College Algebra: Key Concepts and Methods Study Guide

Quadratic Equations

Discriminant and Types of Solutions

The discriminant of a quadratic equation determines the number and type of solutions. For a quadratic equation in the form $ax^2 + bx + c = 0$, the discriminant is given by $D = b^2 - 4ac$.

• If $D > 0$: Two distinct real solutions.

• If $D = 0$: One real solution (a repeated root).

• If $D < 0$: Two complex (non-real) solutions.

Example: For $x^2 - 4x + 3 = 0$, $D = (-4)^2 - 4(1)(3) = 16 - 12 = 4 > 0$, so there are two real solutions.

Methods for Solving Quadratic Equations

Quadratic equations can be solved using several methods:

• Factoring: Express the equation as a product of binomials and set each factor to zero.

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

• Completing the Square: Rearrange the equation to form a perfect square trinomial.

• Graphing: Find the x-intercepts of the parabola represented by the equation.

Example: Solve $x^2 + 6x + 9 = 0$ by factoring: $(x + 3)^2 = 0 \Rightarrow x = -3$.

Functions and Their Properties

Domain, Range, Intercepts, and Values from Graphs

The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values).

• Intercepts: Points where the graph crosses the axes. The x-intercept is where $y=0$, and the y-intercept is where $x=0$.

• Evaluating Values: Substitute a given x-value into the function to find the corresponding y-value.

Example: For $f(x) = x^2$, the domain is all real numbers, and the range is $[0, \infty)$.

Even and Odd Functions; Symmetry

Even functions satisfy $f(-x) = f(x)$ and are symmetric about the y-axis. Odd functions satisfy $f(-x) = -f(x)$ and are symmetric about the origin.

• Example of Even: $f(x) = x^2$

• Example of Odd: $f(x) = x^3$

Piecewise Functions

A piecewise function is defined by different expressions for different intervals of the domain.

• Example: $f(x) = \begin{cases} x^2 & x < 0 \\ x + 1 & x \geq 0 \end{cases}$

Linear Equations and Lines

Forms of Linear Equations

Lines can be written in several forms:

• Slope-Intercept Form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Point-Slope Form: $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line.

Example: A line through $(2, 3)$ with slope $4$is$y - 3 = 4(x - 2)$.

Parallel and Perpendicular Lines

Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals.

• Parallel: If $m_1 = m_2$, lines are parallel.

• Perpendicular: If $m_1 \cdot m_2 = -1$, lines are perpendicular.

Example: A line perpendicular to $y = 2x + 1$ has slope $-\frac{1}{2}$.

Transformations of Functions

Translations and Transformations

Functions can be shifted, stretched, or reflected:

• Translation: $f(x) \to f(x - h) + k$ shifts the graph $h$ units right and $k$ units up.

• Reflection: $f(x) \to -f(x)$ reflects over the x-axis.

• Stretch/Compression: $f(x) \to af(x)$ stretches vertically by $a$.

Example: $f(x) = (x - 2)^2 + 3$ is $f(x) = x^2$ shifted right 2 units and up 3 units.

Absolute Value Function Transformations

The absolute value function $f(x) = |x|$ can be transformed similarly.

• Example: $f(x) = |x - 1| + 2$ is shifted right 1 unit and up 2 units.

Inverse Functions

Finding and Graphing Inverses

The inverse of a function $f(x)$, denoted $f^{-1}(x)$, reverses the roles of x and y. To find the inverse, solve $y = f(x)$ for $x$ in terms of $y$.

• Graphing: The graphs of $f(x)$ and $f^{-1}(x)$ are symmetric about the line $y = x$.

Example: For $f(x) = 2x + 3$, $y = 2x + 3 \Rightarrow x = \frac{y - 3}{2}$, so $f^{-1}(x) = \frac{x - 3}{2}$.

Distance and Circles

Distance Between Two Points

The distance between points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

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

Example: Between $(1, 2)$ and $(4, 6)$: $d = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{9 + 16} = 5$.

Equation and Graph of a Circle

The standard form of a circle with center $(h, k)$ and radius $r$ is:

• $(x - h)^2 + (y - k)^2 = r^2$

Graphing: Plot the center at $(h, k)$ and draw all points $r$ units away.

Example: $(x - 2)^2 + (y + 1)^2 = 9$ is a circle centered at $(2, -1)$ with radius $3$.

Summary Table: Methods and Properties

Topic	Key Formula/Property	Example
Quadratic Discriminant	$D = b^2 - 4ac$	$x^2 - 4x + 3 = 0$, $D = 4$
Quadratic Formula	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$	$x^2 + 2x + 1 = 0$
Distance Formula	$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$	$(1,2)$ and $(4,6)$: $d = 5$
Circle Equation	$(x - h)^2 + (y - k)^2 = r^2$	Center $(2, -1)$, $r = 3$
Line Forms	$y = mx + b$, $y - y_1 = m(x - x_1)$	Slope $4$, point$(2,3)$

Additional info: Academic context and examples have been expanded for clarity and completeness.