College Algebra Final Exam Review: Step-by-Step Guidance

Q1a. Solve the quadratic equation $x^2 - 20x = -99$ using (a) factoring (when possible) and (b) the quadratic formula. Check your answers with a graph and explain how the graph confirms your answers.

Background

Topic: Quadratic Equations

This question tests your ability to solve quadratic equations by factoring and by using the quadratic formula. It also asks you to interpret the solutions graphically.

Key Terms and Formulas:

• Quadratic Equation: $ax^2 + bx + c = 0$

• Factoring: Expressing the quadratic as a product of two binomials.

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Step-by-Step Guidance

1. Rewrite the equation in standard form: Move all terms to one side so it looks like $ax^2 + bx + c = 0$.

2. Attempt to factor the quadratic, if possible. Look for two numbers that multiply to $c$ and add to $b$.

3. If factoring is not possible, identify $a$, $b$, and $c$ and set up the quadratic formula.

4. Calculate the discriminant $b^2 - 4ac$ to determine the nature of the roots.

5. Set up the expressions for the solutions using the quadratic formula, but do not compute the final values yet.

Try solving on your own before revealing the answer!

Q1b. Solve $-5x^2 + 40x = 0$ by factoring and the quadratic formula. Check with a graph.

Background

Topic: Quadratic Equations

This problem reinforces solving quadratics by factoring and the quadratic formula, and interpreting solutions graphically.

Key Terms and Formulas:

• Factoring: Look for a greatest common factor first.

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Step-by-Step Guidance

1. Rewrite the equation in standard form: $-5x^2 + 40x = 0$.

2. Factor out the greatest common factor from both terms.

3. Set each factor equal to zero to find possible solutions.

4. Alternatively, identify $a$, $b$, and $c$ for the quadratic formula and set up the formula.

5. Write the expressions for the solutions, but do not compute the final values yet.

Try solving on your own before revealing the answer!

Q1c. Solve $x^2 - 144 = 0$ by factoring and the quadratic formula. Check with a graph.

Background

Topic: Quadratic Equations

This question involves solving a quadratic that is a difference of squares.

Key Terms and Formulas:

• Difference of Squares: $a^2 - b^2 = (a - b)(a + b)$

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Step-by-Step Guidance

1. Recognize the equation as a difference of squares and factor accordingly.

2. Set each factor equal to zero to find the solutions.

3. Alternatively, identify $a$, $b$, and $c$ for the quadratic formula and set up the formula.

4. Write the expressions for the solutions, but do not compute the final values yet.

Try solving on your own before revealing the answer!

Q1d. Solve $3x^2 - 6x - 9 = 0$ by factoring and the quadratic formula. Check with a graph.

Background

Topic: Quadratic Equations

This problem involves solving a quadratic equation that may or may not be factorable with integer coefficients.

Key Terms and Formulas:

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Factoring: Look for common factors or factor by grouping if possible.

Step-by-Step Guidance

1. Write the equation in standard form.

2. Check for a common factor and factor if possible.

3. If factoring is not possible, identify $a$, $b$, and $c$ for the quadratic formula.

4. Set up the quadratic formula with the identified coefficients.

5. Write the expressions for the solutions, but do not compute the final values yet.

Try solving on your own before revealing the answer!

Q1e. Solve $x^2 - 2x + 3 = 0$ by factoring and the quadratic formula. Check with a graph.

Background

Topic: Quadratic Equations

This equation may not factor with real roots, so the quadratic formula is especially useful here.

Key Terms and Formulas:

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Discriminant: $b^2 - 4ac$ (determines the nature of the roots)

Step-by-Step Guidance

1. Write the equation in standard form.

2. Attempt to factor, but if not possible, proceed to the quadratic formula.

3. Identify $a$, $b$, and $c$ and set up the quadratic formula.

4. Calculate the discriminant to determine if the roots are real or complex.

5. Write the expressions for the solutions, but do not compute the final values yet.

Try solving on your own before revealing the answer!

Q2a. Rewrite $f(x) = x^2 + 6x - 2$ in vertex form. Find the vertex, axis of symmetry, maximum or minimum, and intervals of increase/decrease. Sketch the graph to confirm.

Background

Topic: Quadratic Functions and Vertex Form

This question tests your ability to convert a quadratic from standard to vertex form and analyze its properties.

Key Terms and Formulas:

• Standard Form: $f(x) = ax^2 + bx + c$

• Vertex Form: $f(x) = a(x - h)^2 + k$

• Vertex: $(h, k)$

• Axis of Symmetry: $x = h$

Step-by-Step Guidance

1. Start with the standard form and complete the square to rewrite in vertex form.

2. Identify the values of $h$ and $k$ from the vertex form.

3. State the axis of symmetry using $x = h$.

4. Determine if the vertex is a maximum or minimum based on the sign of $a$.

5. Describe the intervals where the function is increasing or decreasing.

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Q2b. Rewrite $f(x) = x^2 - 4x + 8$ in vertex form. Find the vertex, axis of symmetry, maximum or minimum, and intervals of increase/decrease. Sketch the graph to confirm.

Background

Topic: Quadratic Functions and Vertex Form

This question is similar to the previous one, focusing on converting to vertex form and analyzing the function's properties.

Key Terms and Formulas:

• Standard Form: $f(x) = ax^2 + bx + c$

• Vertex Form: $f(x) = a(x - h)^2 + k$

• Vertex: $(h, k)$

• Axis of Symmetry: $x = h$

Step-by-Step Guidance

1. Start with the standard form and complete the square to rewrite in vertex form.

2. Identify the values of $h$ and $k$ from the vertex form.

3. State the axis of symmetry using $x = h$.

4. Determine if the vertex is a maximum or minimum based on the sign of $a$.

5. Describe the intervals where the function is increasing or decreasing.

Try solving on your own before revealing the answer!

Q3. For $f(x) = 2x(3x-1)^2(x+6)^3(-x+5)$, find the zeros and their multiplicities. Determine if the graph touches or crosses the x-axis at each zero. What is the degree and leading term? Sketch the end behavior.

Background

Topic: Polynomial Functions

This question tests your understanding of zeros, multiplicities, and the end behavior of polynomials.

Key Terms and Formulas:

• Zero: Value of $x$ where $f(x) = 0$

• Multiplicity: The exponent on each factor

• Degree: Sum of the exponents of all factors

• Leading Term: The term with the highest degree

Step-by-Step Guidance

1. Set each factor equal to zero to find the zeros.

2. Identify the multiplicity of each zero by looking at the exponent on each factor.

3. Determine whether the graph touches (even multiplicity) or crosses (odd multiplicity) the x-axis at each zero.

4. Add up the exponents to find the degree of the polynomial.

5. Multiply the leading coefficients from each factor to find the leading term and describe the end behavior.

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Q4. For $f(x) = 3 - x^2$ and $g(x) = 2x + 1$, find:

• a. $f(g(3))$

• b. $g(f(-1))$

• c. $f(g(x))$

• d. $g(f(x))$

Background

Topic: Function Composition

This question tests your ability to compose functions and evaluate them at specific values.

Key Terms and Formulas:

• Composition: $(f \circ g)(x) = f(g(x))$

Step-by-Step Guidance

1. For each part, substitute the inner function into the outer function as indicated.

2. For (a) and (b), evaluate the inner function at the given value, then use that result in the outer function.

3. For (c) and (d), write the composition as a new function in terms of $x$.

4. Do not compute the final values yet; set up the expressions for the student to finish.

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Q5. For $f(x) = x^3 + 1$ and $g(x) = \sqrt[3]{x - 1}$:

• a. Show that these functions are inverses by finding $f(g(x))$ and $g(f(x))$.

• b. Graph both functions and explain why the graph demonstrates they are inverses.

Background

Topic: Inverse Functions

This question tests your understanding of function inverses and their graphical relationship.

Key Terms and Formulas:

• Inverse: $f(g(x)) = x$ and $g(f(x)) = x$

• Graphical Test: Inverse functions are reflections over the line $y = x$

Step-by-Step Guidance

1. Substitute $g(x)$ into $f(x)$ and simplify to see if you get $x$.

2. Substitute $f(x)$ into $g(x)$ and simplify to see if you get $x$.

3. For the graph, sketch both functions and the line $y = x$ to observe the reflection property.

4. Explain how the graph confirms the inverse relationship.

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Q6. You inherit $15,000 and invest it at 9% annual interest. Find the amount after 30 years if interest is accrued:

• a. Annually

• b. Monthly

• c. Continuously

Background

Topic: Exponential Growth and Compound Interest

This question tests your ability to use compound interest formulas for different compounding periods.

Key Terms and Formulas:

• Annual Compounding: $A = P(1 + \frac{r}{n})^{nt}$

• Continuous Compounding: $A = Pe^{rt}$

• $P$ = principal, $r$ = annual rate (decimal), $n$ = number of periods per year, $t$ = years

Step-by-Step Guidance

1. Identify $P = 15,000$, $r = 0.09$, $t = 30$.

2. For (a), set $n = 1$ and write the formula for annual compounding.

3. For (b), set $n = 12$ and write the formula for monthly compounding.

4. For (c), use the continuous compounding formula.

5. Set up the expressions for each case, but do not compute the final values yet.

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Q7. Solve for the variable:

• a. $3 = \log_x 125$

• b. $3 = \log x$

• c. $0 = \ln(x)$

Background

Topic: Logarithmic and Exponential Equations

This question tests your ability to solve equations involving logarithms and natural logarithms.

Key Terms and Formulas:

• Logarithm: $\log_b a = c \implies b^c = a$

• Natural Logarithm: $\ln(x) = y \implies e^y = x$

Step-by-Step Guidance

1. For each equation, rewrite the logarithmic equation in its equivalent exponential form.

2. Solve for the variable in terms of exponents.

3. Set up the expressions for the solutions, but do not compute the final values yet.

Try solving on your own before revealing the answer!

Q8. Convert each exponential equation to a logarithmic equation:

• a. $64 = 4^x$

• b. $3^{-2} = \frac{1}{9}$

Background

Topic: Exponential and Logarithmic Equations

This question tests your ability to rewrite exponential equations in logarithmic form.

Key Terms and Formulas:

• Exponential to Logarithmic: $a = b^c \implies \log_b a = c$

Step-by-Step Guidance

1. For each equation, identify the base, exponent, and result.

2. Rewrite the equation in logarithmic form using the definition.

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Q9. Express as a sum and/or difference of logarithms (no exponents):

• a. $\log \left( \frac{1000x^3y^4}{wz^2} \right)$

• b. $\log_3 \left( \sqrt{\frac{27x^4y^3}{5}} \right)$

Background

Topic: Properties of Logarithms

This question tests your ability to expand logarithmic expressions using properties of logs.

Key Terms and Formulas:

• Product Rule: $\log_b(MN) = \log_b M + \log_b N$

• Quotient Rule: $\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N$

• Power Rule: $\log_b(M^k) = k \log_b M$

• Root Rule: $\log_b(\sqrt[n]{M}) = \frac{1}{n} \log_b M$

Step-by-Step Guidance

1. Apply the quotient rule to separate numerator and denominator.

2. Apply the product rule to expand products in the numerator and denominator.

3. Apply the power rule to bring exponents in front of the logs.

4. For roots, use the root rule to rewrite as a fractional exponent.

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Q10. Solve the following exponential equations:

• a. $e^{3t} = 9$

• b. $2^x = 18$

• c. $2^{x-2} = 16^x$

Background

Topic: Exponential Equations

This question tests your ability to solve equations where the variable is in the exponent.

Key Terms and Formulas:

• Take the natural logarithm or logarithm of both sides to bring down exponents.

• Properties of exponents: $a^{b} = e^{b \ln a}$

Step-by-Step Guidance

1. For each equation, take the logarithm of both sides to isolate the variable.

2. Use properties of logarithms to simplify the equation.

3. Solve for the variable, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Q11. Solve the logarithmic equation: $2\ln(\sqrt{x}) - \ln(6x-1) = 0$

Background

Topic: Logarithmic Equations

This question tests your ability to use properties of logarithms to combine and solve logarithmic equations.

Key Terms and Formulas:

• Power Rule: $k \ln a = \ln a^k$

• Product/Quotient Rule: $\ln a - \ln b = \ln \left( \frac{a}{b} \right)$

• Exponentiate both sides to solve for $x$.

Step-by-Step Guidance

1. Use the power rule to rewrite $2\ln(\sqrt{x})$ as a single logarithm.

2. Combine the logarithms using the quotient rule.

3. Set the resulting logarithm equal to zero and exponentiate both sides to solve for $x$.

4. Set up the equation for $x$, but do not compute the final value yet.

Try solving on your own before revealing the answer!