Lesson 1 – A.1.1 – Function Characteristics Calculus - Santowski 9/4/2015 Calculus - Santowski 1

Lesson Objectives 1. Review characteristics of functions - like domain, range, max, min, intercepts 2. Extend application of function models 3. Introduce new function concepts pertinent to Calculus - concepts like intervals of increase, decrease, concavity, end behaviour, rate of change 9/4/2015Calculus - Santowski 2

Fast Five 1. Name the type of function: f(x) = x 3 2. Find f(2) for f(x) = x 3 3. Name the type of function: g(x) = 3 x 4. Find g(2) for g(x) = 3 x 5. Sketch the graph of h(x) = x 2 6. Find h -1 (2) for h(x) = x 2 7. At what values is t(x) = (x - 4)/(x - 3) undefined 8. Sketch a graph of a linear function with a positive y-intercept and a negative slope 9. Evaluate sin(  /2) - cos(  /3) 10. Sixty is 30% of what number? 9/4/2015Calculus - Santowski 3

(A) Function Characteristics Terminology to review: Domain Range Symmetry Roots, zeroes Turning point Maximum, minimum Increase, decrease End behaviour 9/4/2015Calculus - Santowski 4

(A) Function Characteristics Domain: the set of all possible x values (independent variable) in a function Range: the set of all possible function values (dependent variable, or y values) to evaluate a function: substituting in a value for the variable and then determining a function value. Ex f(3) finite differences: subtracting consecutive y values or subsequent y differences 9/4/2015Calculus - Santowski 5

(A) Function Characteristics zeroes, roots, x-intercepts: where the function crosses the x axes (y-value is 0) y-intercepts: where the function crosses the y axes (x-value is 0) direction of opening: in a quadratic, curve opens up or down symmetry: whether the graph of the function has "halves" which are mirror images of each other 9/4/2015Calculus - Santowski 6

(A) Function Characteristics turning point: points where the direction of the function changes maximum: the highest point on a function minimum: the lowest point on a function local vs absolute: a max can be a highest point in the entire domain (absolute) or only over a specified region within the domain (local). Likewise for a minimum. 9/4/2015Calculus - Santowski 7

(A) Function Characteristics increase: the part of the domain (the interval) where the function values are getting larger as the independent variable gets higher; if f(x 1 ) < f(x 2 ) when x 1 < x 2 ; the graph of the function is going up to the right (or down to the left) decrease: the part of the domain (the interval) where the function values are getting smaller as the independent variable gets higher; if f(x 1 ) > f(x 2 ) when x 1 < x 2 ; the graph of the function is going up to the left (or down to the right) "end behaviour": describing the function values (or appearance of the graph) as x values getting infinitely large positively or infinitely large negatively 9/4/2015Calculus - Santowski 8

(B) Working with the Function Characteristics The slides that follow simply review all functions that you have seen to date in previous courses You are expected to become proficient with a method of GRAPHICALLY determining that which is being asked of you (Use TI-89) Work with your partners through the following exercises: 9/4/2015Calculus - Santowski 9

(B) Working with the Function Characteristics For the quadratic functions, determine the following: f(x) = -½x² - 3x - 4.5f(x) = 2x² - x + 4 (1) Leading coefficient(2) degree (3) domain and range (4) evaluating f(-2) (5) zeroes or roots (6) y-intercept (7) Symmetry(8) turning points (9) maximum values (local and absolute) (10) minimum values (local and absolute) (11) intervals of increase and intervals of decrease (12) end behaviour (+x) and end behaviour (-x) 9/4/2015Calculus - Santowski 10

(B) Working with the Function Characteristics For the cubic functions, determine the following: f(x) = x 3 - 5x² + 3x + 4 f(x) =-2x 3 + 8x² - 5x + 3 f(x) = -3x 3 -15x² - 9x + 27 (1) Leading coefficient(2) degree (3) domain and range (4) evaluating f(-2) (5) zeroes or roots (6) y-intercept (7) Symmetry(8) turning points (9) maximum values (local and absolute) (10) minimum values (local and absolute) (11) intervals of increase and intervals of decrease (12) end behaviour (+x) and end behaviour (-x) 9/4/2015Calculus - Santowski 11

(B) Working with the Function Characteristics For the quartic functions, determine the following: f(x)= -2x 4 -4x 3 +3x²+6x+9 f(x)= x 4 -3x 3 +3x²+8x+5 f(x) = ½x 4 -2x 3 +x²+x+1 (1) Leading coefficient(2) degree (3) domain and range (4) evaluating f(-2) (5) zeroes or roots (6) y-intercept (7) Symmetry(8) turning points (9) maximum values (local and absolute) (10) minimum values (local and absolute) (11) intervals of increase and intervals of decrease (12) end behaviour (+x) and end behaviour (-x) 9/4/2015Calculus - Santowski 12

(B) Working with the Function Characteristics For the Root Functions, determine the: (1) Leading coefficient(2) degree (3) domain and range (4) evaluating f(-2) (5) zeroes or roots (6) y-intercept (7) Symmetry(8) turning points (9) maximum values (local and absolute) (10) minimum values (local and absolute) (11) intervals of increase and intervals of decrease (12) end behaviour (+x) and end behaviour (-x) 9/4/2015Calculus - Santowski 13

(B) Working with the Function Characteristics For the Rational Functions, determine the: (1) Leading coefficient(2) degree (3) domain and range (4) evaluating f(-2) (5) zeroes or roots (6) y-intercept (7) Symmetry(8) turning points (9) maximum values (local and absolute) (10) minimum values (local and absolute) (11) intervals of increase and intervals of decrease (12) end behaviour (+x) and end behaviour (-x) 9/4/2015Calculus - Santowski 14

(B) Working with the Function Characteristics For the Exponential Functions, determine the: (1) Leading coefficient(2) degree (3) domain and range (4) evaluating f(-2) (5) zeroes or roots (6) y-intercept (7) Symmetry(8) turning points (9) maximum values (local and absolute) (10) minimum values (local and absolute) (11) intervals of increase and intervals of decrease (12) end behaviour (+x) and end behaviour (-x) 9/4/2015Calculus - Santowski 15

(B) Working with the Function Characteristics For the logarithmic functions, determine the: (1) Leading coefficient(2) degree (3) domain and range (4) evaluating f(-2) (5) zeroes or roots (6) y-intercept (7) Symmetry(8) turning points (9) maximum values (local and absolute) (10) minimum values (local and absolute) (11) intervals of increase and intervals of decrease (12) end behaviour (+x) and end behaviour (-x) 9/4/2015Calculus - Santowski 16

(B) Working with the Function Characteristics For the trig functions, determine the: (1) Leading coefficient(2) degree (3) domain and range (4) evaluating f(-2) (5) zeroes or roots (6) y-intercept (7) Symmetry(8) turning points (9) maximum values (local and absolute) (10) minimum values (local and absolute) (11) intervals of increase and intervals of decrease (12) end behaviour (+x) and end behaviour (-x) 9/4/2015Calculus - Santowski 17

(C) “A” Level Function Questions 1. If and if f(2) = 2, find the value of f(1). 2. Suppose that Then let g 1 (x) = f(x) and g 2 (x) = f(f(x)) and so on such that g n (x) = f( ….. (f(x)) …) …. where f occurs n times here Develop a general formula for g n (x) and suggest a method for proving that your general formula is true for all cases of n. 9/4/2015Calculus - Santowski 18