College Algebra: Study Guide for Test 1 (Chapters 1 & 2)

Graphs, Functions, and Models

Distance and Midpoint

Understanding the distance and midpoint between two points is fundamental in coordinate geometry and is often used to analyze graphs and geometric figures.

• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ in the plane is given by: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Midpoint Formula: The midpoint of the segment connecting $(x_1, y_1)$ and $(x_2, y_2)$ is: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Example: For points (2, 3) and (6, 7):

  • Distance: $d = \sqrt{(6-2)^2 + (7-3)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}$

  • Midpoint: $M = (\frac{2+6}{2}, \frac{3+7}{2}) = (4, 5)$

Circles: Center, Radius, and Equation

The equation of a circle is derived from the distance formula and describes all points equidistant from a fixed center.

• Standard Form: The equation of a circle with center $(h, k)$ and radius $r$ is: $(x - h)^2 + (y - k)^2 = r^2$

• Finding Center and Radius: Given an equation in standard form, the center is $(h, k)$ and the radius is $r$.

• From Graph: Identify the center and a point on the circle to determine the radius.

• Example: $(x - 3)^2 + (y + 2)^2 = 16$ has center $(3, -2)$ and radius $4$.

Functions: Domain, Range, and Evaluation

Domain and Range

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).

• Simple Functions: For $f(x) = x^2$, domain is all real numbers; range is $[0, \infty)$.

• From Equations: Exclude values that make denominators zero or result in negative values under even roots.

• From Graphs: Estimate the lowest and highest x-values (domain) and y-values (range) shown on the graph.

• Example: $f(x) = \frac{1}{x-2}$ has domain $(-\infty, 2) \cup (2, \infty)$; range $(-\infty, 0) \cup (0, \infty)$.

Evaluating Functions

To evaluate a function, substitute the given input value into the function's formula.

• Example: If $f(x) = 2x + 3$, then $f(4) = 2(4) + 3 = 11$.

Graphing Functions and the Vertical Line Test

Graphing helps visualize the behavior of functions. The vertical line test determines if a graph represents a function: if any vertical line crosses the graph more than once, it is not a function.

• Example: The graph of $y = x^2$ passes the vertical line test; the graph of a circle does not.

Linear Functions and Equations of Lines

Linear Functions from Tables

Linear functions have a constant rate of change. Given a table of values, check if the difference in y-values divided by the difference in x-values is constant.

• Formula: $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Example: If $x$ increases by 1 and $y$ increases by 2 each time, $m = 2$.

Average Rate of Change

The average rate of change of a function between $x = a$ and $x = b$ is the change in $f(x)$ divided by the change in $x$.

• Formula: $\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

• Example: For $f(x) = x^2$, from $x=1$ to $x=3$: $\frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{2} = 4$

Finding Equations of Lines

To find the equation of a line, use the slope-intercept form or point-slope form, depending on the information given.

• Slope-Intercept Form: $y = mx + b$

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

• Parallel Lines: Have the same slope.

• Perpendicular Lines: Slopes are negative reciprocals: $m_1 \cdot m_2 = -1$.

• Example: Line through (2, 3) with slope 4: $y - 3 = 4(x - 2)$

Linear Word Problems and Finding Zeroes

Linear word problems often involve translating real-world situations into equations. The zero of a linear function is the x-value where $f(x) = 0$.

• Finding Zero: Set $f(x) = 0$ and solve for $x$.

• Example: $f(x) = 2x - 6$; $0 = 2x - 6 \implies x = 3$.

More on Functions

Increasing, Decreasing, and Constant Intervals

From a graph, identify where the function is rising (increasing), falling (decreasing), or flat (constant).

• Increasing: As $x$ increases, $f(x)$ increases.

• Decreasing: As $x$ increases, $f(x)$ decreases.

• Constant: $f(x)$ remains the same as $x$ increases.

Relative Maximum and Minimum

A relative maximum is a point where the function reaches a local highest value; a relative minimum is a local lowest value.

• Identified from Graph: Look for peaks (maxima) and valleys (minima).

Piecewise Defined Functions

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

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

• To evaluate, determine which interval $x$ falls into and use the corresponding formula.

Operations with Functions

Functions can be added, subtracted, multiplied, or divided (except where the denominator is zero).

• Sum: $(f+g)(x) = f(x) + g(x)$

• Difference: $(f-g)(x) = f(x) - g(x)$

• Product: $(fg)(x) = f(x) \cdot g(x)$

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

The Difference Quotient

The difference quotient is a formula that measures the average rate of change of a function and is foundational in calculus.

• Formula: $\frac{f(x+h) - f(x)}{h}$

• Example: For $f(x) = x^2$: $\frac{(x+h)^2 - x^2}{h} = \frac{x^2 + 2xh + h^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h$

Composition of Functions

Composition involves applying one function to the result of another: $(f \circ g)(x) = f(g(x))$.

• Evaluating: Substitute $g(x)$ into $f$.

• Domain: The domain of $f \circ g$ consists of all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.

• Example: If $f(x) = x^2$ and $g(x) = x+1$, then $(f \circ g)(x) = f(g(x)) = (x+1)^2$.

Transformations of Functions

Transformations change the position or shape of a function's graph. Common transformations include translations, reflections, stretches, and compressions.

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

• Horizontal Shift: $f(x - h)$ shifts right by $h$ units.

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

• Vertical Stretch/Compression: $a f(x)$ stretches if $|a| > 1$, compresses if $0 < |a| < 1$.

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