College Algebra Study Notes: Linear Equations, Circles, Functions, and Transformations

Linear Equations and Their Properties

Distance Between Two Points

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane is given by the distance formula:

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

• Example: $\sqrt{52} = \sqrt{4 \times 13} = 2\sqrt{13}$

Midpoint Formula

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

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

• Example: $\left( \frac{-1.45 + 2}{2}, \frac{3 - 1}{2} \right) = (0.275, 1)$

Slope of a Line

The slope $m$ of a line passing through $(x_1, y_1)$ and $(x_2, y_2)$ is:

• Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Example: $m = \frac{-1 - 3}{5 - 1} = \frac{-4}{4} = -1$

Point-Slope Form of a Line

The equation of a line with slope $m$ passing through $(x_1, y_1)$:

• Formula: $y - y_1 = m(x - x_1)$

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

Finding Intercepts

• x-intercept: Set $y = 0$ and solve for $x$.

• y-intercept: Set $x = 0$ and solve for $y$.

• Example: For $4x - 5y = -20$, set $y = 0$: $4x = -20 \Rightarrow x = -5$ (x-intercept: $(-5, 0)$). Set $x = 0$: $-5y = -20 \Rightarrow y = 4$ (y-intercept: $(0, 4)$).

Equations of Circles

Standard Form of a Circle

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

• Formula: $(x - h)^2 + (y - k)^2 = r^2$

• Example: $(x - 0)^2 + (y - 1)^2 = 4$ (center $(0, 1)$, radius $2$)

Completing the Square

To write a general quadratic equation in standard form, complete the square:

• Example: $x^2 + y^2 - 2x + 4y + 4 = 0$ becomes $(x - 1)^2 + (y + 2)^2 = 9$ (center $(1, -2)$, radius $3$)

Functions and Their Properties

Definition of a Function

A function is a relation in which each input (x-value) maps to exactly one output (y-value).

• Vertical Line Test: A graph represents a function if no vertical line intersects it more than once.

• Domain and Range:

  • Domain (D): Set of all possible input values (x-values).

  • Range (R): Set of all possible output values (y-values).

  • Example: $D: (-\infty, \infty)$, $R: [2, \infty)$

Interval Notation

• Open Interval: $(a, b)$ excludes endpoints.

• Closed Interval: $[a, b]$ includes endpoints.

• Union: $(-\infty, -3) \cup (-3, 1) \cup (1, \infty)$

Evaluating Functions

• Substitute: Replace $x$ with the given value.

• Example: If $f(x) = \frac{2x+1}{2x+2}$, then $f(1) = \frac{2 \times 1 + 1}{2 \times 1 + 2} = \frac{3}{4}$

Finding Intercepts of Functions

• x-intercept: Set $y = 0$ and solve for $x$.

• y-intercept: Set $x = 0$ and solve for $y$.

• Example: For $y = \frac{x+1}{x+2}$, x-intercept: $x = -1$, y-intercept: $y = \frac{1}{2}$

Function Transformations

Shifting and Stretching Graphs

Transformations change the position or shape of a graph:

• Vertical Shift: $f(x) + k$ shifts up by $k$ units.

• Horizontal Shift: $f(x + h)$ shifts left by $h$ units.

• Vertical Stretch: $a f(x)$ stretches by factor $a$.

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

• Example: $y = x^3$ shifted left 1, stretched vertically by 2, reflected over x-axis, and shifted up 3: $y = -2(x+1)^3 + 3$

Symmetry of Functions

• Even Function: $f(-x) = f(x)$ (symmetric about y-axis)

• Odd Function: $f(-x) = -f(x)$ (symmetric about origin)

• Example: $f(x) = \frac{x^2}{2x+1}$ is neither even nor odd

Quadratic Functions and Their Properties

Standard Form and Vertex

• Standard Form: $y = ax^2 + bx + c$

• Vertex: $x = -\frac{b}{2a}$

• Example: For $y = x^2 - 4x + 1$, vertex at $x = 2$

Finding Maximum and Minimum Values

• Local Minimum: Lowest point in a small region

• Absolute Minimum: Lowest point on the entire graph

• Example: For $y = x^2 - 4x + 1$, absolute minimum at $x = 2$

Summary Table: Function Transformations

Transformation	Equation	Effect
Vertical Shift Up	$f(x) + k$	Add $k$ to all y-values
Horizontal Shift Left	$f(x + h)$	Subtract $h$ from all x-values
Vertical Stretch	$a f(x)$	Multiply all y-values by $a$
Reflection over x-axis	$-f(x)$	Flip graph over x-axis

Additional info:

• Some examples and explanations were expanded for clarity and completeness.

• Graph sketches referenced in the notes were described in text for accessibility.