Function Gallery: Some Basic Functions and Their Properties Linear Equation y = m+b Linear Equation y = -m + b This Eample: y = 3 + 3 This Eample: y = - + 0 Domain (-, ) Domain (-, ) Range (-, ) Range (-, ) No Symmetry Symmetric about the origin Linear Equation: Horizontal Line This Eample: y = 4 Linear Equation: Vertical Line This Eample: = 4 Domain (-, ) Domain {4} Range {4} Range (-, ) Symmetric about the y-ais Symmetric about the -ais = y Absolute Value Function: Domain (-, ) y = Absolute Value Equation: Domain [0, ) Range [0, ) Range (-, ) Increasing on (0, ) Decreasing on (-, 0) Increasing on Decreasing on - = y Absolute Value Equation + - = y This Eample: y - = Absolute Value Equation: Tetbook: Pre-Calculus: Function s & Graphs by Mark Dugopolski 00 pgs. 119, 14, 0,, 68,301 1
Quadratic Function y Quadratic Function y Domain (-, ) Domain (-, ) Range [ 0, ) Range (-, 0] Increasing on [ 0, ) Increasing on (-, 0) Decreasing on (-, 0] Decreasing on [0, ) Symmetric about the y-ais Symmetric about the -ais Quadratic Function y Quadratic Function y Radical Function y This Eample: y 16 Radical Function y This Eample: y 4 Rational Functions y 5 This Eample: y 1 1 Eponential Function y e This eample y Tetbook: Pre-Calculus: Function s & Graphs by Mark Dugopolski 00 pgs. 119, 14, 0,, 68,301
Square Root Function y Square Root Function y This Eample: y This Eample: y 3 Domain [ 0, ) Domain [ 3, ) Range [ 0, ) Range [ 0, ) Square Root Function: y This Eample: y Square Root Function: y 3 This Eample: y 3 Cubic Function y 3 Cubic Function y 3 3 Cubic Function y Domain (-, ) Domain (-, ) This Eample: y 3 4 Range (-, ) Range (-, ) Increasing on (-, ) Decreasing on (-, ) Symmetric about the origin Symmetric about the origin Cube Root Function y 3 Cube Root Function y 3 This Eample: y 5 This Eample: y 3 Note: a cube root function is the inverse of a cubic function Quartic or Fourth-Degree Function y 4 Quartic or Fourth Degree Function y 4 3 4 3 This Eample: y 7 6 This Eample: y 4 6 4 1 4 Tetbook: Pre-Calculus: Function s & Graphs by Mark Dugopolski 00 pgs. 119, 14, 0,, 68,301 3
y Parabolas Translations & Transformations y a b c or y a( h) k y y y () ( 1 / ) ( 4) 4 () 4 Tetbook: Pre-Calculus: Function s & Graphs by Mark Dugopolski 00 pgs. 119, 14, 0,, 68,301 4
Logarithmic Functions Basic Logarithmic Graph y ln( ) Below are some different eamples of some basic logarithmic functions and their graphs. Tetbook: Pre-Calculus: Function s & Graphs by Mark Dugopolski 00 pgs. 119, 14, 0,, 68,301 5
Eponential Functions Basic Eponential Graph y Range [ 0, ) Domain (-, ) e Below are some different eamples of some basic eponential functions and their graphs. y = + 3 y = 3 y = y = 3 y = + 3 y = +3 y = +3 y = 3 y = 3 Tetbook: Pre-Calculus: Function s & Graphs by Mark Dugopolski 00 pgs. 119, 14, 0,, 68,301 6
Rational Functions y 1 The graph of a rational function is called a hyperbola Rational Function: y 5 1 Rational Function: y 3 8 5 6 Rational Function: y 1 Rational Function y 3 5 6 Rational Function y 3 1 Rational Function y Tetbook: Pre-Calculus: Function s & Graphs by Mark Dugopolski 00 pgs. 119, 14, 0,, 68,301 7
Trigonometric Functions The Table below outlines each change for each trigonometric ratio. Tetbook: Pre-Calculus: Function s & Graphs by Mark Dugopolski 00 pgs. 119, 14, 0,, 68,301 8
Asymptotes Use the steps below to find asymptotes: Asymptotes: Factor and reduce the rational function first, If a factor is eliminated in that reduction, it determines a hole in the graph. Vertical Asymptotes To find the Vertical Asymptotes and Domain set the denominator equal to zero and solve. Eample: Find the domain and vertical asymptotes y 8 Domain: -4, Vertical Asymptotes: = -4, Horizontal Asymptotes To Find the Horizontal Asymptotes, compare the degree of the leading coefficients in the numerator and the denominator. If the degrees of the numerator and denominator are the same, p() = q() Then, the horizontal asymptote is the ratio of the leading coefficients. Eample: 1 Because the degree of and is the same, then the H.A. is found by finding the 3 ratio of the leading coefficients, which in this eample is 1 which equals. So the Horizontal Asymptote is y = If the numerator s degree is less than the denominator, p() < q() then the -ais is the horizontal asymptote and the equation is y = 0 Eample: because the degree of is less than the degree of then the HA is y = 0 4 If the numerator is greater than the denominator, p() > q() then there s no Horizontal Asymptote. Slant/Oblique Asymptotes To Find the Slant/Oblique Asymptote compare the degree of the leading coefficients in the numerator and denominator. If the numerator's degree is greater (by a margin of 1) you have a slant asymptote which you will find by doing long division. Eample: 3 5 because the degree of is greater than the degree of then you have a slant asymptote. You must use long division or synthetic division to find the slant asymptote. Step 1: 3 5 Step : The answer is 1 3 Step 3: The Horizontal Asymptote is y = -1 Note: you can not have a slant asymptote and a horizontal asymptote together. Tetbook: Pre-Calculus: Function s & Graphs by Mark Dugopolski 00 pgs. 119, 14, 0,, 68,301 9
Function Shifts and Transformations If you've been doing your graphing by hand, you've probably started noticing some relationships between the equations and the graphs. The topic of function transformation makes these relationships more eplicit. C o Let's start with the function notation for the basic quadratic: f() =. A function transformation takes whatever basic function f() and then "transforms" it, which is a fancy way of saying that you change the formula a bit and thereby move the graph around. For instance, the graph for + 3 looks like this This is three units higher than the basic quadratic. That is, + 3 is f() + 3. We added a "3" outside the basic squaring function f() = to go from the basic quadratic to the transformed function + 3. This is always true: To move a function up, you add outside the function. That is, f() + b is f() moved up b units. Moving the function down works the same way; f() b is f() moved down b units. On the other hand, ( + 3) looks like this: In this graph, f() has been moved over three units to the left. That is, f( + 3) = ( + 3) is f() shifted three units to the left. This is always true: To shift a function left, add inside the function's argument. That is, f( + b) gives f() shifted b units to the left. Shifting to the right works the same way; f( b) is f() shifted b units to the right. Tetbook: Pre-Calculus: Function s & Graphs by Mark Dugopolski 00 pgs. 119, 14, 0,, 68,301 10