College Algebra Final Exam Study Guide: Key Concepts and Worked Examples

Quick Reference Rules and Formulas

• Absolute Value Inequality: If $|A| > b$, then $A > b$ or $A < -b$. If $|A| < b$, then $-b < A < b$.

• Systems with $x^2$ and $y^2$: Let $u = x^2$ and $v = y^2$, solve as a linear system, then take square roots at the end.

• Exponential Equation: To solve $a^x = b$, use $x = \frac{\ln(b)}{\ln(a)}$ or $x = \frac{\log(b)}{\log(a)}$.

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

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

• Circle (Standard Form): $(x - h)^2 + (y - k)^2 = r^2$

• Function Check: A relation is a function if each $x$ is matched with only one $y$.

• Inverse of a Linear Function: Replace $f(x)$ with $y$, switch $x$ and $y$, then solve for $y$.

• Rational Equations: Find restrictions first. Any answer that makes a denominator zero is rejected.

Worked Examples by Topic

1. Absolute Value Inequalities

Absolute value inequalities are solved by splitting into two cases, reflecting the definition of absolute value.

• Example: Solve $|3x - 3| > 9$

• Case 1: $3x - 3 > 9 \implies x > 4$

• Case 2: $3x - 3 < -9 \implies x < -2$

• Final Answer: $(-\infty, -2) \cup (4, \infty)$

2. Systems of Equations with Quadratic Variables

When both $x^2$ and $y^2$ appear, substitute $u = x^2$, $v = y^2$ to create a linear system.

• Example: Solve

  • $2x^2 + 5y^2 = 13$

  • $5x^2 - 3y^2 = 17$

• Let $u = x^2$, $v = y^2$:

• Solve:

  • $2u + 5v = 13$

  • $5u - 3v = 17$

• Solution: $u = 4$, $v = 1$

• So $x = \pm2$, $y = \pm1$

• Final Answer: $\{(2, 1), (2, -1), (-2, 1), (-2, -1)\}$

3. Exponential Equations

To solve equations of the form $a^x = b$, use logarithms to isolate $x$.

• Example: $7^x = 11$

• $x = \frac{\ln(11)}{\ln(7)} \approx 1.23$

• Final Answer: $x = \frac{\ln(11)}{\ln(7)} \approx 1.23$

4. Equations of Lines

Lines can be written in point-slope or slope-intercept form.

• Given: Slope $m = -2$, point $(-2, -\frac{1}{2})$

• Point-slope: $y + \frac{1}{2} = -2(x + 2)$

• Slope-intercept: $y = -2x - \frac{9}{2}$

• Final Answer: Point-slope: $y + \frac{1}{2} = -2(x + 2)$; Slope-intercept: $y = -2x - \frac{9}{2}$

5. Systems of Three Variables

Systems with three variables can be solved using substitution or elimination.

• Example:

  • $2x - 8y + 5z = -1$

  • $x + 2y - z = 0$

  • $8x - y - z = -16$

• Solution: $x = -1$, $y = 3$, $z = 5$

• Final Answer: $(-1, 3, 5)$

6. Graphing Exponential Functions

To graph $f(x) = 2^x$, create a table of values and plot the points.

x	f(x)
-2	1/4
-1	1/2
0	1
1	2
2	4

The graph is an increasing exponential curve.

7. Vertical Line Test for Functions

A graph represents a function if every vertical line intersects it at most once.

• Example: An upward-opening parabola passes the vertical line test.

• Conclusion: $y$ is a function of $x$.

8. Radical Equations

To solve equations with radicals, isolate the radical and square both sides. Always check for extraneous solutions.

• Example: $\sqrt{4x + 29} = x + 6$

• Square both sides: $4x + 29 = (x + 6)^2$

• Expand: $4x + 29 = x^2 + 12x + 36$

• Rearrange: $x^2 + 8x + 7 = 0$

• Factor: $(x + 1)(x + 7) = 0$

• Possible solutions: $x = -1$, $x = -7$

• Check both: Only $x = -1$ is valid.

• Final Answer: $\{-1\}$

9. Completing the Square for Circles

To write the equation of a circle in standard form, complete the square for both $x$ and $y$ terms.

• Example: $x^2 + y^2 + 4x + 4y + 7 = 0$

• Group: $(x^2 + 4x) + (y^2 + 4y) = -7$

• Complete the square: Add $4$to$x$terms and $4$ to $y$ terms.

• Standard form: $(x + 2)^2 + (y + 2)^2 = 1$

• Center: $(-2, -2)$; Radius: $1$

10. Graphing Constant Functions

The graph of $f(x) = 4$ is a horizontal line at $y = 4$.

11. Distance Between Two Points

Use the distance formula to find the length between two points in the plane.

• Example: $(-3, -4)$ and $(3, -3)$

• $d = \sqrt{(3 - (-3))^2 + (-3 - (-4))^2} = \sqrt{6^2 + 1^2} = \sqrt{37} \approx 6.08$

12. Solving Quadratics by Factoring

Factor out the greatest common factor (GCF) and set each factor to zero.

• Example: $2x^2 + 10x = 0$

• Factor: $2x(x + 5) = 0$

• Solutions: $x = 0$, $x = -5$

13. Logarithmic and Exponential Forms

Convert between exponential and logarithmic forms using the definition: $a^b = c \iff \log_a(c) = b$.

• Example: $2^4 = x$ becomes $\log_2(x) = 4$

14. Quadratic Formula with Complex Solutions

When the discriminant is negative, solutions are complex numbers.

• Example: $x^2 - 2x + 17 = 0$

• $x = \frac{2 \pm \sqrt{4 - 68}}{2} = \frac{2 \pm 8i}{2} = 1 \pm 4i$

15. Compound Inequalities

Solve each part of the inequality and express the solution in interval notation.

• Example: $-8 < 3x + 1 \leq 7$

• Subtract $1$:$-9 < 3x \leq 6$

• Divide by $3$:$-3 < x \leq 2$

• Final Answer: $(-3, 2]$

16. Determining Functions from Relations

A relation is not a function if any $x$-value is paired with more than one $y$-value.

• Example: $\{(1,4), (1,8), (9,4), (9,8)\}$

• $x = 1$ maps to both $4$and $8$; not a function.

• Domain: $\{1, 9\}$; Range: $\{4, 8\}$

17. Inverse of a Linear Function

To find the inverse, switch $x$ and $y$ and solve for $y$.

• Example: $f(x) = 5x + 3$

• Let $y = 5x + 3$; switch: $x = 5y + 3$

• Solve: $y = \frac{x - 3}{5}$

• Inverse: $f^{-1}(x) = \frac{x - 3}{5}$

• Check: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$

18. Exponential and Logarithmic Conversion

Convert between logarithmic and exponential forms as needed.

• Example: $2 = \log_b(25)$

• Exponential form: $b^2 = 25$

• Possible base: $b = 5$ (since $b > 0$, $b \neq 1$)

19. Rational Equations and Restrictions

Always check for values that make denominators zero and reject such solutions.

• Example: $\frac{8x}{x + 1} = 4 - \frac{8}{x + 1}$

• Restriction: $x + 1 \neq 0 \implies x \neq -1$

• Solve: $x = -1$ (but this is not allowed)

• Final Answer: No solution / $\emptyset$

Answer Formats

• Interval notation: $(-\infty, -2) \cup (4, \infty)$

• Ordered pair: $(2, -1)$

• Ordered triple: $(-1, 3, 5)$

• Set of ordered pairs: $\{(2, 1), (2, -1), (-2, 1), (-2, -1)\}$

• Empty set: $\emptyset$

• Exact radical answer: $\sqrt{37}$

• Complex number answer: $1 \pm 4i$

• Logarithmic form: $\log_2(x) = 4$

• Exponential form: $b^2 = 25$

• Circle standard form: $(x + 2)^2 + (y + 2)^2 = 1$

Test-Day Tips

• For equations with radicals, rational expressions, or absolute value, always write the rule first, solve carefully, and check whether the answer is allowed (i.e., does not violate any restrictions).