College Algebra: Functions, Graphs, and Transformations Study Guide

Functions and Their Properties

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically written in the form f(x) = ax^2 + bx + c. They are fundamental in algebra and have a characteristic parabolic graph.

• Domain: The set of all possible input values (x) for which the function is defined. For quadratic functions, the domain is usually all real numbers: $(-\infty, \infty)$.

• Range: The set of all possible output values (f(x)). For $f(x) = -3x^2 + 6x - 1$, since the leading coefficient is negative, the parabola opens downward, and the range is $(-\infty, k]$, where k is the maximum value (vertex).

• Vertex: The point $(h, k)$ where the parabola reaches its maximum or minimum. For $f(x) = ax^2 + bx + c$, the vertex is at $x = -\frac{b}{2a}$.

• Axis of Symmetry: The vertical line passing through the vertex, given by $x = -\frac{b}{2a}$.

• Direction: If $a > 0$, the parabola opens upward; if $a < 0$, it opens downward.

• Intercepts:

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

  • x-intercepts: Solve $f(x) = 0$ for x.

• Difference Quotient: Measures the average rate of change of the function over an interval. For $f(x)$, the difference quotient over $[a, b]$ is $\frac{f(b) - f(a)}{b - a}$.

Example: For $f(x) = -3x^2 + 6x - 1$:

• Vertex: $x = -\frac{6}{2 \times -3} = 1$, so $f(1) = -3(1)^2 + 6(1) - 1 = 2$; vertex is (1, 2).

• Axis of symmetry: $x = 1$.

• y-intercept: $f(0) = -1$.

• Parabola opens downward ($a = -3$).

Function Notation and Operations

Functions can be combined and manipulated using various operations:

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

• Addition/Subtraction: $(f + g)(x) = f(x) + g(x)$, $(f - g)(x) = f(x) - g(x)$.

• Multiplication: $(f \cdot g)(x) = f(x) \cdot g(x)$.

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

Example: If $f(x) = -3x^2 + 6x - 1$ and $g(x) = \sqrt{x}$:

• $(f + g)(x) = -3x^2 + 6x - 1 + \sqrt{x}$

• $(f \circ g)(x) = f(g(x)) = -3(\sqrt{x})^2 + 6\sqrt{x} - 1 = -3x + 6\sqrt{x} - 1$

Solving Equations and Inequalities

Equations and inequalities involving functions are solved by finding the values of x that satisfy the given conditions.

• Solving $f(x) = k$: Set the function equal to k and solve for x.

• Solving $f(x) < k$ or $f(x) > k$: Find the intervals where the function is less than or greater than k.

• Average Value: For $f(x)$ over $[a, b]$, $\text{Average} = \frac{1}{b-a} \int_a^b f(x) dx$ (for continuous functions).

Piecewise Functions

Definition and Graphing

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

• Example: $f(x) = \begin{cases} -2x - 3 & x < -2 \\ x^2 & -2 \leq x < 2 \\ x + 5 & x \geq 2 \end{cases}$

• To graph a piecewise function, plot each segment over its specified interval.

• Evaluate the function at specific points by determining which interval the input falls into.

Example: Compute $f(-7)$, $f(-1)$, $f(3)$:

• $f(-7) = -2(-7) - 3 = 14 - 3 = 11$

• $f(-1) = (-1)^2 = 1$

• $f(3) = 3 + 5 = 8$

Transformations of Functions

Graphing and Transformations

Transformations change the position or shape of a function's graph.

• Vertical Shifts: $f(x) + k$ shifts the graph up by k units.

• Horizontal Shifts: $f(x - h)$ shifts the graph right by h units.

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

• Stretching/Compressing: $af(x)$ stretches (if $|a| > 1$) or compresses (if $0 < |a| < 1$) vertically.

Example: For $f(x) = 3\sqrt{x-4} + 5$, the graph of $\sqrt{x}$ is shifted right by 4 units, stretched vertically by 3, and shifted up by 5.

Function Composition and Domains

Composing Functions

When composing functions, the domain of the composite function is determined by the domains of the individual functions and the composition itself.

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

• Example: If $f(x) = -3x^2 + 6x - 1$ and $g(x) = \sqrt{x}$, the domain of $f \circ g$ is $x \geq 0$.

Circle Equations

Standard Form and Properties

The equation of a circle in standard form is $x^2 + y^2 + Dx + Ey + F = 0$. To identify the center and radius, rewrite in the form $(x - h)^2 + (y - k)^2 = r^2$.

• Center: $(h, k)$

• Radius: $r$

Example: For $x^2 + y^2 + 6x + 2y + 6 = 0$:

• Complete the square for x and y:

• $x^2 + 6x + y^2 + 2y = -6$

• $x^2 + 6x = (x + 3)^2 - 9$

• $y^2 + 2y = (y + 1)^2 - 1$

• So, $(x + 3)^2 + (y + 1)^2 = 4$

• Center: $(-3, -1)$, Radius: $2$

HTML Table: Function Operations and Domains

Operation	Formula	Domain
Addition	$(f + g)(x) = f(x) + g(x)$	Intersection of domains of f and g
Subtraction	$(f - g)(x) = f(x) - g(x)$	Intersection of domains of f and g
Multiplication	$(f \cdot g)(x) = f(x) \cdot g(x)$	Intersection of domains of f and g
Division	$(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$	Intersection of domains of f and g, $g(x) \neq 0$
Composition	$(f \circ g)(x) = f(g(x))$	All x in domain of g such that $g(x)$ is in domain of f

Additional info:

• Some questions involve finding inverse functions and determining one-to-one properties. A function is one-to-one if each output is produced by exactly one input.

• Rain gutter problem (application): Involves maximizing area using algebraic modeling and optimization.

• Transformations and compositions are key for understanding how functions behave under various operations.