Increasing and Decreasing Functions

Section 2.6: Increasing and decreasing functions. Chapter 2: Functions, Linear equations, and inequalities Using a Graph to Determine Where a Function is Increasing, Decreasing, or Constant A function is increasing over an open interval provided the y-coordinates of the points in the interval get larger, or equivalently the graph gets higher as it moves from left to right over the interval. A function is decreasing over an open interval provided the 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 of the points in the interval get smaller, or equivalently the graph gets lower as it moves from left to right over the interval. A point is a local maximum point provided it is higher than any point close to it. (Technically a point is a local maximum point if the graph changes from increasing to decreasing at that point.) The local maximum value is the y-coordinate of the local maximum point. A point is a local minimum point provided it is lower than any point close to it. (Technically a point is a local minimum point if the graph changes from decreasing to increasing at that point.) The local minimum value is the y-coordinate of the local minimum point. Here is a visual representation of what is written above. Here is another, example of what is written above. Notice this graph has a local maximum point, but it does not have a local minimum. Section 2.6: Increasing and decreasing functions. Chapter 2: Functions, Linear equations, and inequalities For example: Use the graph of f(x) to determine: a) interval(s) where the graph is increasing. b) interval(s) where the graph is decreasing. c) the coordinates of local maximum point, if any d) the local maximum value e) the coordinates of the local minimum point if any f) the local minimum value • To determine the intervals where a graph is increasing and decreasing: break graph into intervals in terms of 𝑥, using only round parenthesis and determine if the graph is getting higher or lower in the interval. First interval: goes from the left edge of the graph which has an 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of 𝑥 = −∞ to the point (−2,16) which has an 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of 𝑥 = −2 First interval (−∞, −2) Second interval goes from the point (−2, 16) 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑥 = −2 to the point (2, −16) 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑥 = 2 A point is a local maximum point provided it is higher than any point close to it. (Technically a point is a local maximum point if the graph changes from increasing to decreasing at that point.) The local maximum value is the y-coordinate of the local maximum point. 𝑐) (−2,16) 𝑖𝑠 𝑎 𝑙𝑜𝑐𝑎𝑙 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑝𝑜𝑖𝑛𝑡, d) the local maximum value is 𝑦 = 16 when x =2 Second interval (−2,3) Third interval goes from the point (2, −16) 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑥 = 2. to the right edge of the graph 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑥 = ∞ Third interval (2, ∞) • Determine if the graph is increasing (getting higher) or decreasing (getting lower) in each interval. A point is a local minimum point provided it is lower than any point close to it. (Technically a point is a local minimum point if the graph changes from decreasing to First interval (−∞, −2) increasing increasing at that point.) Second interval (−2,1) decreasing Third interval (2, ∞) increasing The local minimum value is the y-coordinate of the local minimum point. Answer: a) increasing (−∞, −2) ∪ (2, ∞) b) decreasing (−2,2) 𝑒) (2, −16)𝑖𝑠 𝑎 𝑙𝑜𝑐𝑎𝑙 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑝𝑜𝑖𝑛𝑡, f) the local minimum value is 𝑦 = −16 𝑤ℎ𝑒𝑛 𝑥 = −2. Section 2.6: Increasing and decreasing functions. Chapter 2: Functions, Linear equations, and inequalities Determine whether a function is increasing or decreasing given data in table form. There are two ways to determine if a function is increasing or decreasing given a table. 1) Plot the points and examine the graph. Increasing – if graph gets higher as it moves from left to right Decreasing – if graph gets lower as it moves from left to right 2) Look at the relative size of the numbers in the f(x) column Increasing – if the values in the f(x) column are getting larger Decreasing – if the values in the f(x) column are getting smaller For the table below, select whether the table represents a function that is increasing or decreasing, 𝑥 𝑓(𝑥) There are two ways to determine the answer. 1 200 2 190 1) Plot the points 3 170 4 140 5 100 6 50 • The points on the graph are clearly getting lower, the function is decreasing 2) Look at the relative size of the numbers in the f(x) column. • Each of the values in the f(x) column is getting smaller. The function is decreasing. Answer: Decreasing Section 2.6: Increasing and decreasing functions. Chapter 2: Functions, Linear equations, and inequalities For the table below, select whether the table represents a function that is increasing or decreasing, 𝑥 𝑔(𝑥) 1 10 2 15 3 25 4 40 5 60 6 85 There are two ways to determine the answer. 1) plot the points • The points are clearly getting higher. The graph is increasing. 2) look at the relative size of the numbers in the g(x) column. • Each of the values in the f(x) column is getting larger. The function is increasing. Answer: Increasing Section 2.6: Increasing and decreasing functions. Chapter 2: Functions, Linear equations, and inequalities Use A Graph to Locate the Absolute Maximum and Absolute Minimum • • • • The absolute maximum is the highest point over the entire domain of a function or relation. The absolute maximum value is the 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of the absolute maximum point. The absolute minimum is the lowest point over the entire domain of a function or relation. The absolute minimum value is the 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of the absolute minimum point. Find the • • • • Coordinates of the absolute maximum point. Value of the absolute maximum Coordinates of the absolute minimum point Value of the absolute minimum • • • • Coordinates of the absolute maximum point. (2,2) Value of the absolute maximum: absolute maximum value is 𝑓(𝑥) = 2 which occurs when 𝑥 = 2 Coordinates of the absolute minimum point (0, −2) Value of the absolute minimum : absolute minimum value is 𝑓(𝑥) = −2 which occurs when 𝑥 = 0 #1 – 10: Find the a) interval(s) where the graph is increasing. b) interval(s) where the graph is decreasing. c) the coordinates of local maximum point, if any d) the local maximum value e) the coordinates of the local minimum point if any f) the local minimum value 1) 2) Section 2.6: Rates of change, increasing and decreasing functions. Chapter 2: Functions, Linear equations, and inequalities #1 – 10: Find the a) interval(s) where the graph is increasing. b) interval(s) where the graph is decreasing. c) the coordinates of local maximum point, if any d) the local maximum value e) the coordinates of the local minimum point if any f) the local minimum value 3) 5) 4) 6) #1 – 10: Find the a) interval(s) where the graph is increasing. b) interval(s) where the graph is decreasing. c) the coordinates of local maximum point, if any d) the local maximum value e) the coordinates of the local minimum point if any f) the local minimum value 7) 9) 8) 10) #11 – 12: Find the a) interval(s) where the graph is increasing. b) interval(s) where the graph is decreasing. c) interval(s) where graph is constant d) the coordinates of local maximum point if any e) the local maximum value f) the coordinates of the local minimum point if any g) the local minimum value 11) 12) #13-16: Determine if the function defined in the table is increasing or decreasing 13) x f(x) 1 5 2 10 3 20 4 40 5 70 14) x g(x) 1 15 2 25 3 33 4 38 5 40 15) x h(x) 1 100 2 90 3 70 4 40 5 0 16) x k(x) 1 100 2 70 3 50 4 40 5 35 #17 – 20: a) b) c) d) Coordinates of the absolute maximum point. Value of the absolute maximum Coordinates of the absolute minimum point Value of the absolute minimum 17) 18) 19) 20)