Honors Algebra II Unit 2 Review: Functions and Their Graphs

Key Concepts

• Functions map each input value (domain) to a unique output value (range)
• Vertical line test determines if a relation is a function
  • If any vertical line intersects the graph more than once, it is not a function
• Functions can be represented using equations, graphs, or tables
• Domain represents the set of all possible input values for a function
• Range represents the set of all possible output values for a function
• One-to-one functions have a unique output value for each input value and pass the horizontal line test
• Even and odd functions exhibit symmetry about the y-axis or origin, respectively

Types of Functions

• Linear functions have a constant rate of change and are represented by the equation $y = mx + b$
  • $m$ represents the slope, and $b$ represents the y-intercept
• Quadratic functions are represented by the equation $y = ax^2 + bx + c$
  • The graph of a quadratic function is a parabola
• Exponential functions have a constant growth or decay factor and are represented by the equation $y = ab^x$
  • $a$ represents the initial value, and $b$ represents the growth or decay factor
• Logarithmic functions are the inverse of exponential functions and are represented by the equation $y = \log_b(x)$
• Polynomial functions are the sum of terms with non-negative integer exponents
• Rational functions are the quotient of two polynomial functions
• Trigonometric functions (sine, cosine, tangent) relate angles to side lengths in right triangles

Function Notation and Terminology

• Function notation $f(x)$ represents the output value of a function $f$ for a given input value $x$
• $f(a)$ represents the output value when the input is $a$
• $f^{-1}(x)$ represents the inverse function of $f(x)$
  • If $f(a) = b$, then $f^{-1}(b) = a$
• Piecewise functions are defined by different equations for different intervals of the domain
• Increasing functions have output values that increase as the input values increase
• Decreasing functions have output values that decrease as the input values increase
• Constant functions have the same output value for all input values

Graphing Techniques

• To graph a function, create a table of input and output values, then plot the points on a coordinate plane
• Identify key features of a graph, such as intercepts, symmetry, and asymptotes
  • x-intercepts occur when $y = 0$, and y-intercepts occur when $x = 0$
• Use the slope-intercept form $y = mx + b$ to graph linear functions
• Complete the square or use the vertex form $y = a(x - h)^2 + k$ to graph quadratic functions
  • $(h, k)$ represents the vertex of the parabola
• Use transformations to graph functions more efficiently
• Identify the domain and range of a function from its graph

Transformations of Functions

• Transformations alter the position, shape, or orientation of a function's graph
• Vertical translations shift the graph up or down and are represented by $y = f(x) + k$
  • Positive $k$ values shift the graph up, while negative $k$ values shift the graph down
• Horizontal translations shift the graph left or right and are represented by $y = f(x - h)$
  • Positive $h$ values shift the graph right, while negative $h$ values shift the graph left
• Vertical stretches or compressions multiply the output values by a constant $a$ and are represented by $y = af(x)$
  • $|a| > 1$ stretches the graph vertically, while $0 < |a| < 1$ compresses the graph vertically
• Horizontal stretches or compressions multiply the input values by a constant $\frac{1}{b}$ and are represented by $y = f(bx)$
  • $|b| > 1$ compresses the graph horizontally, while $0 < |b| < 1$ stretches the graph horizontally
• Reflections flip the graph across the x-axis or y-axis
  • $y = -f(x)$ reflects the graph across the x-axis, while $y = f(-x)$ reflects the graph across the y-axis

Function Operations and Composition

• Function addition $(f + g)(x) = f(x) + g(x)$ adds the output values of two functions for each input value
• Function subtraction $(f - g)(x) = f(x) - g(x)$ subtracts the output values of two functions for each input value
• Function multiplication $(f \cdot g)(x) = f(x) \cdot g(x)$ multiplies the output values of two functions for each input value
• Function division $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$ divides the output values of two functions for each input value, provided $g(x) \neq 0$
• Function composition $(f \circ g)(x) = f(g(x))$ applies one function to the output of another function
  • To find $(f \circ g)(a)$, first evaluate $g(a)$, then use the result as the input for $f$

Real-World Applications

• Linear functions model situations with constant rates of change (uniform motion, cost per unit)
• Quadratic functions model projectile motion, profit optimization, and area problems
• Exponential functions model population growth, compound interest, and radioactive decay
• Logarithmic functions model sound intensity (decibels), earthquake magnitude (Richter scale), and pH levels
• Trigonometric functions model periodic phenomena (sound waves, tides, seasons)
• Piecewise functions model situations with different rates or conditions for different intervals (tax brackets, shipping rates)

Common Mistakes and How to Avoid Them

• Confusing function notation $f(x)$ with multiplication $f \cdot x$
  • Remember that $f(x)$ represents the output value of the function $f$ for the input value $x$
• Misinterpreting the order of operations in function composition
  • Evaluate the inner function first, then use the result as the input for the outer function
• Incorrectly applying transformations to functions
  • Pay attention to the sign and location of the constants in the transformation equations
• Failing to identify the correct domain and range of a function
  • Consider any restrictions on the input values and observe the behavior of the graph
• Misidentifying even and odd functions
  • Check for symmetry about the y-axis (even) or origin (odd)
• Forgetting to distribute negative signs when expanding or factoring expressions
  • Be careful when applying the distributive property or factoring out common factors
• Incorrectly canceling terms in rational expressions
  • Only cancel factors common to both the numerator and denominator, not individual terms