Asymptotes, Holes and Graphs
Asymptotes, Holes and Graphs
An asymptote occurs where the function is getting infinitely close to a line on the graph but never touches the line. Horizontal asymptotes may cross the line from time-to-time; it is the end behavior we are concerned with. There are three types of asymptotes: Horizontal, Vertical and Oblique. Oblique asymptotes, will most likely, not be used in your calculus course but vertical and horizontal will be used frequently in order to graph functions. Hole in a Graph 𝑥! − 4 𝑓 𝑥 = ⇒ 𝑥≠2 𝑥−2
Three General Cases for Horizontal Asymptotes Since there are so many conditions and situations for asymptotes and the methods learned in algebra are so minimal to what is used in calculus, we will come back to this later. Case 1 𝑥 ! + 𝑥 !!! + ⋯ 𝑛 > 𝑚 ⇒ 𝐻𝐴: 𝑦 = 0 𝑓 𝑥 = ! !!! 𝑥 +𝑥 +⋯ Case 2 𝑥 ! + 𝑥 !!! + ⋯ 𝑛 < 𝑚 ⇒ 𝐻𝐴: 𝑛𝑜𝑛𝑒 𝑓 𝑥 = ! !!! 𝑥 +𝑥 +⋯ 𝑎 Case 3 a𝑥 ! + 𝑥 !!! + ⋯ 𝑛 = 𝑚 ⇒ 𝐻𝐴: 𝑦 = 𝑓 𝑥 = 𝑏 b𝑥 ! + 𝑥 !!! + ⋯
Ex. 1 Horizontal and Vertical
𝑓 𝑥 =
𝐻𝐴:
𝑥! + 𝑥 + 1 𝑥! + 𝑥! + 𝑥 + 1
𝑦 = 0,
𝑉𝐴:
𝑥 = −2
𝑥! + 𝑥! + 𝑥 + 1 𝑓 𝑥 = 𝑥! + 𝑥 + 1
Ex. 2 Oblique
𝑁𝑜 𝐻𝐴,
𝑂𝐴:
𝐻𝐴:
3 𝑦= , 2
𝑉𝐴:
1 𝑥 =−! 2
Inequalities
𝑓 𝑥 𝑓 𝑥
< 𝑎 ⇒ −𝑎 < 𝑓 𝑥 < 𝑎 or 𝑓 𝑥 < 𝑎 and 𝑓 𝑥 > −𝑎 ≤ 𝑎 ⇒ −𝑎 ≤ 𝑓 𝑥 ≤ 𝑎 or 𝑓 𝑥 ≤ 𝑎 and 𝑓 𝑥 ≥ −𝑎
Interest Formulas
𝐴 = 𝐴! 𝑒 !"
𝑃 = 𝑃!
𝑟 12
𝑟 1 − 1 + 12
Physics Formulas
(rate)(time)=distance 𝑟𝑡 = 𝑑
!!
𝑦 = 𝑥
3𝑥 ! + 𝑥 𝑓 𝑥 = ! 2𝑥 + 1
Ex. 3 Horizontal and Vertical
L=Loan 𝑃 = Monthly Payment 𝑟 = Interest rate for annual 𝑡 = Loan length in months
Symmetry
By Point 𝑥-axis For every point 𝑥, 𝑦 there is a 𝑥, −𝑦 𝑦-axis For every point 𝑥, 𝑦 there is a −𝑥, 𝑦 origin For every point 𝑥, 𝑦 there is a −𝑥, −𝑦 Testing 𝑥-axis: Replace each 𝑦 with a – 𝑦, if the same equation results, it is symmetric. 𝑦-axis: Replace each 𝑥 with a – 𝑥, if the same equation results, it is symmetric. Origin: Replace each 𝑥, 𝑦 with a −𝑥, – 𝑦, if the same equation results, it is symmetric.
Variations (Proportionality) 𝑘 is the constant of proportionality
𝑦 is proportional to x: y= 𝑘𝑥
𝑦 is inversely proportional to ! 𝑥: 𝑦 = !
Common Graphs and Formulas 𝑦 = 𝑥!
𝑦 = 𝑥!
_________________________________________________________________________________________________________
𝑦= 𝑥
1 𝑦= 𝑥
_________________________________________________________________________________________________________ ! 𝑦=𝑒 𝑦 = ln 𝑥
_________________________________________________________________________________________________________ 𝑥! + 𝑦! = 1 𝑥 ! − 𝑦 ! = 1
_________________________________________________________________________________________________________ 𝑦! − 𝑥! = 1 𝑦 = 𝑥! − 1
An asymptote occurs where the function is getting infinitely close to a line on the graph but never touches the line. Horizontal asymptotes may cross the line from time-to-time; it is the end behavior we are concerned with. There are three types of asymptotes: Horizontal, Vertical and Oblique. Oblique asymptotes, will most likely, not be used in your calculus course but vertical and horizontal will be used frequently in order to graph functions. Hole in a Graph 𝑥! − 4 𝑓 𝑥 = ⇒ 𝑥≠2 𝑥−2
Three General Cases for Horizontal Asymptotes Since there are so many conditions and situations for asymptotes and the methods learned in algebra are so minimal to what is used in calculus, we will come back to this later. Case 1 𝑥 ! + 𝑥 !!! + ⋯ 𝑛 > 𝑚 ⇒ 𝐻𝐴: 𝑦 = 0 𝑓 𝑥 = ! !!! 𝑥 +𝑥 +⋯ Case 2 𝑥 ! + 𝑥 !!! + ⋯ 𝑛 < 𝑚 ⇒ 𝐻𝐴: 𝑛𝑜𝑛𝑒 𝑓 𝑥 = ! !!! 𝑥 +𝑥 +⋯ 𝑎 Case 3 a𝑥 ! + 𝑥 !!! + ⋯ 𝑛 = 𝑚 ⇒ 𝐻𝐴: 𝑦 = 𝑓 𝑥 = 𝑏 b𝑥 ! + 𝑥 !!! + ⋯
Ex. 1 Horizontal and Vertical
𝑓 𝑥 =
𝐻𝐴:
𝑥! + 𝑥 + 1 𝑥! + 𝑥! + 𝑥 + 1
𝑦 = 0,
𝑉𝐴:
𝑥 = −2
𝑥! + 𝑥! + 𝑥 + 1 𝑓 𝑥 = 𝑥! + 𝑥 + 1
Ex. 2 Oblique
𝑁𝑜 𝐻𝐴,
𝑂𝐴:
𝐻𝐴:
3 𝑦= , 2
𝑉𝐴:
1 𝑥 =−! 2
Inequalities
𝑓 𝑥 𝑓 𝑥
< 𝑎 ⇒ −𝑎 < 𝑓 𝑥 < 𝑎 or 𝑓 𝑥 < 𝑎 and 𝑓 𝑥 > −𝑎 ≤ 𝑎 ⇒ −𝑎 ≤ 𝑓 𝑥 ≤ 𝑎 or 𝑓 𝑥 ≤ 𝑎 and 𝑓 𝑥 ≥ −𝑎
Interest Formulas
𝐴 = 𝐴! 𝑒 !"
𝑃 = 𝑃!
𝑟 12
𝑟 1 − 1 + 12
Physics Formulas
(rate)(time)=distance 𝑟𝑡 = 𝑑
!!
𝑦 = 𝑥
3𝑥 ! + 𝑥 𝑓 𝑥 = ! 2𝑥 + 1
Ex. 3 Horizontal and Vertical
L=Loan 𝑃 = Monthly Payment 𝑟 = Interest rate for annual 𝑡 = Loan length in months
Symmetry
By Point 𝑥-axis For every point 𝑥, 𝑦 there is a 𝑥, −𝑦 𝑦-axis For every point 𝑥, 𝑦 there is a −𝑥, 𝑦 origin For every point 𝑥, 𝑦 there is a −𝑥, −𝑦 Testing 𝑥-axis: Replace each 𝑦 with a – 𝑦, if the same equation results, it is symmetric. 𝑦-axis: Replace each 𝑥 with a – 𝑥, if the same equation results, it is symmetric. Origin: Replace each 𝑥, 𝑦 with a −𝑥, – 𝑦, if the same equation results, it is symmetric.
Variations (Proportionality) 𝑘 is the constant of proportionality
𝑦 is proportional to x: y= 𝑘𝑥
𝑦 is inversely proportional to ! 𝑥: 𝑦 = !
Common Graphs and Formulas 𝑦 = 𝑥!
𝑦 = 𝑥!
_________________________________________________________________________________________________________
𝑦= 𝑥
1 𝑦= 𝑥
_________________________________________________________________________________________________________ ! 𝑦=𝑒 𝑦 = ln 𝑥
_________________________________________________________________________________________________________ 𝑥! + 𝑦! = 1 𝑥 ! − 𝑦 ! = 1
_________________________________________________________________________________________________________ 𝑦! − 𝑥! = 1 𝑦 = 𝑥! − 1
