College Algebra: Functions, Graphs, Transformations, and Applications – Study Guide

Functions and Their Graphs

Identifying Key Features of Graphs

Understanding the graphical behavior of functions is fundamental in College Algebra. Key features include relative minima and maxima, domain, range, and intercepts.

• Relative Minima/Maxima: Points where the function reaches local lowest or highest values.

• Domain: The set of all possible input values (x-values) for which the function is defined.

• Range: The set of all possible output values (y-values) the function can take.

• Intercepts: Points where the graph crosses the axes.

Example: For a given graph, relative minima at $(-8,-4), (0,0), (5,0)$; domain $[-10,7]$; range $[-10,10]$; $f(x)=0$ at $x=-10, 0, 5$.

Quadratic Functions

Vertex and X-Intercepts

Quadratic functions are polynomials of degree 2 and have parabolic graphs. The vertex is the highest or lowest point, and x-intercepts are where the function equals zero.

• Standard Form: $f(x) = ax^2 + bx + c$

• Vertex Formula: $x = -\frac{b}{2a}$

• X-Intercepts: Solve $ax^2 + bx + c = 0$ using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Example: For $f(x) = x^2 + 6x + 2$, vertex at $x = -3$, $f(-3) = 11$; x-intercepts: $x = -6 \pm 2\sqrt{7}$.

Piecewise and Absolute Value Functions

Graphing and Transformations

Piecewise functions are defined by different expressions over different intervals. Absolute value functions have a characteristic 'V' shape and can be transformed by shifting, stretching, or reflecting.

• Absolute Value Function: $f(x) = |x|$

• Reflection: $g(x) = -\frac{1}{2}|x|$ reflects across the x-axis and stretches horizontally by a factor of 2.

• Shifting: $h(x) = 5|x| - 2 + 3$ shifts the graph up by 3 and stretches vertically by 5.

Example: $f(x) = |x|$ reflected and stretched: $g(x) = -\frac{1}{2}|x|$.

Polynomial Functions

Zeros, Multiplicity, and End Behavior

Polynomials can be analyzed by their zeros (roots), multiplicity, and how they behave as $x$ approaches infinity.

• Zero: Value of $x$ where $f(x) = 0$.

• Multiplicity: The number of times a zero occurs; determines if the graph touches or crosses the axis.

• End Behavior: Determined by the leading term as $x \to \infty$ or $x \to -\infty$.

Example Table:

Zero(s)	Multiplicity	Touch or Cross
x = -3	2	Touch
x = 1	1	Cross

Transformations of Functions

Describing and Applying Transformations

Transformations include shifting, reflecting, stretching, and compressing graphs. These are applied to parent functions to create new graphs.

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

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

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

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

Example: $g(x) = -f(x+2)$ is $f(x)$ reflected across the x-axis and shifted left by 2.

Rational Functions

Domain, Asymptotes, and Intercepts

Rational functions are ratios of polynomials. Their graphs may have vertical and horizontal asymptotes, and their domains exclude values that make the denominator zero.

• Domain: All real numbers except where denominator is zero.

• Vertical Asymptote: Set denominator equal to zero and solve for $x$.

• Horizontal Asymptote: Determined by degrees of numerator and denominator.

• X-Intercept: Set numerator equal to zero and solve for $x$.

Example: For $f(x) = \frac{x^2 + 6x + 5}{2x^2 + 7x + 5}$:

• Domain: $x \neq -1, -\frac{5}{2}$

• Vertical Asymptotes: $x = -1, x = -\frac{5}{2}$

• Horizontal Asymptote: $y = \frac{1}{2}$

• X-Intercept: $x = -5$

Modeling and Applications

Revenue and Maximization Problems

Algebraic models are used to represent real-world scenarios, such as maximizing revenue or profit.

• Revenue Function: $R(x) = x \left(350 - \frac{1}{2}x \right)$

• Maximizing Revenue: Find vertex of quadratic revenue function.

• Price for Maximum Revenue: Substitute $x$ value into price function.

Example: Maximum revenue at $x = 150$, $R(150) = 57500$.

Scatterplots and Regression

Identifying Relationships

Scatterplots are used to visualize relationships between variables. The pattern may be linear, nonlinear, or neither.

• Linear Relationship: Points form a straight line.

• Nonlinear Relationship: Points curve or do not align linearly.

• No Relationship: Points are scattered randomly.

Example Table:

Scatterplot	Relationship
a	Neither
b	Linear
c	Nonlinear

Domain of Radical Functions

Finding the Domain

For functions involving square roots, the domain is restricted to values that make the radicand non-negative.

• Radical Function: $f(x) = \sqrt{x^2 + 3x + 2}$

• Domain: Solve $x^2 + 3x + 2 \geq 0$

Example: Domain is $(-\infty, -2] \cup [-1, \infty)$.

Inequalities Involving Rational Expressions

Solving Rational Inequalities

Rational inequalities require finding where the expression is less than or equal to zero, considering critical points and intervals.

• Example Inequality: $\frac{(x-1)(x+2)^2}{x(x-3)} \leq 0$

• Solution: Find zeros and undefined points, test intervals.

Example: Solution is $[-2,0) \cup (0,3)$.

Summary Table: Key Concepts

Concept	Definition	Example
Domain	Set of input values	$(-\infty, -2] \cup [-1, \infty)$
Range	Set of output values	$[-10,10]$
Vertex	Highest/lowest point of parabola	$(-3,11)$
Asymptote	Line graph approaches but never touches	$x = -1$, $y = \frac{1}{2}$
Zero	Where $f(x) = 0$	$x = -5$