MATH section. Curve Sketching Page of 9 The step by step procedure below is for regular rational and polynomial functions. If a function contains radical or trigonometric term, then proceed carefully because the steps below must be modified. Step Determine if the function is rational or polynomial. If the function is a polynomial then the domain is (, and skip to step Step : Determine if the rational function is proper or improper. If it is proper, then the horizontal asymptote is, typically y (for justification, take the function and apply limit as, and go to step. Step : Do a long polynomial division (numerator divided by denominator to break up the rational epression into a simple polynomial and a fraction at the end. The simple polynomial is your asymptote (not vertical. Step : Find the vertical asymptotes. If you have a real number solution, then the solution(ss is/are the vertical asymptote. The domain is all real numbers with values of the asymptotes removed. Step : Fin-intercept (set, if eists. Step 6: Find the first and second derivatives if necessary. Step 7: Draw lines one for first derivative and the other for second derivative. Each line must be the same amount as the intervals of the domain. Step 8: Compute the Critical Points (by setting first derivative equal to and Inflection Points (by setting second derivative equal to. If you have real number solution, then label on the lines created in step 7. Step 9: Only when a critical number is unique (same value does not a solution of inflection points, apply the second derivative test on the value. With this information we can find if it is a local maimum or minimum and predict the behavior surrounding this critical value. Otherwise, use a full test (take a test point on the interval and compute the value of its first and second derivatives to find the behavior. Step : Compute the y value of all critical and inflection points and sketch the graph. To illustrate the steps above, eercise and are show with steps listed.
MATH section. Curve Sketching Page of 9 y + Step : Function is polynomial. Domain: (, Step : y-intercept: y + ( ( y + [ ] [ ] 6 Step 6: 6[] [ ] 6 6 Step 7: Step 8: Critical points 6 ( 6 Inflection points 66 6( Step 9: 6 6( > local min. Local min. 6 6( < local ma. Local ma.
MATH section. Curve Sketching Page of 9 Now that we determine all missing information with our knowledge of local minimum and maimum. Local min. Local ma. Step : y ( ( (, + y ( ( 8 6 (, 6 + + y ( ( (, + + (,6 (, (, y + 9 Steps and : Improper rational function ( 9 Step : + + 9 + + y + + 9 + 9 ( + + 9 9 We have a horizontal asymptote of y
MATH section. Curve Sketching Page of 9 Step : + 9 no solution Domain: (, Step : Step 6: Step 7: ( y ( + 9 { + } [ ]( + 9 ( [ ] []( 9 ( [] {9} 8 ( + 9 ( + 9 ( + 9 ( + 9 { } (( + 9 { [ ] } 8 [] ( 9 ( 9 8( 9 []( 9 + + + + ( + 9 + ( + 9 ( + 9 8{ 9 } 8{9 } Step 8: Critical points 8 8 8 ( + 9 ( + 9 Inflection points 8{9 } 8{9 } ( + 9 + 8(( 8{9 } ( + 9 8( ( + ( Step 9: 8{9 ( } > (( + 9 local min.
MATH section. Curve Sketching Page of 9 Local min. Now that we determine all missing information with our knowledge of local minimum and maimum. Local min. Step : ( y (, ( + 9 y ( (, + 9 + 9 y + 9 + 9 (, (, (, (,
MATH section. Curve Sketching Page 6 of 9 Additional eamples: 6 y domain: (,. y-int.: y [ ] [] [ ] ( + ( + ( ( + ( + ( Local ma. Local min. ( < local ma. ( > local min. y + (, y ( ( (, y ( ( (, (, (, (,
MATH section. Curve Sketching Page 7 of 9 8 y ( domain: (,, y-int: y ( ( ( ( {[] (( + ( ( } ( {[]( [ ( ]} + ( { 8 } ( { 9 } ( ( ( + ( ( + ( ( ( + ( + ( ( + ( ( ( { 9 } ( + ( + ( ( ( + ( { 9 + } + (( + ( (+ ( Local ma. { } ( ( 9( < local ma. at ( ( ( > { } ( 9 <
MATH section. Curve Sketching Page 8 of 9 at (( ( < { } ( ( 9( < The values and are inflection points because we have a concavity change; also the slope of the tangent line at these points are. y ( ( (, y ( ( ( (, y ( ( ( (, y 9 6 ( 9 9 9 9 9 9 6 y ( 9 9 9 9 9 (,,, (, 9 (, 9 (, (, y + + + + Horizontal asymptote: lim + + + + y Vertical asymptote: an Domain: (, (, y-int: none
MATH section. Curve Sketching Page 9 of 9 + + 6 + 6 + + 6 + 6 + 6 6 Local min.. ( ( + 6 > local ma. at ( < ( ( + 6 > ( y + + (, ( ( y + + + ( ( (, + + y + + + ( ( 9 (, 7 9 6 7 + + 9 9 9 7 (, 9 (,
MATH section. Curve Sketching Page of 9 y 9 Horizontal asymptote: lim lim lim 9 9 9 y 9 + Vertical asymptote: ( + ( Domain: (, (, (, y-int: y ( 9 ( 9 ( 9 []( 9 [ ] 9 9 (( 9 { } ( 9 [ ] ( 9 ( 9 ( 9 ( 9 [ ]( 9 ( 9[] { + 9+ + 8} { + 7} ( 9 ( 9 9 9 9 no solution, no critical points. ( 9 ( 9 ( + 9 { + 7} { + 7} { + 7} + 7 ( 9 ( 9 no solution at ( 9 (( 9 < ( {( + 7} < (( 9 y ( (, 7 ( 9 69 7
MATH section. Curve Sketching Page of 9 at ( 9 (( 9 < ({( + 7} < (( 9 ( y (, 8 ( 9 9 8 at ( 9 < (( 9 ({( + 7} > (( 9 ( y (, 7 ( 9 6 9 7 6 y Horizontal asymptote: lim lim lim y + + Vertical asymptote: ( ( + + no solution Domain: (, (, y-int.: y ( ( ( []( [ ] (( { } ( [ 6 ] ( ( ( ( 6 ( [ ]( ( [] 6 { + + + } 6 { + } ( ( ( + ( + ( ( + + + + ( + no solution
MATH section. Curve Sketching Page of 9 6 { + } 6 { + } ( + 6 6 ( + ( + 6 { + } ( + (6 + Local ma. 6 + < local ma. at ( < (( 6( {( + } > (( ( y (, 7 ( 8 7 y (, ( y (, (, (, (
MATH section. Curve Sketching Page of 9 8 y Since we have one term which is an even root function, the domain will be the algebraic statement inside the radical sign greater than and equal to zero. Domain: [, y-int.: y [] Inflection pints: no solution, none < local ma. ( y ( ( (, Local ma. (,
MATH section. Curve Sketching Page of 9 y + ( + Since we have one term which is an even root function, the domain will be the algebraic statement inside the radical sign greater than and equal to zero. ( + ( + ( + (, (, (, ( + + ( + neg. POS. POS. ( + ( neg. neg. POS. ( + POS. neg. POS. Domain: (, ] [, y-int. y This Function has a horizontal asymptote. + + + ( + lim ( + lim lim lim + + + + + + y lim lim + + + + + + + + + ( ( [] + + ( + + ( + + + + + ( ( + ( [] + + + + + ( + + ( + + + ( ( ( + ( + + + + + + Critical points + + + + ( + + + + (+ + + + no sol., none + + + (+ + + no solution, none
MATH section. Curve Sketching Page of 9 at ( + < ( + ( < ( + y + + + (, at ( + > ( + ( < ( ( ( + y + ( ( ( (, y + ( ( ( (, y + + (, y ( Since we have one term which is an even root function, the domain will be the algebraic statement inside the radical sign greater than and equal to zero. ( + ( + + Domain:, y-int. y (, (, (, ( + neg. POS. POS ( POS POS neg. ( ( + neg. POS neg.
MATH section. Curve Sketching Page 6 of 9 ( [] ( + ( ( ( (( ( ( ( ( ( [ ] ( ( ( ( ( ( + + 6 ( + ( ( + ( 6 ( 6 + ( 6 ( + ( discard discard Local min. Local ma.
MATH section. Curve Sketching Page 7 of 9 ( 6( > local min. ( ( (, ( 6( < ( ( local ma. y (, y ( ( ( (, y ( ( (, endpoints: y (, y (, (, (, (, (, y Since all terms only contains, the domain is (, points carefully. The y-int.: y and we must investigate critical and inflection ( 9 + + + 9 9( 9( 9 + + + 9 9 and are not defined at, so we must remove from our derivatives chart.
MATH section. Curve Sketching Page 8 of 9 Local min. ( + > local min. 9 ( at ( > ( ( + < 9 ( ( ( ( (, y y ( ( (, y ( ( ( ( 6 (, 6 (, A very small concavity change. It looks like a straight line. (, 6 (,
MATH section. Curve Sketching Page 9 of 9 6 y + ( + Since this function is an odd root, the domain is (,. The y-int.: y ( ( + ( + + (( + ( + ( + ( + [ ] ( + ( ( + ( + + ( + + + + + + ( + ( + ( + ( + ( + ( + ( + ( + ( + + + ( + ( + + and are not defined at, so we must remove from our derivatives chart. at ( ( + at > ( > ( ( +
MATH section. Curve Sketching Page of 9 > < + + at ( ( > > ( + ( + By the results above is an inflection point. Also, the function is continuous at. Therefore, is also an inflection point (this one is an inflection point with a vertical tangent line. y + + (, y + ( (, (, (, 8 y + cos Domain: (,. The y-int.: y ( + cos( + y [] [ sin (] sin + [cos (] cos sin sin + k sin + k cos cos + k
MATH section. Curve Sketching Page of 9 Since our function and its derivatives are periodic, the derivative chart below is based on one period (from + k to + k where k is any integer. + k + k + k + k + k + k + k For the full tests, we should ignore the part of + k. at 6 sin > cos 6 6 < 6 at 6 6 sin > 6 6 6 cos 6 > By the results above, we have an inflection point with the slope tangent line zero at and in general + k. y k cos k k k ( k, k + k + + + + + + + + y k cos k k k ( k, k + k + + + + + + + + The picture of the graph to the right is a plot of the function y + cos on the interval [, ]. (, ( +, + ( +, + (, (, (,
MATH section. Curve Sketching Page of 9 y tan < < Since ta n is continuous and single piece on (,, our domain is (, and we have vertical asymptotes: and. The y-int.: y [] [sec (] sec [sec (sec tan (] sec tan cos sec cos cos sec ± sec cos discard sin sec tan sin cos cos sin sin discard sec tan cos Local min. Local ma. > sec tan < sec tan ( y tan (, y ( tan( (, y tan ( (, local min. local ma.
MATH section. Curve Sketching Page of 9 (, (, (, y sec + tan < < Both terms sec and tan are single piece and continuous on the interval (,. So the domain of (, is good and we have one vertical asymptote. No y-int. due to domain. [sec tan (] [sec (] sec tan sec + + {[sec tan (](tan + (sec [sec (]} + [sec (sec tan (] {sec tan + sec } + sec tan sec + sec tan + sec tan sec (sec + sec tan + tan sec (sec + tan sin + cos sectan+ sec sin sin cos + cos cos sec tan + sec sin + sin + cos discard discard cos cos cos sin + cos sec (sec + tan sin sin cos + cos cos sec (sec + tan (+ sin sin cos discard discard + cos cos cos This function has no critical and inflection points on the given interval.
MATH section. Curve Sketching Page of 9 at sec tan + sec ( ( + ( > sec sec + tan + ( > Just to get a starting point of the graph, I computed the value of the function when. y sec( tan( ( ( (, + + sin y + cos Since both sin and cos are well defined on (, we just have to check for vertical asymptotes for our domain. + cos cos no solution, no vertical asymptotes Domain: (,. The y-int.: y + + + [cos (]( + cos (sin [ sin (] cos + cos + sin cos + ( cos ( cos ( cos (( + cos + + + [( sin (] ( cos ( cos ( cos ( sin ( sin ( + cos {[]( + cos (cos + []} sin { cos } ( + cos ( + cos
MATH section. Curve Sketching Page of 9 cos + + cos ( + cos cos cos + + k + k ( + cos cos sin { cos } ( + cos sin cos sin { cos } sin cos ( + cos sin { cos } + k + k + k Local ma. Local min. + k + k + k + k + k y y sin cos < local ma. + cos + sin cos > local min. + cos + sin + k ( + k, + cos k + + sin k + ( + k, + cos k + + + k + k
MATH section. Curve Sketching Page 6 of 9 y y + k + k ( + k ( k ( + k ( k sin ( (+ k, + cos + + ( sin ( ( + k, + cos + + ( ( + k, ( + k, ( + k, (, + k + y Improper rational function + + + + ( + + y + + ( + + + ( This function has a slanted asymptote of y + Vertical asymptote: Domain: (, (, The y-int.: y
MATH section. Curve Sketching Page 7 of 9 [ + ]( ( + + [] ( + ( + + + 8 [+ 8] ( ( + 8 ( ( ( {[+ ]( ( + 8[]} ( (( {( + 88 ( + 8} {} 6 ( ( ( + 8 (8 ± (8 ( ( ( + 8 ( + 8 no real number solution ( none 6 6 ( ( 6 none at ( + 8( < (( 6 (( < + ( ( y (, ( at ( + 8( < (( ( ( 8 6 (( > + + y (, (
MATH section. Curve Sketching Page 8 of 9 y ( + Improper rational function (+ + + + + + y + ( + + + ( + ( + + + ( + This function has a slanted asymptote of y Vertical asymptote: ( + + Domain: (, (, The y-int.: y (( + ( + ( + [ ] ( + ( ( + ( ( + {[]( + []} { + } { + } + ( + ( + (( + [ + 6 ] ( + ( + ( + ( ( + {[ + ]( + ( + []} 6 ( + {( + + ( + } {} 6 ( + ( + ( + + ( + + + + ( + ( ( + 6 6 6 ( + ( + Local ma.
MATH section. Curve Sketching Page 9 of 9 at 6( (( + ( + ( (( + < local ma. > at + > + By these results is an inflection point. ( 7 7 y (, (( + ( y (, (( + 6( > (( + 6 < + (, (, 7