Please help me answering the math problems?
Posted: Fri Jul 08, 2022 5:37 am
Please help me answering the math problems?
AVERAGE RATE OF CHANGE COMMON CORE ALGEBRA II When we model using functions, we are very often interested in the rate that the output is changing when compared to a change in the input. Exercise #1: The function f(x) is shown graphed to the right. (a) Evaluate each of the following based on the graph: (i) ƒ (0) (ii) ƒ (4) (iii) ƒ(7) (iv) f(13) (b) Find the change in the function, Af, over each of the following domain intervals. Find this by subtraction and show the change on the graph. (i) 0≤x≤4 (ii) 4≤x≤7 (iii) 7 ≤x≤13 (c) Why can't you simply compare the changes in f from part (b) to determine over which interval the function is changing the fastest? (i) 0≤x≤4 (d) Calculate the average rate of change for the function over each of the intervals and determine which interval has the greatest rate. (ii) 4≤x≤7 (iii) 7 ≤x≤13 (e) Using a straightedge, draw in the lines whose slopes you found in part (d) by connecting the points shown on the graph. The average rate of change gives a measurement of what property of the line?
The average rate of change is an exceptionally important concept in mathematics because it gives us a way to quantify how fast a function changes on average over a certain domain interval. Although we used its formula in the last exercise, we state it formally here: AVERAGE RATE OF CHANGE For a function over the domain interval a ≤x≤ b, the function's average rate of change is calculated by: A change in the output f(b)-f(a) Ax change in the input b-a Exercise #2: Consider the two functions f(x)=5x+7 and g(x)=2x² +1. (a) Calculate the average rate of change for both functions over the following intervals. Do your work carefully and show the calculations that lead to your answers. (i) -2≤x≤3 (ii) 1≤x≤5 (b) The average rate of change for / was the same for both (i) and (ii) but was not the same for g. Why is that? Exercise #3: The table below represents a linear function. Fill in the missing entries. x y 15 11 -5 1 22 45 Mon Couston Cone ACER II + IT Tver 42 LIVED FUNCTION FRATION D Turm Acum 1- Com rex 1 M Name: (a) -3≤x≤-1 FLUENCY 1. For the function g(x) given in the table below, calculate the average rate of change for each of the following intervals. AVERAGE RATE OF CHANGE COMMON CORE ALGEBRA II HOMEWORK (a) 0≤x≤2 g(x) -3 M -1 -2 (b)-1≤x≤6 4 13 6 12 (b) 2≤x≤4 9 (d) Explain how you can tell from the answers in (a) through (c) that this is not a table that represents a linear function. Date: 5 2. Consider the simple quadratic function f(x)=x². Calculate the average rate of change of this function over the following intervals: (d) Clearly the average rate of change is getting larger at x gets larger. How is this reflected in the graph offshown sketched to the right? (c)-3≤x≤9 (c) 4≤x≤6 Costany Cone ALCI. II. Hver i Lovran Fee FousTICIVE AND THE ALCE I+ Mot
3. Which has a greater average rate of change over the interval -2 ≤x≤4, the function g(x)=16x-3 or the function f(x)=2x²? Provide justification for your answer. APPLICATIONS 4. An object travels such that its distance, d, away from its starting point is shown as a function of time, t, in seconds, in the graph below. (a) What is the average rate of change of d over the interval 5 ≤t≤7? Include proper units in your answer. (b) The average rate of change of distance over time (what you found in part (a)) is known as the average speed of an object. Is the average speed of this object greater on the interval 0≤t≤5 or 11≤t ≤14 ? Justify. Distance (feet) M (14,79) (11,64) (7,64) I (5,20) Time (seconds) REASONING 5. What makes the average rate of change of a linear function different from that of any other function? What the special name we give to the average rate of change a linear function? M
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The average rate of change is an exceptionally important concept in mathematics because it gives us a way to quantify how fast a function changes on average over a certain domain interval. Although we used its formula in the last exercise, we state it formally here: AVERAGE RATE OF CHANGE For a function over the domain interval a ≤x≤ b, the function's average rate of change is calculated by: A change in the output f(b)-f(a) Ax change in the input b-a Exercise #2: Consider the two functions f(x)=5x+7 and g(x)=2x² +1. (a) Calculate the average rate of change for both functions over the following intervals. Do your work carefully and show the calculations that lead to your answers. (i) -2≤x≤3 (ii) 1≤x≤5 (b) The average rate of change for / was the same for both (i) and (ii) but was not the same for g. Why is that? Exercise #3: The table below represents a linear function. Fill in the missing entries. x y 15 11 -5 1 22 45 Mon Couston Cone ACER II + IT Tver 42 LIVED FUNCTION FRATION D Turm Acum 1- Com rex 1 M Name: (a) -3≤x≤-1 FLUENCY 1. For the function g(x) given in the table below, calculate the average rate of change for each of the following intervals. AVERAGE RATE OF CHANGE COMMON CORE ALGEBRA II HOMEWORK (a) 0≤x≤2 g(x) -3 M -1 -2 (b)-1≤x≤6 4 13 6 12 (b) 2≤x≤4 9 (d) Explain how you can tell from the answers in (a) through (c) that this is not a table that represents a linear function. Date: 5 2. Consider the simple quadratic function f(x)=x². Calculate the average rate of change of this function over the following intervals: (d) Clearly the average rate of change is getting larger at x gets larger. How is this reflected in the graph offshown sketched to the right? (c)-3≤x≤9 (c) 4≤x≤6 Costany Cone ALCI. II. Hver i Lovran Fee FousTICIVE AND THE ALCE I+ Mot
3. Which has a greater average rate of change over the interval -2 ≤x≤4, the function g(x)=16x-3 or the function f(x)=2x²? Provide justification for your answer. APPLICATIONS 4. An object travels such that its distance, d, away from its starting point is shown as a function of time, t, in seconds, in the graph below. (a) What is the average rate of change of d over the interval 5 ≤t≤7? Include proper units in your answer. (b) The average rate of change of distance over time (what you found in part (a)) is known as the average speed of an object. Is the average speed of this object greater on the interval 0≤t≤5 or 11≤t ≤14 ? Justify. Distance (feet) M (14,79) (11,64) (7,64) I (5,20) Time (seconds) REASONING 5. What makes the average rate of change of a linear function different from that of any other function? What the special name we give to the average rate of change a linear function? M