Precalculus Practice Test 2 – Step-by-Step Guidance

Q1. For each function, determine the y-intercept, x-intercept, equation of the asymptote, domain, range, and sketch the graph.

Background

Topic: Functions and Their Graphs (Linear, Exponential, Logarithmic)

This question tests your understanding of how to analyze and graph different types of functions, including finding intercepts, asymptotes, domain, and range.

Key Terms and Formulas:

• Y-intercept: The point where the graph crosses the y-axis ($x=0$).

• X-intercept: The point where the graph crosses the x-axis ($y=0$).

• Asymptote: A line that the graph approaches but never touches.

• Domain: All possible $x$-values for the function.

• Range: All possible $y$-values for the function.

Step-by-Step Guidance (for part a: $y = -(3x-2) + 4$)

1. Rewrite the function in a more familiar form, if needed. For example, expand or simplify the expression.

2. To find the y-intercept, set $x=0$ and solve for $y$.

3. To find the x-intercept, set $y=0$ and solve for $x$.

4. Identify if there are any asymptotes (for linear functions, there are none; for exponential/logarithmic, look for horizontal or vertical asymptotes).

5. Determine the domain and range based on the type of function and any restrictions (e.g., logarithms require positive arguments).

Try solving on your own before revealing the answer!

Q2. If $9500 is invested at an interest rate of 5.2% per year, compounded semi-annually, find the value of the investment after 12 years. Round your answer to the nearest dollar.

Background

Topic: Exponential Growth – Compound Interest

This question tests your ability to use the compound interest formula to calculate the future value of an investment.

Key Formula:

$A(t) = P \left(1 + \frac{r}{n}\right)^{nt}$

• $A(t)$ = amount after $t$ years

• $P$ = principal (initial amount)

• $r$ = annual interest rate (as a decimal)

• $n$ = number of times compounded per year

• $t$ = number of years

Step-by-Step Guidance

1. Identify the values: $P = 9500$, $r = 0.052$, $n = 2$, $t = 12$.

2. Plug these values into the compound interest formula.

3. Calculate $\frac{r}{n}$ and $nt$ separately.

4. Set up the expression for $A(t)$, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Q3. Solve each equation for $x$.

Background

Topic: Exponential and Logarithmic Equations

This question tests your ability to solve equations involving exponents and logarithms, often requiring properties of logarithms or exponentials.

Key Terms and Formulas:

• Exponential Equations: Use logarithms to solve for the variable in the exponent.

• Logarithmic Equations: Use properties of logarithms to combine or expand, then exponentiate to solve for the variable.

• Key Properties:

  • $\log_b(a) = c \implies a = b^c$

  • $e^{\ln x} = x$

  • $\ln(ab) = \ln a + \ln b$

Step-by-Step Guidance (for part a: $4^x = 8^{x-1}$)

1. Express both sides with the same base if possible (e.g., $4$and $8$ as powers of $2$).

2. Rewrite the equation using the new base.

3. Set the exponents equal to each other, since the bases are the same.

4. Solve the resulting linear equation for $x$.

Try solving on your own before revealing the answer!

Q4. A bacteria culture starts with 3200 bacteria and grows according to $P(t) = P_0 e^{kt}$. After 4 hours, the population is 5000.

Background

Topic: Exponential Growth Models

This question tests your ability to model population growth using exponential functions and to solve for unknown parameters.

Key Formula:

$P(t) = P_0 e^{kt}$

• $P_0$ = initial population

• $k$ = growth rate

• $t$ = time

Step-by-Step Guidance (for part a)

1. Plug in the known values: $P_0 = 3200$, $P(4) = 5000$, $t = 4$.

2. Set up the equation: $5000 = 3200 e^{4k}$.

3. Divide both sides by $3200$ to isolate the exponential term.

4. Take the natural logarithm of both sides to solve for $k$.

Try solving on your own before revealing the answer!

Q5. A sample of a chemical substance decays exponentially according to $A(t) = A_0 e^{kt}$. A sample starts with 240 grams. After 6 days, 150 grams remain.

Background

Topic: Exponential Decay Models

This question tests your ability to model decay processes and solve for decay constants and half-life.

Key Formula:

$A(t) = A_0 e^{kt}$

• $A_0$ = initial amount

• $k$ = decay constant (will be negative)

• $t$ = time

Step-by-Step Guidance (for part a)

1. Plug in the known values: $A_0 = 240$, $A(6) = 150$, $t = 6$.

2. Set up the equation: $150 = 240 e^{6k}$.

3. Divide both sides by $240$ to isolate the exponential term.

4. Take the natural logarithm of both sides to solve for $k$.

Try solving on your own before revealing the answer!

Q6. The magnitude $M$ of an earthquake is modeled by $M = \log \left(\frac{I}{S}\right)$.

Background

Topic: Logarithmic Scales (Richter Scale)

This question tests your understanding of logarithmic models and how to interpret ratios using logarithms.

Key Terms and Formulas:

• $M$ = magnitude

• $I$ = intensity of the earthquake

• $S$ = standard reference intensity

• $M = \log \left(\frac{I}{S}\right)$

Step-by-Step Guidance (for part a)

1. Set $M = 6.3$ and write the equation: $6.3 = \log \left(\frac{I}{S}\right)$.

2. Rewrite the equation in exponential form to solve for $\frac{I}{S}$.

Try solving on your own before revealing the answer!

Q7. Let $t = \frac{5\pi}{6}$.

Background

Topic: Unit Circle and Trigonometric Functions

This question tests your understanding of reference angles, terminal points, and exact trigonometric values on the unit circle.

Key Terms and Formulas:

• Reference Number: The acute angle formed with the x-axis.

• Terminal Point: The point $(x, y)$ on the unit circle corresponding to angle $t$.

• Trigonometric Values: $\cos t$, $\sin t$, $\csc t$.

Step-by-Step Guidance (for part a)

1. Determine which quadrant $t = \frac{5\pi}{6}$ is in.

2. Find the reference angle by subtracting from $\pi$ if in the second quadrant.

3. Use the unit circle to find the coordinates of the terminal point.

4. Use these coordinates to find the exact values of the trigonometric functions.

Try solving on your own before revealing the answer!

Q8. The point $\left(-\frac{\sqrt{5}}{3}, -\frac{2}{3}\right)$ lies on the unit circle and is the terminal point determined by the real number $t$. Find the exact values of $\sin t$, $\cos t$, $\tan t$, $\csc t$, $\sec t$, $\cot t$.

Background

Topic: Trigonometric Functions on the Unit Circle

This question tests your ability to use coordinates on the unit circle to find all six trigonometric functions.

Key Terms and Formulas:

• On the unit circle, $\cos t = x$ and $\sin t = y$.

• $\tan t = \frac{y}{x}$

• $\csc t = \frac{1}{\sin t}$

• $\sec t = \frac{1}{\cos t}$

• $\cot t = \frac{x}{y}$

Step-by-Step Guidance

1. Identify $x$ and $y$ from the given point.

2. Write $\sin t$ and $\cos t$ directly from $y$ and $x$.

3. Set up the expressions for $\tan t$, $\csc t$, $\sec t$, and $\cot t$ using the definitions above.

Try solving on your own before revealing the answer!

Q9. Consider the function $y = -3 \cos(2x - \pi) + 1$.

Background

Topic: Graphs of Trigonometric Functions

This question tests your understanding of amplitude, period, phase shift, and midline for cosine functions, as well as graphing transformations.

Key Terms and Formulas:

• Amplitude: $|a|$ in $y = a \cos(bx + c) + d$

• Period: $\frac{2\pi}{|b|}$

• Phase Shift: $-\frac{c}{b}$

• Midline: $y = d$

Step-by-Step Guidance

1. Identify $a$, $b$, $c$, and $d$ from the function.

2. Calculate the amplitude using $|a|$.

3. Find the period using $\frac{2\pi}{|b|}$.

4. Determine the phase shift using $-\frac{c}{b}$.

5. Write the equation of the midline as $y = d$.

Try solving on your own before revealing the answer!

Q10. For the given graph, answer questions about amplitude, midline, period, and write a possible equation for $y = f(x)$.

Background

Topic: Interpreting Graphs of Trigonometric Functions

This question tests your ability to read key features from a trigonometric graph and write an equation that models it.

Key Terms and Formulas:

• Amplitude: Half the distance between maximum and minimum values.

• Midline: The horizontal line halfway between the maximum and minimum.

• Period: The horizontal length of one complete cycle.

• General Equation: $y = a \sin(bx + c) + d$ or $y = a \cos(bx + c) + d$

Step-by-Step Guidance

1. Identify the maximum and minimum values from the graph to find amplitude and midline.

2. Determine the period by measuring the distance between repeating points (e.g., two consecutive maxima).

3. Use these values to write a possible equation for the function, choosing sine or cosine as appropriate.

Try solving on your own before revealing the answer!