Functions and Their Graphs Advanced Math Chapter 2

Linear Equations in Two Variables Advanced Math Section 2.1

Advanced Math chapter 23 Slope-intercept form m is slope b is y-intercept

Advanced Math chapter 24 Slope

Advanced Math chapter 25 Vertical and horizontal lines Vertical line have undefined slope Example: x = 6 Horizontal lines have zero slope Example: y = 2

Advanced Math chapter 26 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 27 Point-slope form Point (x 1, y 1 ) 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 28 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 29 Parallel lines Have equal slopes

Advanced Math chapter 210 Perpendicular lines Have slopes that are negative reciprocals of each other

Advanced Math chapter 211 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 212 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

Functions Advanced Math Section 2.2

Advanced Math chapter 214 Relation When two quantities are related to each other

Advanced Math chapter 215 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 216 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 217 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 218 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 219 Examples Is it a function?

Advanced Math chapter 220 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 221 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 222 Examples Find the following:

Advanced Math chapter 223 Example Piecewise function

Advanced Math chapter 224 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 225 Examples Find the domain of each function

Advanced Math chapter 226 Domain limitations Domains can be limited by physical context Examples Length, radius, volume, etc. can’t be zero or negative

Advanced Math chapter 227 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. 24-2xxx

Example Evaluate the difference function Advanced Math chapter 228

Analyzing graphs of functions Advanced Math Section 2.3

Advanced Math chapter 230 Graphs of functions f(x) = y Domain is possible x values Range is possible y values Examples: Exercises 2 and 4

Advanced Math chapter 231 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 232 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 233 Increasing functions A function is increasing over an interval if, for any x 1 and x 2 in the interval, x 1 < x 2 implies that f(x 1 ) < f(x 2 ) The graph is going up

Advanced Math chapter 234 Decreasing functions A function is increasing over an interval if, for any x 1 and x 2 in the interval, x 1 f(x 2 ) The graph is going down

Advanced Math chapter 235 Constant functions A function is constant over an interval if, for any x 1 and x 2 in the interval, f(x 1 ) = f(x 2 ) The graph is horizontal

Advanced Math chapter 236 Examples Exercises 32, 34

Advanced Math chapter 237 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 238 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 239 Example Estimate any relative maxima or relative minima

Advanced Math chapter 240 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 x 1 = 1 to x 2 = 5

Advanced Math chapter 241 Even functions Symmetric with respect to the y-axis

Advanced Math chapter 242 Odd functions Symmetric with respect to the origin

Advanced Math chapter 243 Examples Determine whether the following functions are even, odd, or neither.

A Library of parent functions Advanced Math Section 2.4

Advanced Math chapter 245 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 246 Example Write the linear function for which f(5) = - 4 and f(-2) = 17

Advanced Math chapter 247 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 248 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 249 Squaring function

Advanced Math chapter 250 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 251 Cubic function

Advanced Math chapter 252 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 253 Square Root function

Advanced Math chapter 254 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 255 Reciprocal function

Advanced Math chapter 256 Step functions Resemble stairsteps Most famous is greatest integer function

Advanced Math chapter 257 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 258 Greatest integer function

Advanced Math chapter 259 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

Transformations of functions Advanced Math Section 2.5

Advanced Math chapter 261 Vertical shift upward

Advanced Math chapter 262 Vertical shift downward

Advanced Math chapter 263 Horizontal shift to the right

Advanced Math chapter 264 Horizontal shift to the left

Advanced Math chapter 265 Reflection in the x-axis

Advanced Math chapter 266 Reflection in the y-axis

Advanced Math chapter 267 Examples Exercises 6, 12

Advanced Math chapter 268 Nonrigid transformations Cause distortion Stretch Shrink

Advanced Math chapter 269 Vertical stretch

Advanced Math chapter 270 Vertical shrink

Advanced Math chapter 271 Horizontal stretch

Advanced Math chapter 272 Horizontal shrink

Advanced Math chapter 273 Examples Exercises 24, 38, 44, 48

Combinations of functions: Composite functions Advanced Math Section 2.6

Advanced Math chapter 275 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 276 Finding the sum

Advanced Math chapter 277 Finding the difference

Advanced Math chapter 278 Finding the product

Advanced Math chapter 279 Finding the quotient

Advanced Math chapter 280 Finding the quotient

Advanced Math chapter 281 Composition of functions Domain is limited by both functions Use the most limited domain

Advanced Math chapter 282 Examples

Inverse functions Advanced Math Section 2.7

Advanced Math chapter 284 Inverse function Switch domain and range Interchange x and y values

Advanced Math chapter 285 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 286 Verifying inverse functions For inverse functions

Advanced Math chapter 287 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 288 Examples Determine whether the two functions are inverses of each other.

Advanced Math chapter 289 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 290 Examples Sketch the graphs of f and g and show that they are inverse functions. graph 1graphgraph 2

Advanced Math chapter 291 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 292 One-to-One function Passes the horizontal line test Has and 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 293 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 294 Finding inverses algebraically 1.Use the horizontal line test to decide whether f has an inverse 2.In the equation for f(x), replace f(x) with y 3.Switch x and y, and solve for y Replace y with f -1 (x). 4.Verify that they are inverses.

Advanced Math chapter 295 Examples Find the inverses of the following functions algebraically