Precalculus Practice Final Study Guide: Key Concepts and Problem Types

Equations and Inequalities

Solving Linear Equations and Inequalities

Linear equations and inequalities are foundational in precalculus, involving variables to the first power and basic algebraic manipulation.

• Linear Equation: An equation of the form $ax + b = cx + d$.

• Linear Inequality: An inequality such as $ax + b < cx + d$.

• Absolute Value Equations/Inequalities: Equations or inequalities involving $|ax + b|$ require considering both positive and negative cases.

• Example: Solve $2x + 3 = 4x - 2$.

  • Subtract $2x$ from both sides: $3 = 2x - 2$

  • Add $2$to both sides: $5 = 2x$

  • Divide by $2$:$x = 2.5$

• Example: Solve $|7 - 2x| = 3$.

  • Set $7 - 2x = 3$ and $7 - 2x = -3$.

  • First case: $7 - 2x = 3$ → $2x = 4$ → $x = 2$

  • Second case: $7 - 2x = -3$ → $2x = 10$ → $x = 5$

Functions and Their Properties

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the domain.

• Definition: $f(x) = \begin{cases} x+1 & x \leq 2 \\ 3 & 2 < x \leq 4 \\ -2x+7 & x > 4 \end{cases}$

• Key Points:

  • Identify intervals and corresponding expressions.

  • Graph each piece over its interval.

Function Operations and Domain

Functions can be added, subtracted, multiplied, divided, and composed. The domain of the resulting function is the intersection of the domains of the original functions, considering any new restrictions.

• Sum: $(f+g)(x) = f(x) + g(x)$

• Quotient: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

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

• Example: If $f(x) = 2x + 3$, $g(x) = 3x - 1$, then $(f \circ g)(x) = f(g(x)) = 2(3x - 1) + 3 = 6x - 2 + 3 = 6x + 1$

Quadratic Functions

Vertex, Axis of Symmetry, and Graphing

Quadratic functions are of the form $f(x) = ax^2 + bx + c$. Their graphs are parabolas.

• Vertex: The point $(h, k)$ where $h = -\frac{b}{2a}$ and $k = f(h)$.

• Axis of Symmetry: The vertical line $x = h$.

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

• Discriminant: $D = b^2 - 4ac$ determines the nature of the roots.

  • $D > 0$: Two real roots

  • $D = 0$: One real root

  • $D < 0$: Two complex roots

• Example: For $f(x) = x^2 + 2x - 3$:

  • Vertex: $h = -1$, $k = -4$

  • Axis of symmetry: $x = -1$

  • Discriminant: $D = 2^2 - 4(1)(-3) = 4 + 12 = 16$ (two real roots)

  • Roots: $x = \frac{-2 \pm \sqrt{16}}{2} = \frac{-2 \pm 4}{2}$ → $x = 1$, $x = -3$

End Behavior of Polynomials

The end behavior of a polynomial function describes how the function behaves as $x$ approaches $\infty$ or $-\infty$.

• Leading Term Test: The term with the highest degree determines end behavior.

• Example: For $f(x) = 7x^5 - 9x^4 + 2x - 1$, as $x \to \infty$, $f(x) \to \infty$; as $x \to -\infty$, $f(x) \to -\infty$.

Linear Functions and Geometry

Finding the Equation of a Line

To find the equation of a line parallel to a given line and passing through a point, use the point-slope form and the fact that parallel lines have equal slopes.

• General Form: $y = mx + b$

• Parallel Lines: Same slope $m$

• Example: Find the equation of the line through $(3, -2)$ parallel to $2x - y = 5$.

  • Rewrite $2x - y = 5$ as $y = 2x - 5$ (slope $m = 2$)

  • Use point-slope: $y + 2 = 2(x - 3)$ → $y = 2x - 8$

Midpoint Formula

The midpoint of a segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is:

• Formula: $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Example: Midpoint of $(10, 2)$ and $(-2, 6)$ is $\left( \frac{10 + (-2)}{2}, \frac{2 + 6}{2} \right) = (4, 4)$

Transformations of Functions

Describing Transformations

Transformations include shifts, stretches, compressions, and reflections. The order of operations is important.

• Example: $y = -4|x - 2| + 3$

  • Shift right by 2 units

  • Vertical stretch by 4

  • Reflect over the x-axis

  • Shift up by 3 units

Complex Numbers

Simplifying Complex Expressions

Complex numbers are of the form $a + bi$, where $i = \sqrt{-1}$.

• Addition/Subtraction: Combine real and imaginary parts.

• Multiplication/Division: Use conjugates for division.

• Example: $\frac{4 + i}{7 - 3i}$

  • Multiply numerator and denominator by the conjugate $7 + 3i$.

  • Result: $\frac{(4 + i)(7 + 3i)}{(7 - 3i)(7 + 3i)}$

Polynomials: Division and Factoring

Polynomial Long Division and Synthetic Division

These methods are used to divide polynomials and factor them.

• Long Division: Divide as with numbers, subtracting multiples of the divisor.

• Synthetic Division: Shortcut for division by $x - c$.

• Example: Divide $x^3 + 2x^2 - 17x - 10$ by $x - 5$.

Rational Functions

Asymptotes, Intercepts, and Graphing

Rational functions are quotients of polynomials. Their graphs may have vertical and horizontal asymptotes.

• Vertical Asymptotes (VA): Values of $x$ that make the denominator zero.

• Horizontal Asymptotes (HA): Determined by degrees of numerator and denominator.

• Y-intercept: Set $x = 0$.

• X-intercept: Set numerator equal to zero.

• Example: $f(x) = \frac{3x}{(x+1)(x-2)}$

  • VA: $x = -1$, $x = 2$

  • HA: $y = 0$ (degree numerator < denominator)

  • Y-intercept: $f(0) = 0$

  • X-intercept: $3x = 0$ → $x = 0$

Logarithms and Exponentials

Properties and Solving Equations

Logarithms and exponentials are inverse functions. Their properties allow simplification and solving equations.

• Definition: $x = \log_b a$ means $b^x = a$

• Properties:

  • $\log_b (xy) = \log_b x + \log_b y$

  • $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$

  • $\log_b (x^k) = k \log_b x$

• Example: Rewrite $2\log_a m - 3\log_a n$ as a single logarithm:

  • $2\log_a m = \log_a m^2$

  • $3\log_a n = \log_a n^3$

  • So, $\log_a m^2 - \log_a n^3 = \log_a \left(\frac{m^2}{n^3}\right)$

Inverse Functions

Finding the inverse of a function involves solving for $x$ in terms of $y$ and swapping variables.

• Example: $f(x) = 2x + 3$

  • Let $y = 2x + 3$

  • Solve for $x$: $x = \frac{y - 3}{2}$

  • Inverse: $f^{-1}(x) = \frac{x - 3}{2}$

Summary Table: Types of Problems

Topic	Example Problem	Key Concept
Linear Equations	$2x + 3 = 4x - 2$	Solving for $x$
Absolute Value	$|7 - 2x| = 3$	Two cases for solution
Quadratic Functions	$f(x) = x^2 + 2x - 3$	Vertex, axis, roots
Piecewise Functions	Graph $f(x)$ as defined	Multiple expressions
Rational Functions	$f(x) = \frac{3x}{(x+1)(x-2)}$	Asymptotes, intercepts
Logarithms	$x = \log_3 81$	Inverse of exponentials
Function Operations	$(f \circ g)(x)$	Composition
Complex Numbers	$\frac{4 + i}{7 - 3i}$	Simplification

*Additional info: Some explanations and examples have been expanded for clarity and completeness beyond the original questions.*