Graphs, Functions, and Models

Understanding and Graphing Functions

Functions are mathematical relationships where each input has exactly one output. Graphing functions helps visualize their behavior and key features.

• Graphing Basic Functions: Plot points for given values of x and connect them smoothly to reveal the function's shape.

• Symmetry: A function may be symmetric with respect to the y-axis (even function), x-axis, or origin (odd function). Test symmetry by substituting -x for x or -y for y in the equation.

• Piecewise Functions: Defined by different expressions over different intervals. Graph each piece over its specified domain.

• Transformations: Shifting, reflecting, stretching, or compressing the graph of a function. For example, $f(x) + c$ shifts up, $f(x - h)$ shifts right.

Example: Graph $f(x) = x^2$ and $f(x) = (x-3)^2 - 5$ to see a right shift by 3 and a downward shift by 5.

More on Functions

Function Operations and Composition

Functions can be combined using addition, subtraction, multiplication, division, and composition.

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

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

• Inverse Functions: If $f(g(x)) = x$ and $g(f(x)) = x$, then f and g are inverses. To find the inverse, solve for x in terms of y and interchange variables.

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

Quadratic Functions and Equations; Inequalities

Quadratic Functions and Their Properties

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

• Vertex: The highest or lowest point, found at $x = -\frac{b}{2a}$.

• Axis of Symmetry: The vertical line $x = -\frac{b}{2a}$.

• Intercepts: Set $x=0$ for y-intercept, $f(x)=0$ for x-intercepts.

• Solving Quadratics: Use factoring, completing the square, or the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Inequalities: Solve quadratic inequalities by finding zeros and testing intervals.

Example: Solve $x^2 - 4x + 3 = 0$ by factoring: $(x-1)(x-3)=0$ so $x=1,3$.

Polynomial Functions and Rational Functions

Polynomial and Rational Equations

Polynomials are sums of terms with non-negative integer exponents. Rational functions are ratios of polynomials.

• End Behavior: Determined by the leading term for large $|x|$.

• Asymptotes: Vertical asymptotes occur where the denominator is zero; horizontal or oblique asymptotes depend on degrees of numerator and denominator.

• Solving Rational Equations: Multiply both sides by the least common denominator (LCD) to clear fractions.

Example: Solve $\frac{2}{x-1} = 3$ by multiplying both sides by $(x-1)$.

Exponential Functions and Logarithmic Functions

Exponential Growth and Decay

Exponential functions have the form $f(x) = ab^{x}$, where $b > 0$ and $b \neq 1$. Logarithmic functions are the inverses of exponentials.

• Exponential Growth: $P(t) = P_0 e^{kt}$, where $k > 0$.

• Exponential Decay: $P(t) = P_0 e^{-kt}$, where $k > 0$.

• Logarithms: $y = \log_b x$ means $b^y = x$.

• Properties: $\log_b(xy) = \log_b x + \log_b y$, $\log_b(x^k) = k \log_b x$.

Example: Solve $2^x = 8$ by taking $\log_2$ of both sides: $x = 3$.

Systems of Equations and Matrices

Solving Systems of Equations

Systems of equations can be solved by substitution, elimination, or using matrices.

• Substitution: Solve one equation for a variable and substitute into the other.

• Elimination: Add or subtract equations to eliminate a variable.

• Matrices: Write the system as $AX = B$ and solve using inverse matrices or row reduction.

Example: Solve $2x + y = 5$, $x - y = 1$ by elimination: Add equations to get $3x = 6$, so $x=2$, $y=1$.

Conic Sections

Circles, Parabolas, and More

Conic sections include circles, ellipses, parabolas, and hyperbolas, each with standard equations.

• Circle: $(x-h)^2 + (y-k)^2 = r^2$

• Parabola: $y = a(x-h)^2 + k$

• Vertex: The center for circles, the turning point for parabolas.

Example: The circle with center (0,0) and radius 5: $x^2 + y^2 = 25$.

Sequences, Series, and Combinatorics

Arithmetic and Geometric Sequences

Sequences are ordered lists of numbers. Series are sums of sequences.

• Arithmetic Sequence: $a_n = a_1 + (n-1)d$

• Geometric Sequence: $a_n = a_1 r^{n-1}$

• Summation: $S_n = \frac{n}{2}(a_1 + a_n)$ for arithmetic, $S_n = a_1 \frac{1 - r^n}{1 - r}$ for geometric ($r \neq 1$)

Example: Find the 5th term of $2, 4, 6, ...$: $a_5 = 2 + 4 \times 1 = 6$.

Application Problems

Word Problems and Modeling

Many algebraic concepts are applied to real-world problems, such as cost modeling, population growth, and coin problems.

• Cost Functions: $C(x) = \text{fixed cost} + \text{variable cost} \times x$

• Population Growth: $P(t) = P_0 e^{kt}$

• Coin Problems: Set up equations based on the value and number of coins.

Example: If the setup fee is $225 and each shirt costs $6.75, $C(x) = 225 + 6.75x$.

Tables: Exponential Growth Example

Population Growth Table

The following table shows population growth over time, useful for modeling with exponential functions.

Additional info: Use the exponential growth formula $P(t) = P_0 e^{kt}$ to model the data.

Key Concepts and Strategies

• Always identify the type of function before solving or graphing.

• Check domains and ranges for all functions.

• Label all intercepts, vertices, and asymptotes on graphs.

• For word problems, define variables clearly and set up equations before solving.