Functions and Their Graphs Advanced Math Chapter 2

Linear Equations in Two Variables Advanced Math Section 2.1

Slope-intercept form m is slope b is y-intercept Advanced Math chapter 2

Slope Advanced Math chapter 2

Vertical and horizontal lines Vertical line have undefined slope Example: x = 6 Horizontal lines have zero slope Example: y = 2 Advanced Math chapter 2

Examples Find the slope of the line connecting points (1, 3) and (-2, 1) Draw a line through the point (2, 1) that has a slope of -1 Advanced Math chapter 2

Point-slope form Point (x1, y1) is on the line Most useful for finding the equation of a line when you know the slope and a point on the line Advanced Math chapter 2

Examples Write the equation of a line with a slope of 2 that goes through the point (3, 4). Write the equation of a line that goes through the point (-5, 3) that has a slope of -12. Advanced Math chapter 2

Parallel lines Have equal slopes Advanced Math chapter 2

Perpendicular lines Have slopes that are negative reciprocals of each other Advanced Math chapter 2

Example Find the slope-intercept forms of the equations of the lines that pass through the point (3, 4) and are (a) parallel and (b) perpendicular to the line 4x + 3y = 7 Advanced Math chapter 2

Slope A ratio if both axes have the same unit of measure Example: slope of a ramp A rate of change if they have different units of measure Example: straight-line depreciation Advanced Math chapter 2

Functions Advanced Math Section 2.2

Relation When two quantities are related to each other Advanced Math chapter 2

Function Special type of relation For every x value there is exactly one y value It’s ok if two or more x values have the same y value Advanced Math chapter 2

Ways to represent a function Verbally A sentence Numerically A table or list of ordered pairs Graphically Points on a graph Algebraically An equation Advanced Math chapter 2

Testing for functions Decide whether each input value is matched with exactly one output value Or each value in the domain is matched with exactly one value in the range Examples: exercises 5 and 7 Advanced Math chapter 2

Testing for functions algebraically Solve the equation for y If each x value corresponds to only one y value, it is a functions If any x value has more than one y value, it’s not a function Advanced Math chapter 2

Examples Is it a function? Advanced Math chapter 2

Function notation f(x) Read “f of x” Tells you that x is the independent variable and f(x) is the dependent variable Advanced Math chapter 2

Finding the value of a function Find the value of the function at whatever is inside the parentheses Replace each x is the original equation with whatever is inside the parentheses Advanced Math chapter 2

Examples Find the following: Advanced Math chapter 2

Example Piecewise function Advanced Math chapter 2

Implied domain Set of all real numbers for which the expression is defined Not stated In general, excludes values that would cause division by zero or that would result in the even root of a negative number Advanced Math chapter 2

Examples Find the domain of each function Advanced Math chapter 2

Domain limitations Domains can be limited by physical context Examples Length, radius, volume, etc. can’t be zero or negative Advanced Math chapter 2

Example An open box is to be made from a square piece of material 24 centimeters on a side by cutting equal squares from the corners and turning up the sides. Write the volume, V as a function of x and determine its domain. 24-2x x Advanced Math chapter 2

Example Evaluate the difference quotient Advanced Math chapter 2

Analyzing graphs of functions Advanced Math Section 2.3

Graphs of functions f(x) = y Domain is possible x values Range is possible y values Examples: Exercises 2 and 4 Advanced Math chapter 2

Vertical line test y is a function of x if no vertical line intersects the graph at more than one point Examples: exercises10, 12, 14 Advanced Math chapter 2

Zeros of a function Where f(x) = 0 x-intercepts To find them, set f(x) = 0 and solve for x Example: Find the zeros of the function Advanced Math chapter 2

Increasing functions A function is increasing over an interval if, for any x1 and x2 in the interval, x1 < x2 implies that f(x1) < f(x2) The graph is going up Advanced Math chapter 2

Decreasing functions A function is increasing over an interval if, for any x1 and x2 in the interval, x1 < x2 implies that f(x1) > f(x2) The graph is going down Advanced Math chapter 2

Constant functions A function is constant over an interval if, for any x1 and x2 in the interval, f(x1) = f(x2) The graph is horizontal Advanced Math chapter 2

Examples Exercises 32, 34 Advanced Math chapter 2

Relative minimum AKA local minimum Lowest point of the graph on some interval f(a) is a relative minimum if f(a) ≤ f(x) Approximate from a graph (for now) Advanced Math chapter 2

Relative maximum AKA local maximum Highest point of the graph on some interval f(a) is a relative maximum if f(a) ≥ f(x) Approximate from a graph (for now) Advanced Math chapter 2

Example Estimate any relative maxima or relative minima Advanced Math chapter 2

Average rate of change The slope of the line between two points on a curve (a secant line) Example: Find the average rate of change of the following function from x1 = 1 to x2 = 5 Advanced Math chapter 2

Even functions Symmetric with respect to the y-axis Advanced Math chapter 2

Odd functions Symmetric with respect to the origin Advanced Math chapter 2

Examples Determine whether the following functions are even, odd, or neither. Advanced Math chapter 2

A Library of parent functions Advanced Math Section 2.4

Linear functions Slope m y-intercept (0, b) Domain is all real numbers Range is all real numbers x-intercept (-b/m, 0) Graph is increasing if m > 0, decreasing if m < 0 Advanced Math chapter 2

Example Write the linear function for which f(5) = - 4 and f(-2) = 17 Advanced Math chapter 2

Two special linear functions Constant function f(x) = c Horizontal line Domain is all real numbers Range is a single value, c Identity function f(x) = x Slope of 1 Passes through origin Advanced Math chapter 2

Squaring function Domain is all real numbers Range is all nonnegative real numbers Even function Intercept at (0, 0) Decreasing on (-∞, 0), increasing on (0, ∞) Symmetric with respect to y-axis Relative minimum and (0, 0) Advanced Math chapter 2

Squaring function Advanced Math chapter 2

Cubic function Domain is all real numbers Range is all real numbers Odd function Intercept at (0, 0) Increasing on (-∞, ∞) Symmetric with respect to the origin Advanced Math chapter 2

Cubic function Advanced Math chapter 2

Square Root function Domain is all nonnegative real numbers Range is all nonnegative real numbers Intercept at (0, 0) Increasing on the interval (0, ∞) Advanced Math chapter 2

Square Root function Advanced Math chapter 2

Reciprocal function Domain is all real numbers, x ≠ 0 Range is all real numbers, y ≠ 0 Odd function No intercepts Decreasing on intervals (-∞, 0) and (0, ∞) Symmetric with respect to origin Advanced Math chapter 2

Reciprocal function Advanced Math chapter 2

Step functions Resemble stairsteps Most famous is greatest integer function Advanced Math chapter 2

Greatest integer function Domain is all real numbers Range is all integers y-intercept at (0, 0) x-intercepts in the interval [0, 1) Constant between each pair of consecutive integers Jumps vertically one unit at each integer value Advanced Math chapter 2

Greatest integer function Advanced Math chapter 2

Parent functions It is important that you are familiar with the 8 parent functions on page 219. This will help you analyze more complicated graphs in section 2.5 Advanced Math chapter 2

Transformations of functions Advanced Math Section 2.5

Rigid transformations The graph shifts, but doesn’t change size Slides around or flips, but doesn’t get distorted Advanced Math chapter 2

Vertical shift upward Advanced Math chapter 2

Vertical shift downward Advanced Math chapter 2

Horizontal shift to the right Advanced Math chapter 2

Horizontal shift to the left Advanced Math chapter 2

Reflection in the x-axis Advanced Math chapter 2

Reflection in the y-axis Advanced Math chapter 2

Examples Exercises 6 (skip c and g), 12, 38, 44, 48 Advanced Math chapter 2

Nonrigid transformations Cause distortion Stretch Shrink Advanced Math chapter 2

Vertical stretch Each y multiplied by c Advanced Math chapter 2

Vertical shrink Each y multiplied by c Advanced Math chapter 2

Horizontal stretch Each y multiplied by f(c) Advanced Math chapter 2

Horizontal shrink Each y multiplied by f(c) Advanced Math chapter 2

Examples Exercises 6c and g, 24, 28, 42 Advanced Math chapter 2

Combinations of functions: Composite functions Advanced Math Section 2.6

Arithmetic combinations Addition Subtraction Multiplication Division Domain of the combination consists of all real numbers that are common to the domains of the original functions. When dividing, the denominator can’t equal 0 Advanced Math chapter 2

Finding the sum Advanced Math chapter 2

Finding the difference Advanced Math chapter 2

Finding the product Advanced Math chapter 2

Finding the quotient Advanced Math chapter 2

Finding the quotient Advanced Math chapter 2

Composition of functions Domain is limited by both functions The inside function can’t equal something that’s not in the domain of the outside function. Advanced Math chapter 2

Examples Advanced Math chapter 2

Inverse functions Advanced Math Section 2.7

Inverse function Switch domain and range Interchange x and y values Advanced Math chapter 2

Finding inverse functions informally Think of how to “undo” the function f -1(x) always indicates the inverse of f(x), never the reciprocal Advanced Math chapter 2

Verifying inverse functions For inverse functions Advanced Math chapter 2

Inverse functions If g is the inverse of f, then f is also the inverse of g They are inverses of each other Advanced Math chapter 2

Examples Determine whether the two functions are inverses of each other. Advanced Math chapter 2

Graphs of inverse functions Reflect in the line y = x If (a, b) is on the graph of a function, then (b, a) is on the graph of its inverse. Advanced Math chapter 2

Examples Sketch the graphs of f and g and show that they are inverse functions. Advanced Math chapter 2

Horizontal Line Test A function has an inverse if and only if no horizontal line intersects the graph at more than one point Like the vertical line test, but with a horizontal line Advanced Math chapter 2

One-to-One function Passes the horizontal line test Has an inverse Each y corresponds to exactly one x [Also, each x corresponds to exactly one y (passes the vertical line test) or it wouldn’t be a function at all] Advanced Math chapter 2

Examples Exercises 26, 30, 32 First, is it a function (vertical line test) Next, does it have an inverse (horizontal line test) Advanced Math chapter 2

Finding inverses algebraically Use the horizontal line test to decide whether f has an inverse In the equation for f(x), replace f(x) with y Switch x and y, and solve for y Replace y with f -1(x). Verify that they are inverses. Advanced Math chapter 2

Examples Find the inverses of the following functions algebraically Advanced Math chapter 2