Worked Examples: SAT Strategy
Example 1: Easy (Interpreting Constants)
Question: The function $P(t) = 1,800(1.02)^t$ gives the estimated number of marine mammals in a certain area, where $t$ is the number of years since a study began. What is the best interpretation of $P(0) = 1,800$ in this context?
SAT Strategy: Recognize the standard exponential form $y = a(b)^x$, where $a$ is the initial value (when $x=0$) and $b$ is the growth/decay factor.
- $P(0)$ represents the value of the function when $t = 0$.
- Since $t$ is the number of years since the study began, $t = 0$ means the exact moment the study started.
- Answer: The estimated number of marine mammals in the area was 1,800 when the study began.
Example 2: Medium (Translating Percentages to Bases)
Question: At the time an article was first featured on a website, there were 40 comments. At the end of each hour, the number of comments had
increased by 190% of the number of comments at the end of the previous hour. Which equation represents this model?
SAT Strategy: A percentage increase is always added to 100% (or $1$).
- The initial amount is $a = 40$.
- An increase of $190%$ means the new amount is $100% + 190% = 290%$ of the previous amount.
- Convert $290%$ to a decimal base $b$ by dividing by 100: $b = 2.90$.
- Substitute into $C = a(b)^t$.
- Answer: $C = 40(2.9)^t$.
Example 3: Hard (Fractional Exponents and Doubling Time)
Question: The function $f(t) = 60,000(2)^{\frac{t}{410}}$ gives the number of bacteria in a population $t$ minutes after an initial observation. How much time, in minutes, does it take for the number of bacteria in the population to double?
SAT Strategy: When an exponential equation is in the form $y = a(2)^{\frac{t}{k}}$, the value $k$ is the exact time it takes for the population to double.
- For the population to double, the multiplier must be exactly $2^1$.
- This means the exponent must equal 1: $\frac{t}{410} = 1$.
- Multiply both sides by 410: $t = 410$.
- Answer: 410.
Practice Questions
Level 1: Easy (Foundational SAT)
- The function $f$ is defined by $f(x) = 270(0.1)^x$. What is the value of $f(0)$? (Grid-in)
- The population of Greenville increased by 7% from 2015 to 2016. If the 2016 population is $k$ times the 2015 population, what is the value of $k$?
- A) $0.07$
- B) $0.93$
- C) $1.07$
- D) $7$
- The function $h$ is defined by $h(x) = 5(2)^x$. What is the value of $h(3)$? (Grid-in)
- A company opens an account with an initial balance of $\$36,100.00$. The account earns 5% interest annually, and no additional deposits or withdrawals are made. Which equation could define $A(t)$, the account balance $t$ years after the account is opened?
- A) $A(t) = 36,100(1.05)^t$
- B) $A(t) = 36,100(0.05)^t$
- C) $A(t) = 36,100(1.5)^t$
- D) $A(t) = 36,100(0.95)^t$
- What is the $y$-intercept of the graph of $y = 12(0.8)^x$ in the $xy$-plane?
- A) $(0, 0.8)$
- B) $(0, 9.6)$
- C) $(0, 12)$
- D) $(12, 0)$
Level 2: Medium (Standard SAT)
- For $x > 0$, the function $f$ is defined as follows: $f(x)$ equals $201%$ of $x$. Which of the following could describe this function?
- A) Decreasing exponential
- B) Decreasing linear
- C) Increasing exponential
- D) Increasing linear
- The table below shows selected values for the function $g$.
{|c|c|}
$x$ & $g(x)$
$0$ & $3$
$1$ & $12$
$2$ & $48$
Which of the following equations defines $g(x)$?
- A) $g(x) = 3(4)^x$
- B) $g(x) = 4(3)^x$
- C) $g(x) = 3x + 9$
- D) $g(x) = 9x + 3$
- The function $h$ is defined by $h(x) = 10(3)^x$. In the $xy$-plane, the graph of $y = p(x)$ is the result of shifting the graph of $y = h(x)$ down 2 units. What is the $y$-coordinate of the $y$-intercept of the graph of $p$? (Grid-in)
- A radioactive substance has a half-life of 5 days. If the initial amount of the substance is 100 grams, which equation models the amount remaining, $R$, after $d$ days?
- A) $R = 100(0.5)^{5d}$
- B) $R = 100(0.5)^{\frac{d}{5}}$
- C) $R = 100(5)^{\frac{d}{0.5}}$
- D) $R = 50(0.5)^d$
- $2^{3x - 1} = 32$. What is the value of $x$? (Grid-in)
Level 3: Hard (Advanced SAT)
- The function $f(x) = 5,470(0.64)^{\frac{x}{12}}$ gives the value, in dollars, of a piece of equipment after $x$ months of use. If the value of the equipment decreases each year by $p%$ of its value the preceding year, what is the value of $p$? (Grid-in)
- The functions $f$ and $g$ are defined for $x \ge 0$ by the equations $f(x) = 18(1.25)^x + 41$ and $g(x) = 9(0.73)^x$. Which of the following equations displays, as a constant or coefficient, the maximum value of the function it defines?
- A) I only
- B) II only
- C) I and II
- D) Neither I nor II
- Which of the following expressions is equivalent to $2^{x+3} - 2^x$?
- A) $2^3$
- B) $3(2^x)$
- C) $7(2^x)$
- D) $8(2^x)$
- An exponential function $f$ is given by $f(x) = a(b)^x$, where $a$ and $b$ are positive constants. If $f(0) = 5$ and $f(2) = 45$, what is the value of $a + b$? (Grid-in)
- If $3^{x^2 - 4x} = 1$, what is the positive solution for $x$? (Grid-in)
Level 4: Very Hard (Slightly beyond SAT)
- A savings account earns $8%$ annual interest, compounded yearly. If the initial deposit is $\$500$, the amount $A$ in the account after $t$ years is $A = 500(1.08)^t$. Which of the following equations is equivalent and shows the approximate monthly interest rate?
- A) $A = 500(1.0064)^{12t}$
- B) $A = 500(1.08)^{12t}$
- C) $A = 500(0.0064)^{12t}$
- D) $A = 500(1.0064)^{\frac{t}{12}}$
- $P(t) = 260(1.04)^{\frac{t}{6}}$. The function $P$ models the population of a city $t$ years after 2003. According to the model, the population is predicted to increase by $4%$ every $n$ months. What is the value of $n$? (Grid-in)
- The function $y = 20(3)^{\frac{x}{2}}$ relates the variables $x$ and $y$. If the value of $x$ is increased by 4, by what factor is the value of $y$ multiplied?
- A) $3$
- B) $6$
- C) $9$
- D) $81$
- In the $xy$-plane, the graphs of $y = 3^x$ and $y = 2(3^x) - 9$ intersect at the point $(x, y)$. What is the value of $y$? (Grid-in)
- The function $f$ is defined by $f(x) = 2.2^x$. The function $g$ is defined by $g(x) = f(x+3)$. The equation $g(x) = k(2.2^x)$ is true for all $x$, where $k$ is a constant. What is the value of $k$? (Round your answer to the nearest hundredth if necessary). (Grid-in)
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