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

3. If factoring is not possible or to check your work, use the quadratic formula. Identify $a$, $b$, and $c$ from your standard form.

4. Set up the quadratic formula with your values for $a$, $b$, and $c$, but do not compute the final values yet.

5. To check graphically, sketch the parabola $y = x^2 - 20x + 99$ and identify where it crosses the $x$-axis. These $x$-intercepts correspond to your solutions.

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Q1b. Solve the quadratic equation: $-5x^2 + 40x = 0$ 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 is similar to the previous one, focusing on solving a quadratic equation by factoring and the quadratic formula, and interpreting the solutions graphically.

Key Terms and Formulas:

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

• Factoring: Taking out common factors and expressing as a product of binomials.

• 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 $c$ and set up the quadratic formula.

5. Sketch the graph of $y = -5x^2 + 40x$ and observe where it crosses the $x$-axis.

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Q1c. Solve the quadratic equation: $x^2 - 144 = 0$ 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 (Difference of Squares)

This question involves solving a quadratic equation that is a difference of squares, which is a special factoring case.

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. Write the equation in standard form: $x^2 - 144 = 0$.

2. Recognize this as a difference of squares and factor accordingly.

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

4. Alternatively, use the quadratic formula with $a = 1$, $b = 0$, $c = -144$.

5. Sketch the graph of $y = x^2 - 144$ and identify the $x$-intercepts.

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Q1d. Solve the quadratic equation: $3x^2 - 6x - 9 = 0$ 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 asks you to solve a quadratic equation that may not factor easily, so the quadratic formula is especially useful.

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Write the equation in standard form: $3x^2 - 6x - 9 = 0$.

2. Check if the equation can be factored by looking for two numbers that multiply to $a \cdot c$ and add to $b$.

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

4. Set up the quadratic formula with your values, but do not compute the final values yet.

5. Sketch the graph of $y = 3x^2 - 6x - 9$ and observe the $x$-intercepts.

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Q1e. Solve the quadratic equation: $x^2 - 2x + 3 = 0$ 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 (Complex Roots)

This question may result in complex solutions, so using the quadratic formula is important.

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: $x^2 - 2x + 3 = 0$.

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. Sketch the graph of $y = x^2 - 2x + 3$ and observe whether it crosses the $x$-axis.

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Q2a. Rewrite $f(x) = x^2 + 6x - 2$ in vertex form. Find the vertex, axis of symmetry, maximum or minimum, and intervals where the function is increasing or decreasing. Sketch a graph to confirm your answers.

Background

Topic: Quadratic Functions (Vertex Form)

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

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 $f(x) = x^2 + 6x - 2$.

2. Complete the square to rewrite the function in vertex form.

3. Identify the vertex $(h, k)$ from the vertex form.

4. State the axis of symmetry and whether the vertex is a maximum or minimum.

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 where the function is increasing or decreasing. Sketch a graph to confirm your answers.

Background

Topic: Quadratic Functions (Vertex Form)

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

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 $f(x) = x^2 - 4x + 8$.

2. Complete the square to rewrite the function in vertex form.

3. Identify the vertex $(h, k)$ from the vertex form.

4. State the axis of symmetry and whether the vertex is a maximum or minimum.

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

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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 (Zeros, Multiplicity, End Behavior)

This question tests your understanding of how to find zeros of a polynomial, their multiplicities, and how these affect the graph. It also asks about the degree and leading term, which determine end behavior.

Key Terms and Formulas:

• Zero: Value of $x$ where $f(x) = 0$

• Multiplicity: The exponent on each factor; determines if the graph crosses or touches the x-axis at that zero

• Degree: Sum of the exponents of all factors

• Leading Term: The term with the highest degree when expanded

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 for each zero: if the multiplicity is odd, the graph crosses the x-axis; if even, it touches and turns around.

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 use this to describe the end behavior.

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Q4. Given $f(x) = 3 - x^2$ 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, start by evaluating the inner function first.

2. Plug the result into the outer function.

3. For (c) and (d), write the composed 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. Given $f(x) = x^3 + 1$ 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 inverse functions and how to verify two functions are inverses algebraically and graphically.

Key Terms and Formulas:

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

• Graph of inverses: Symmetry about 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 check for symmetry.

4. Explain how the symmetry confirms the inverse relationship.

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Q6. Suppose 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 different compound interest formulas to calculate future value.

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 compounding 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 use the annual compounding formula.

3. For (b), set $n = 12$ and use the same formula.

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

5. Set up each formula with the given values, but do not compute the final amounts 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: $y = \log_b x \implies b^y = x$

• Natural Logarithm: $\ln(x) = \log_e x$

Step-by-Step Guidance

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

2. Solve for the variable using algebraic manipulation.

3. Do not compute the final value; leave the solution in a form ready for evaluation.

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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: $a^b = c \implies \log_a c = b$

Step-by-Step Guidance

1. Identify the base, exponent, and result in each equation.

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

3. Do not solve for the variable; just set up the logarithmic equation.

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Q9. Express as a sum and/or difference of logarithms (with 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 such as the product, quotient, and power rules.

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: $\log_b (M^k) = k \log_b M$

• Square Root: $\sqrt{A} = A^{1/2}$

Step-by-Step Guidance

1. Break up the numerator and denominator using the product and quotient rules.

2. Apply the power rule to move exponents in front of the logarithms.

3. For square roots, rewrite as exponents and apply the power rule.

4. Do not simplify to the final expanded form; leave the expression set up for the student to finish.

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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, often by taking logarithms or rewriting bases.

Key Terms and Formulas:

• Natural Logarithm: $\ln(x)$

• Logarithm: $\log_b x$

• Exponent Rules: $a^{mn} = (a^m)^n$

Step-by-Step Guidance

1. For each equation, isolate the exponential expression if needed.

2. Take the natural logarithm or logarithm of both sides to bring down the exponent.

3. Solve for the variable using algebraic manipulation.

4. For (c), rewrite $16$as a power of $2$ to have the same base.

5. Do not compute the final value; leave the solution set up for the student to finish.

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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: $k \ln(x) = \ln(x^k)$

• Difference Rule: $\ln(a) - \ln(b) = \ln(\frac{a}{b})$

• Exponent Rule: $\sqrt{x} = x^{1/2}$

Step-by-Step Guidance

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

2. Simplify $(\sqrt{x})^2$ to $x$.

3. Combine the logarithms using the difference rule.

4. Set the resulting logarithm equal to zero and rewrite in exponential form.

5. Do not solve for $x$ yet; leave the equation set up for the student to finish.

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