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- 1. . Sec on 2.2 A Catalogue of Essen al Func ons V63.0121.011, Calculus I Professor Ma hew Leingang New York UniversityAnnouncements First WebAssign-ments are due January 31 First wri en assignment is due February 2 First recita ons are February 3
- 2. Announcements First WebAssign-ments are due January 31 First wri en assignment is due February 2 First recita ons are February 3
- 3. Objectives Iden fy diﬀerent classes of algebraic func ons, including polynomial (linear,quadra c,cubic, etc.), ra onal, power, trigonometric, and exponen al func ons. Understand the eﬀect of algebraic transforma ons on the graph of a func on. Understand and compute the composi on of two func ons.
- 4. Recall: What is a function? Deﬁni on A func on f is a rela on which assigns to to every element x in a set D a single element f(x) in a set E. The set D is called the domain of f. The set E is called the target of f. The set { y | y = f(x) for some x } is called the range of f.
- 5. Four ways to represent a function verbally—by a descrip on in words numerically—by a table of values or a list of data visually—by a graph symbolically or algebraically—by a formula Today the focus is on the diﬀerent kinds of formulas that can be used to represent func ons.
- 6. Classes of Functions linear func ons, deﬁned by slope and intercept, two points, or point and slope. quadra c func ons, cubic func ons, power func ons, polynomials ra onal func ons trigonometric func ons exponen al/logarithmic func ons
- 7. Outline Algebraic Func ons Linear func ons Other polynomial func ons Other power func ons General ra onal func ons Transcendental Func ons Trigonometric func ons Exponen al and logarithmic func ons Transforma ons of Func ons Composi ons of Func ons
- 8. Linear functionsLinear func ons have a yconstant rate of growth and (x2 , y2 )are of the form (x1 , y1 ) ∆y = y2 − y1 f(x) = mx + b. (0, b) ∆x = x2 − x1The slope m represents the ∆y m=“steepness” of the graphed ∆xline, and the intercept b . xrepresents an ini al value ofthe func on.
- 9. Modeling with Linear Functions Example Assume that a taxi costs $2.50 to get in and $0.40 per 1/5 mile. Write the fare f(x) as a func on of distance x traveled.
- 10. Modeling with Linear Functions Example Assume that a taxi costs $2.50 to get in and $0.40 per 1/5 mile. Write the fare f(x) as a func on of distance x traveled. Answer The ini al fare is $2.50, and the change in fare per mile is $0.40/0.2 mi = $2/mi. So if x is in miles and f(x) in dollars, the equa on is f(x) = 2.5 + 2x
- 11. A Biological Example Example Biologists have no ced that the chirping rate of crickets of a certain species is related to temperature, and the rela onship appears to be very nearly linear. A cricket produces 113 chirps per minute at 70 ◦ F and 173 chirps per minute at 80 ◦ F. (a) Write a linear equa on that models the temperature T as a func on of the number of chirps per minute N. (b) If the crickets are chirping at 150 chirps per minute, es mate the temperature.
- 12. Biological Example: Solution Solu on The point-slope form of the equa on for a line is appropriate here: If a line passes through (x0 , y0 ) with slope m, then the line has equa on y − y0 = m(x − x0 ).
- 13. Biological Example: Solution Solu on The point-slope form of the equa on for a line is appropriate here: If a line passes through (x0 , y0 ) with slope m, then the line has equa on y − y0 = m(x − x0 ). 80 − 70 10 1 The slope of our line is = = 173 − 113 60 6
- 14. Biological Example: Solution Solu on The point-slope form of the equa on for a line is appropriate here: If a line passes through (x0 , y0 ) with slope m, then the line has equa on y − y0 = m(x − x0 ). 80 − 70 10 1 The slope of our line is = = 173 − 113 60 6 So an equa on rela ng T and N is 1 1 113 T − 70 = (N − 113) =⇒ T = N − + 70 6 6 6
- 15. Solution continued So an equa on rela ng T and N is 1 1 113 T − 70 = (N − 113) =⇒ T = N − + 70 6 6 6
- 16. Solution continued So an equa on rela ng T and N is 1 1 113 T − 70 = (N − 113) =⇒ T = N − + 70 6 6 6 If N = 150, then 37 T= + 70 = 76 1 ◦ F 6 6
- 17. Other polynomial functions Quadra c func ons take the form f(x) = ax2 + bx + c The graph is a parabola which opens upward if a > 0, downward if a < 0.
- 18. Other polynomial functions Quadra c func ons take the form f(x) = ax2 + bx + c The graph is a parabola which opens upward if a > 0, downward if a < 0. Cubic func ons take the form f(x) = ax3 + bx2 + cx + d
- 19. Other power functions Whole number powers: f(x) = xn . 1 nega ve powers are reciprocals: x−3 = 3 . 1/3 √ x frac onal powers are roots: x = 3 x.
- 20. General rational functions Deﬁni on A ra onal func on is a quo ent of polynomials. Example x3 (x + 3) The func on f(x) = is ra onal. (x + 2)(x − 1) The domain is all real numbers except −2 and 1. The func on is 0 when x = 0 or x = −3.
- 21. Outline Algebraic Func ons Linear func ons Other polynomial func ons Other power func ons General ra onal func ons Transcendental Func ons Trigonometric func ons Exponen al and logarithmic func ons Transforma ons of Func ons Composi ons of Func ons
- 22. Trigonometric functions Sine and cosine Tangent and cotangent Secant and cosecant GeoGebra applets to graph these
- 23. Exponential and logarithmicfunctions exponen al func ons (for example f(x) = 2x ) logarithmic func ons are their inverses (for example f(x) = log2 (x)) GeoGebra applets to graph these
- 24. Outline Algebraic Func ons Linear func ons Other polynomial func ons Other power func ons General ra onal func ons Transcendental Func ons Trigonometric func ons Exponen al and logarithmic func ons Transforma ons of Func ons Composi ons of Func ons
- 25. Transformations of Functions Take the squaring func on and graph these transforma ons: y = (x + 1)2 y = (x − 1)2 y = x2 + 1 y = x2 − 1
- 26. Transformations of Functions Take the squaring func on and graph these transforma ons: y = (x + 1)2 y = (x − 1)2 y = x2 + 1 y = x2 − 1 Observe that if the ﬁddling occurs within the func on, a transforma on is applied on the x-axis. A er the func on, to the y-axis.
- 27. Vertical and Horizontal Shifts Suppose c > 0. To obtain the graph of y = f(x) + c, shi the graph of y = f(x) a distance c units . . . y = f(x) − c, shi the graph of y = f(x) a distance c units . . . y = f(x − c), shi the graph of y = f(x) a distance c units . . . y = f(x + c), shi the graph of y = f(x) a distance c units . . .
- 28. Vertical and Horizontal Shifts Suppose c > 0. To obtain the graph of y = f(x) + c, shi the graph of y = f(x) a distance c units . . . upward y = f(x) − c, shi the graph of y = f(x) a distance c units . . . y = f(x − c), shi the graph of y = f(x) a distance c units . . . y = f(x + c), shi the graph of y = f(x) a distance c units . . .
- 29. Vertical and Horizontal Shifts Suppose c > 0. To obtain the graph of y = f(x) + c, shi the graph of y = f(x) a distance c units . . . upward y = f(x) − c, shi the graph of y = f(x) a distance c units . . . downward y = f(x − c), shi the graph of y = f(x) a distance c units . . . y = f(x + c), shi the graph of y = f(x) a distance c units . . .
- 30. Vertical and Horizontal Shifts Suppose c > 0. To obtain the graph of y = f(x) + c, shi the graph of y = f(x) a distance c units . . . upward y = f(x) − c, shi the graph of y = f(x) a distance c units . . . downward y = f(x − c), shi the graph of y = f(x) a distance c units . . . to the right y = f(x + c), shi the graph of y = f(x) a distance c units . . .
- 31. Vertical and Horizontal Shifts Suppose c > 0. To obtain the graph of y = f(x) + c, shi the graph of y = f(x) a distance c units . . . upward y = f(x) − c, shi the graph of y = f(x) a distance c units . . . downward y = f(x − c), shi the graph of y = f(x) a distance c units . . . to the right y = f(x + c), shi the graph of y = f(x) a distance c units . . . to the le
- 32. Why? Ques on Why is the graph of g(x) = f(x + c) a shi of the graph of f(x) to the le by c?
- 33. Why? Ques on Why is the graph of g(x) = f(x + c) a shi of the graph of f(x) to the le by c? Answer Think about x as me. Then x + c is the me c into the future. To rec fy the future of the graph of f with that of g, pull the graph of f c into the past.
- 34. Illustrating the shift (x, f(x)) . x
- 35. Illustrating the shift Adding c moves x to the right (x, f(x)) . x x+c
- 36. Illustrating the shift (x + c, f(x + c)) Adding c moves x to the right (x, f(x)) But then f is applied . x x+c
- 37. Illustrating the shift (x, f(x + c)) (x + c, f(x + c)) Adding c moves x to the right (x, f(x)) But then f is applied To get the graph of f(x + c), the value f(x + c) must be above x . x x+c
- 38. Illustrating the shift (x, f(x + c)) (x + c, f(x + c)) Adding c moves x to the right (x, f(x)) But then f is applied To get the graph of f(x + c), the value f(x + c) must be above x So we translate backward . x x+c
- 39. Now try these y = sin (2x) y = 2 sin (x) y = e−x y = −ex
- 40. Scaling and ﬂipping c<0 c>0 |c| > 1 |c| < 1 |c| < 1 |c| > 1 f(cx) . . . . H compress, ﬂip H stretch, ﬂip H stretch H compress cf(x) . . . . V stretch, ﬂip V compress, ﬂip V compress V stretch
- 41. Outline Algebraic Func ons Linear func ons Other polynomial func ons Other power func ons General ra onal func ons Transcendental Func ons Trigonometric func ons Exponen al and logarithmic func ons Transforma ons of Func ons Composi ons of Func ons
- 42. Composition of FunctionsCompounding in Succession g◦f x f . g (g ◦ f)(x) f(x)
- 43. Composing Example Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.
- 44. Composing Example Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f. Solu on (f ◦ g)(x) = sin2 x (g ◦ f)(x) = sin(x2 ) Note they are not the same.
- 45. Decomposing Example √ Express x2 − 4 as a composi on of two func ons. What is its domain?
- 46. Decomposing Example √ Express x2 − 4 as a composi on of two func ons. What is its domain? Solu on √ We can write the expression as f ◦ g, where f(u) = u and g(x) = x2 − 4. The range of g needs to be within the domain of f. To insure that x2 − 4 ≥ 0, we must have x ≤ −2 or x ≥ 2.
- 47. Summary There are many classes of algebraic func ons Algebraic rules can be used to sketch graphs

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