \(t\) measured in years: (1) \(f_1(t)=600(1.05)^t\) (2) \(f_2(t)=450(0.88)^t\) (3) \(f_3(t)=600(0.95)^t\) a) Which population is smallest at \(t=0\)? Give its size. b) Which models describe decay? Explain how you know. c) Find \(f_1(20)\). Round to the nearest hundredth.

A quantity follows a model of the form \(B(x)=ca^x\). a) Identify \(c\) and \(a\) for \(B(x)=15.5(0.6)^x\). b) Does the quantity grow or decay as \(x\) increases? Explain. c) Find \(B(5)\). Round to two decimal places.

A wildlife preserve has \(40\) deer at the beginning of a study. After one year, the population has grown to \(48\) deer. Assume the population grows exponentially each year. Write a function \(f(t)\) that models the deer population after \(t\) years.

In an exponential process, the amount \(f(t)\) is multiplied by \(4\) whenever time increases by \(2\) units. Write an equation expressing \(f(t-2)\) using only \(f(t)\) and a numerical factor.

The number of users of a new app doubles each month. At the beginning of the study, the app has \(4500\) users. Write an exponential function \(N(t)\) for the number of users after \(t\) months. Then find the number of users after half a year and after one year.

A protected plant population grows by a factor of \(1.4\) each year. At the beginning of the study, there are \(250\) plants. Write a function \(B(t)\) for the population after \(t\) years. Then find the predicted population after \(5\) years and after \(10\) years. Round to the nearest whole plant.

The approximation \(2^{10}\approx 10^3\) is often used in computing and engineering to estimate powers of \(2\). a) Find the exact value of \(2^{10}\). By what percent does the exact value exceed \(1000\), using \(1000\) as the reference value? b) Suppose computing power doubles every \(2\) years. Use the approximation to explain why computing power increases by about a factor of \(10^3\) in \(20\) years and by about a factor of \(10^9\) in \(60\) years. c) A computer currently performs \(10^{16}\) operations per second. Estimate its performance after \(40\) years under the same growth assumption. Write the result in scientific notation.

Find the base \(a\) of the exponential function \(f(x)=a^x\) if its graph passes through \((-2, 6.25)\).

Exponential functions of the form \(f(x)=b^x\) require restrictions on the base. 1. Explain the problem with evaluating \(f(0.5)\) when \(b=-4\). What general restriction on \(b\) ensures that \(f\) is defined for every real input? 2. Why is \(b=1\) usually excluded from the definition of an exponential function? Describe the function that results in this case. 3. Show algebraically that every graph in the family \(f(x)=b^x\), for every allowed base \(b\), passes through the same point on the y-axis.

The graph of \(f(x)=ab^x\), where \(a,b>0\), passes through \((2, 5)\) and \((3, 20)\). a) Find \(a\) and \(b\). b) Another function \(g(x)=cd^x\) passes through \((2, 5)\) and \((3, 15)\). Without fully calculating \(d\), explain whether \(d\) is greater than or less than \(b\).

A bacterial culture grows under ideal conditions. a) Estimate the doubling time to the nearest half-hour. b) About when does the population reach \(400\) bacteria? Give your answer to the nearest half-hour. c) Estimate the population after \(9\) hours, to the nearest \(50\) bacteria.

A cup of hot tea cools in a room at \(70\,\text{°F}\). In this model, the difference between the tea's temperature and the room temperature decreases by the same percent each minute. a) What was the tea's temperature at the beginning? b) About how long does it take to cool to \(105\,\text{°F}\)? c) Why does the tea get closer to room temperature without cooling below it in this model?

Match equations (a) through (d) with graphs \(p\), \(q\), \(r\), and \(s\). Justify each match using the y-intercept and growth or decay behavior. (a) \(y=3(1.4)^x\) (b) \(y=3(0.6)^x\) (c) \(y=1.5(1.4)^x\) (d) \(y=1.5(0.6)^x\)

The graph shows two bacterial cultures modeled by \(N(t)=N_0a^t\), where \(t\) is measured in hours and \(N(t)\) is measured in thousands of bacteria. Find \(N_0\) and \(a\) for each culture.

The table represents an exponential function. Find the missing output and state the factor for each increase of \(1\) in \(x\). <table> <tr><th>\(x\)</th><td>\(0\)</td><td>\(1\)</td><td>\(2\)</td><td>\(3\)</td></tr> <tr><th>\(f(x)\)</th><td>\(12\)</td><td></td><td>\(27\)</td><td>\(40.5\)</td></tr> </table>

Eli claims that \(f(x)=10(0.6)^x\) eventually becomes negative because the number keeps shrinking. a) Find \(f(2)\). b) Explain why Eli's claim is false for every real \(x\).

A runner records the distance she runs each week. The table gives her weekly totals. <table> <tr><th>Week \(t\)</th><td>\(0\)</td><td>\(1\)</td><td>\(2\)</td></tr> <tr><th>Miles \(D(t)\)</th><td>\(8\)</td><td>\(10\)</td><td>\(12.5\)</td></tr> </table> Suppose she increases her weekly distance by the same percent each week. Write a model \(D(t)\) and predict her distance in week \(3\), rounded to the nearest tenth. Explain why adding \(2\) miles every week would not follow the model.

A bacterial culture is modeled by \(N(t)=200(1.5)^t\), where \(t\) is measured in hours. a) By what factor does the population change over \(2\) hours, \(3\) hours, and \(5\) hours? b) Find the population after half an hour and after \(15\) minutes. Round to the nearest whole bacterium.

Let \(f(x)=5(3)^x\). Determine the factor by which \(f(x)\) is multiplied when a) \(x\) increases by \(1\); b) \(x\) increases by \(4\); c) \(x\) decreases by \(2\); d) \(x\) increases by \(0.5\).

Let \(f(x)=4^x\). 1. Find \(\frac{f(x+1)}{f(x)}\) for \(x=0\) and \(x=2\). What do you notice? 2. Find the factor by which \(f(x)\) is multiplied when \(x\) increases by \(s=1.5\). 3. Find the value of \(s\) for which the function value is multiplied by exactly \(32\). 4. Complete the statement for \(f(x)=a^x\): “When \(x\) increases by \(s\), the function value is multiplied by ...”

Find the positive base \(b\) of the exponential function \(y=b^x\) that passes through each point. a) \(P(4, 81)\) b) \(Q(-3, 0.125)\) c) \(R(1.5, 8)\)

An exponential function has the form \(f(x)=ca^x\), where \(a>0\). Find \(c\) and \(a\) in each case. a) The graph passes through \(A(0, 7)\) and \(B(2, 63)\). b) The graph passes through \(A(0, 200)\) and \(B(-1, 50)\).

Consider the exponential function \(f(x)=a^x\), where \(a>0\) and \(a\ne 1\). a) For one such function, the output is multiplied by \(4\) whenever \(x\) increases by \(2\). Find the base \(a\). b) Explain why \(a=1\) is excluded from the usual definition of an exponential growth or decay function. What would its graph look like? c) Explain why a negative base such as \(a=-2\) does not define a real-valued function for every real \(x\). Compare \(x=2\) and \(x=0.5\).

Let \(g(x)=(0.75)^x\). a) By what factor does the function value change when \(x\) increases by \(2\)? b) By what percent does the function value decrease when \(x\) increases by \(3\)? c) If \(y=g(x)\), express \(g(2x)\) in terms of \(y\).

Let \(f(x)=ca^x\), where \(a>0\) and \(c\ne0\). 1. Show that \(\frac{f(x-4)}{f(x)}\) is independent of \(x\), and express the constant ratio in terms of \(a\). 2. Find the ratio when \(a=0.8\).

Analyze the exponential function \(f(x)=1.5(3)^x\). a) Find its y-intercept. b) Is it increasing or decreasing? Explain. c) Give its range. d) As \(x\) gets smaller and smaller, what happens to \(f(x)\)?

Let \(h(x)=5(2)^{-3x}\). The input \(x\) is increased repeatedly by \(\frac{1}{3}\). Find the factor by which the function value is multiplied at each step.

The graph of an exponential function \(f(x)=ca^x\) passes through \(A(2, 18)\) and \(B(4, 162)\). Find the function, then determine the missing y-coordinates of \(C(0, y_C)\) and \(D(-1, y_D)\).

The table shows values of a function \(g\): <table> <tr><td>\(x\)</td><td>\(-1\)</td><td>\(0\)</td><td>\(1\)</td><td>\(2\)</td></tr> <tr><td>\(g(x)\)</td><td>\(12\)</td><td>\(6\)</td><td>\(3\)</td><td>\(1.5\)</td></tr> </table> a) Identify the function family and write an equation for \(g\). b) Describe the change in the outputs when \(x\) increases by \(1\). c) Find \(g(0.5)\). Round to the nearest hundredth. Explain why this input is allowed even though it is not in the table.

An exponential function has the form \(g(x)=b^x\). Its graph passes through \((0.75, 8)\). Find \(b\).

Consider \(f(x)=4^x\) and \(g(x)=4.1^x\). For \(x>0\), which function grows faster? Justify your answer by comparing their growth factors and describing what happens to \(\frac{g(x)}{f(x)}\) as \(x\) increases.

The figure shows four graphs, labeled \(p\), \(q\), \(r\), and \(s\), of functions of the form \(y=b^x\). Match each graph with one function. Justify your matches using growth or decay, steepness, or key points. \(A(x)=3^x\) \(B(x)=1.5^x\) \(C(x)=0.5^x\) \(D(x)=0.2^x\)

A biologist compares two bacterial cultures. Each starts with one million bacteria. Culture A doubles each hour, \(f(x)=2^x\), while Culture B is halved each hour, \(g(x)=0.5^x\). Population is measured in millions. The model uses \(x\ge 0\), while the graph shows both complete functions. a) Describe the two graphs and explain their symmetry. b) Find the population ratio \(\frac{f(x)}{g(x)}\) at \(x=3\). c) Show that \(f(x)g(x)\) is constant for every \(x\), and state its value.

Find an equation of the form \(y=ab^x\) for each graph \(f\) and \(g\). Use the y-intercept and one other clearly readable point from each graph.

The table shows the number of users of a new app from week \(0\) through week \(3\). <table> <tr><th>Week \(t\)</th><td>\(0\)</td><td>\(1\)</td><td>\(2\)</td><td>\(3\)</td></tr> <tr><th>Users \(y\)</th><td>\(10\)</td><td>\(30\)</td><td>\(90\)</td><td>\(270\)</td></tr> </table> a) Show that the data represent exponential growth, and give the growth factor \(b\). b) Write a function in the form \(y=ab^t\). c) How many users does the model predict at the end of week \(4\)?

Analyze each statement and draw the requested conclusion. a) Suppose \(0.6^m<0.6^n\). What relationship must hold between \(m\) and \(n\)? b) For a positive base \(a\), suppose \(a^5>a^7\). What interval must contain \(a\)? c) Suppose \(2^k=0.2\). Find two consecutive integers between which \(k\) lies.

A frozen cake at \(-18\,^\circ\text{C}\) is placed in a kitchen at \(22\,^\circ\text{C}\). The temperature difference between the cake and the room decreases exponentially. After \(10\) minutes, the difference is \(32\,^\circ\text{C}\). a) Find the initial temperature difference. b) Find the decay factor per minute for the temperature difference, and write a function for the difference after \(t\) minutes. Round the per-minute decay factor to four decimal places. c) Find the cake's temperature after \(30\) minutes. d) Make a table of the temperature difference after \(0\), \(20\), \(40\), and \(60\) minutes. Round each table value to the nearest hundredth.

Find the parameters \(a\) and \(c\) in \(y=ca^x\) so that the graph passes through \(P(1, 4)\) and \(Q(-2, 32)\). Then find \(y_R\) and \(y_S\) for the points \(R(2, y_R)\) and \(S(-3, y_S)\).

Find the equation of an exponential function \(f(x)=ab^x\) whose graph passes through \((-1, 12)\) and \((1.5, 0.375)\).