College Algebra Study Guide: Functions, Graphs, Transformations, and Polynomial/Rational Analysis

Functions and Their Graphs

Function Transformations

Transformations are operations that alter the position or shape of a function's graph. Common transformations include translations (shifts), reflections, stretches, and compressions.

• Vertical Shift: Adding or subtracting a constant to a function, $f(x) + k$, shifts the graph up or down by $k$ units.

• Horizontal Shift: Replacing $x$ with $x - h$ in $f(x)$, $f(x - h)$, shifts the graph right by $h$ units; $f(x + h)$ shifts left by $h$ units.

• Reflection: Multiplying the function by $-1$, $-f(x)$, reflects the graph across the x-axis. Replacing $x$ with $-x$, $f(-x)$, reflects across the y-axis.

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

• Horizontal Stretch/Compression: Replacing $x$ with $bx$ ($b > 1$ compresses, $0 < b < 1$ stretches horizontally).

• Example: The graph of $g(x) = f(x) + 2$ is the graph of $f(x)$ shifted up by 2 units.

Graphing Piecewise and Transformed Functions

Piecewise functions are defined by different expressions over different intervals. Transformations can be applied to each piece.

• Key Point: Apply transformations to each segment of the piecewise function.

• Example: If $f(x)$ is given, $g(x) = f(x - 2) + 1$ shifts $f(x)$ right by 2 units and up by 1 unit.

Quadratic, Square Root, and Cubic Functions

Quadratic Functions and Their Transformations

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

• Vertex Form: $f(x) = a(x - h)^2 + k$; vertex at $(h, k)$.

• Transformations: Shifts, stretches, and reflections can be applied to $x^2$ to obtain other parabolas.

• Example: $g(x) = (x + 2)^2 - 1$ is $x^2$ shifted left by 2 and down by 1.

Square Root and Cubic Functions

Square root functions have the form $f(x) = \\sqrt{x}$, and cubic functions $f(x) = x^3$.

• Transformations: Similar rules apply for shifting, stretching, and reflecting these graphs.

• Example: $g(x) = \\sqrt{x - 2} + 1$ shifts $\\sqrt{x}$ right by 2 and up by 1.

Domain and Range

Finding the Domain

The domain of a function is the set of all possible input values ($x$) for which the function is defined.

• Rational Functions: Exclude values that make the denominator zero.

• Square Root Functions: The radicand must be non-negative.

• Example: For $f(x) = \\frac{1}{x - 2}$, the domain is $(-\\infty, 2) \\cup (2, \\infty)$.

Finding the Range

The range is the set of all possible output values ($y$) of the function.

• Quadratic Functions: If $a > 0$, the range is $[k, \\infty)$; if $a < 0$, the range is $(-\\infty, k]$.

• Rational Functions: Analyze horizontal asymptotes and excluded values.

Inverse Functions

Definition and Properties

An inverse function $f^{-1}(x)$ reverses the effect of $f(x)$. A function has an inverse if and only if it is one-to-one (passes the horizontal line test).

• Finding the Inverse: Solve $y = f(x)$ for $x$ in terms of $y$, then interchange $x$ and $y$.

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

Polynomial Functions

Identifying Polynomial Functions

A polynomial function is an expression of the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_0$ where $n$ is a non-negative integer.

• Degree: The highest power of $x$ in the polynomial.

• Zeros: Values of $x$ where $f(x) = 0$.

• Multiplicity: The number of times a zero is repeated.

• End Behavior: Determined by the leading term $a_nx^n$.

• Example: $f(x) = x^3 - 2x^2 + x$ is a cubic polynomial (degree 3).

Graphing Polynomial Functions

Polynomial graphs are smooth and continuous. The number of turning points is at most $n - 1$ for degree $n$.

• Key Point: The graph crosses the x-axis at zeros of odd multiplicity and touches at zeros of even multiplicity.

Rational Functions

Definition and Properties

A rational function is a function of the form $f(x) = \\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \\neq 0$.

• Vertical Asymptotes: Occur at values of $x$ where $Q(x) = 0$ and $P(x) \\neq 0$.

• Holes: Occur at values where both $P(x)$ and $Q(x)$ are zero (common factors).

• Horizontal Asymptotes: Determined by the degrees of $P(x)$ and $Q(x)$.

• Example: $f(x) = \\frac{1}{x^2 - 4}$ has vertical asymptotes at $x = 2$ and $x = -2$.

Graphing Rational Functions

To graph a rational function, identify asymptotes, holes, intercepts, and plot points between and beyond asymptotes.

• Step-by-Step:

  1. Find the domain.

  2. Determine vertical and horizontal asymptotes.

  3. Find holes (if any).

  4. Plot intercepts and additional points.

  5. Sketch the graph, showing behavior near asymptotes.

Function Operations and Composition

Operations on Functions

Functions can be added, subtracted, multiplied, divided, and composed.

• Sum: $(f + g)(x) = f(x) + g(x)$

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

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

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

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

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

Applications of Polynomial and Rational Functions

Modeling with Functions

Polynomial and rational functions can be used to model real-world scenarios, such as projectile motion or profit maximization.

• Example: The height of a ball thrown upward can be modeled by $h(x) = -0.02x^2 + 0.7x + 6$.

• Profit Maximization: The profit function $P(x) = -0.01x^2 + 3.22x + 400$ can be maximized by finding its vertex.

Tables: Summary of Transformations

Transformation	Algebraic Form	Graph Effect
Vertical Shift	$f(x) + k$	Up by $k$ units if $k > 0$, down if $k < 0$
Horizontal Shift	$f(x - h)$	Right by $h$ units if $h > 0$, left if $h < 0$
Reflection (x-axis)	$-f(x)$	Flips graph over x-axis
Reflection (y-axis)	$f(-x)$	Flips graph over y-axis
Vertical Stretch/Compression	$a f(x)$	Stretches if $|a| > 1$, compresses if $0 < |a| < 1$
Horizontal Stretch/Compression	$f(bx)$	Compresses if $|b| > 1$, stretches if $0 < |b| < 1$

Additional info:

• Some context and explanations have been expanded for clarity and completeness.

• Graph selection and matching questions reinforce understanding of transformations and function properties.

• Domain and range problems use interval notation, which is standard in college algebra.

• Polynomial and rational function analysis includes zeros, multiplicity, asymptotes, and end behavior.