You Try: Find the x-intercepts of f ( x) Find the roots (zeros, x-intercepts) of 2. x x. x 2. x 8 x x 2 8x 4 4x 32

Transcription

1 1 Find the roots (zeros, -intercepts) of a Find the -intercepts of 5 1 We are looking for the solutions to Method 1: Factoring Using Guess & heck Using an rea Model, fill in a Generic Rectangle with the first 3 and last terms. Given is the area (product), the dimensions (factors) have to be and We GUESS which factors will give us a product of 3 and when combined as the middle term will give us 1. Method : ompleting the square ( 6) 3 ( 6) or or Move constant term to other side Simplify Simplify and factor Inverse operations : square root 6 8 Method 3: Quadratic formula Leave space to complete the square 1 omplete the square with = 0 a = 1, b = 1, and c = 3 1b Which of the following could be part of the process of finding the roots of y 4 60? Select all that apply. ) ) ( 4) 4(1)( 60) ) (1) D) 4 ( 4) 4(1)( 60) (1) E) F) 4 ( 4) 60 ( 4 ( ) 60 ( 4) ) -REI.4b Page 1 of 13 M@WUSD 01/15/16

2 nswer the following questions about the graph of the function g() shown below: nswer the following questions about the graph of the function g() shown below: a) What is the y-intercept? The y-intercept is where the graph crosses the y-ais, in this case at the point (0, ) or y =. b) What are the -intercept(s)? The -intercept is where the graph crosses the -ais, in this case at the points ( 4, 0) and (4, 0) or = 4 or 4. c) How has this graph transformed from the mother function, f() =? g() is wider than the graph of f() = (which means the quadratic coefficient has an absolute value less than 1). g() is concave down (which means the quadratic coefficient is negative), while f() = is concave up. g() is also shifted units up from f() = (which means the constant term is +). d) Is the average rate of change from = 0 to = 4 negative, positive, zero, or undefined? If we look only at the portion of the graph between = 0 and = 4, we can see that the graph is decreasing. This means that the average rate of change on this interval is negative. e) What is the approimate value of g( 8)? We are looking at the graph where = 8 and trying to determine what the y-value is at that point. We can approimate that g( 8) 6. f) Is g() increasing or decreasing on 4 0? If we look only at the portion of the graph between = 4 and = 0, we can see that the graph is increasing. F-IF.4 a) What are the y-intercept(s)? b) What are the -intercept(s)? c) How has this graph transformed from the mother function, f() =? d) Is the average rate of change from = 0 to = negative, positive, zero, or undefined? e) What is the approimate value of g( 3)? f) Is g() increasing or decreasing on 0? Page of 13 M@WUSD 01/15/16

3 3 The function f () = is graphed below. 3 cont d D) g() = + 3 g() is of the form f( + 3). The constant is being added to the function s domain. This means the parent function will shift to the left or right. Since the constant is positive, it will shift to the left three units. Using this graph as the parent function f(), sketch a graph of each of the following functions. To check this, I could also create a table. 4 3 g() ) g() = + 3 g() is of the form f() + 3. The constant is being added to the function and not to the function s domain. This means the parent function will shift up (since 3 is positive) three units. 3 F.F.3 Match the function with its graph. 1. f () = To check this, I could also create a table g() ) g() = g() is of the form f( ). The domain of is being transformed to. This means the parent function will reflect over the y-ais. To check this, I could also create a table g() D ) g() = g() is of the form f(). The range of the function is being transformed to its opposites. This means the parent function will reflect over the -ais. E To check this, I could also create a table g() F-F.3 Page 3 of 13 M@WUSD 01/15/16

4 4 The verage Rate of hange of a function f() from = a to = b is f ( b) f ( a) b a change in function values change in 4a 1) Given the function, 4 1 find the average rate of change from = 1 to = 5. ) Given the function, 1 find the average rate of change from = 0 to = 1. To do this, I need to evaluate the function at = 0 and = 1. f (0) (0) 1 f (1) (1) 1 f (0) 0 1 f (0) 1 f (1) 1 f (1) 1 Now, I can substitute those values into the formula. f ( b) f ( a) b a 1 ( 1) The average rate of change of f() from = 0 to = 1 is. ) Given the function, g() = +1 find the average rate of change from = 0 to = 1. To do this, I need to evaluate the function at = 0 and = 1. g(0) = 0 +1 g(1) = 1 +1 g(0) =1+1 g(0) = g(1) = +1 g(1) = 3 Now, I can substitute those values into the formula. g( b) g( a) b a The average rate of change of 1 g() from = 0 to = 1 is 1. 1 ) From = 0 to = 1, what do we notice about f() and g()? On the interval from = 0 to = 1, f() is increasing at a greater rate than g(). Given what we know about linear and eponential functions, they are both increasing everywhere. F-IF.6 ) Given the function, g() = 3 find the average rate of change from = 1 to = 3. 4b 3 1 g( ) Given the functions above, select all of the following intervals on which the average rate of change of f() is greater than the average rate of change of g(). ) 1 0 ) 0 3 ) 1 3 D) 3 Select all that apply. Page 4 of 13 M@WUSD 01/15/16

5 5 nswer the following questions about the graph of the function g() shown below: 5 Two functions, f() and g() are represented below. Function 1: 3 3 Function : g() a) What are the y-intercept(s)? The y-intercept is where the graph crosses the y-ais, in this case at the point (0, 3) or y = 3. b) What are the -intercept(s)? The -intercept is where the graph crosses the -ais. In this case, there is no -intercept, because the eponential function has an asymptote at the line y =. c) What is the average rate of change between = 0 and =? First, if I look at the portion of the graph between = 0 and =, I see that it is increasing, so I know my average rate of change will be positive. I know that I need to evaluate the function at the boundary points. g(0) = 3 and g() = 6. Therefore, I can use the formula for rate of change. g( b) g( a) average rate of change = b a d) Is the function increasing, decreasing, both, or neither? I can see that the function is increasing everywhere. F-IF.4 Select all of the following statements that are TRUE about the functions above. Select all that apply. ) f() and g() have the same -intercept(s). ) f() and g() have the same y-intercept(s). ) f() and g() have the same average rate of change between = 3 and = 1. D) oth functions are decreasing everywhere. E) oth functions have a maimum at (0, ) F-IF.9 Page 5 of 13 M@WUSD 01/15/16

6 Linear functions: f() = m + b You can tell a function is linear if its rate of change is constant. Linear function in a table: f() The rate of change is constant. 6a Match each situation to the type of function that would be used to model it. Linear model Eponential model Neither 1) The amount of grain in a silo is 15 lbs and gains 10 lbs every 3 minutes. Linear functions in proportional situations: With phrases like: a decrease of $0 per month 60 miles an hour 3 bananas for every $5 ) soccer player kicks a ball off a cliff and it hits the ground 5 seconds later. 3) The amount of fish in a lake starts at 13 and triples every years. Eponential functions: a b You can tell a function is eponential when its values are changing at a constant multiplicative rate over constant intervals, or a constant percent rate. 6b Match each eample with the function that could be used to model its behavior Eponential function in a table: f() Here you see that f() is doubling (multiplying by ). This is a 100% increase. D Eponential functions in situations: With phrases like: bacteria triples every minute a 30% increase per month decays by half of its population each hour F-LE.1& 1) The number of people in line for a rollercoaster increases by 5 people every minute. ) car s value increases by 5% every year. 3) The number of cells in a petri dish multiplies five times every hour. 4) teacher s salary increases $1,000 for every year they teach. Page 6 of 13 M@WUSD 01/15/16

7 7 The two types of sequences studied in this course are arithmetic and geometric. rithmetic Sequences The common difference of an arithmetic sequence can be found using. Geometric Sequences The common ratio of a geometric sequence can be found using. 7a Determine whether each sequence below is arithmetic, geometric, or neither. If possible, identify the common difference or ratio. 1. 7, 10, 13, 16,. 1, 4, 9,16, The eplicit formula for an arithmetic sequence is. The eplicit formula for a geometric sequence is. 3. 4, 40, 400, For each sequence below, state whether it is arithmetic, geometric, or neither. If it is an arithmetic or geometric sequence, write its eplicit formula. Eample 1: 19, 15, 11, 7, This sequence is arithmetic because the terms have a common difference. 19, 15, 11, 7, OR To find the eplicit formula: 7b 4. 75, 87, 99, , 3, 3, 6, 1,... Match each sequence to its eplicit formula. Eample : 5, 10, 0, 40, This sequence is geometric because the terms have a common ratio. 5, 10, 0, 40, OR ( ) ( ) ( ) To find the eplicit formula: D 1) 3, 6, 9, 1, ) 3, 6, 1, 4, 3) 1, 3, 9, 7, 4) 1, 18, 15, 1, F-F. Page 7 of 13 M@WUSD 01/15/16

8 8 Graph the following piecewise-defined function., for 1 3 1, for 1 First, we will sketch the first piece of the graph, f() =. We only need the part of this graph when < 1, so we erase the other part and put an open circle at = 1. 8 Graph the following piecewise-defined function., for p ( ), for Then, we will sketch the second piece of the graph, f() = 3 1. We only need the part of this graph when 1, so we erase the other part and put a closed circle at = 1. a) What is p( )? Why? a) What is f( 1)? Why? This is the boundary point so you have to pay close attention to which piece of the function includes = 1. I can see from the graph that f() has a closed circle at the point ( 1, 4), therefore, f( 1) = 4. I could also evaluate the function using the second piece at 1 to show that it would be 4. b) Is f() increasing, decreasing, or neither between = 1 and = 0? Eplain your answer. We can look at the piece of the graph from = 1 to = 0 and see that the graph is increasing. F.IF.7a b) Is p() increasing, decreasing, or neither between = and = 0? Eplain your answer. End of Study Guide Page 8 of 13 M@WUSD 01/15/16

9 1a You Try Solutions: Find the -intercepts of 5 1 1b Which of the following could be part of the process of finding the roots of y 4 60? The binomial can also be thought of as the trinomial + +, where = 0 and = 0. Select all that apply. Method 1: Factoring Using Guess & heck To find the -intercepts, we solve for when f() = 0. Write original epression The last terms have to be and 1. ) ) The first terms are either 5 and or 5 and 5. Use intuition to try one of these pairs and check if the outer and inner terms add to. It checks: ) D) E) 4 ( 4) 4(1)( 60) (1) 4 ( 4) 4(1)( 60) (1) 4 ( 4) 60 ( 4) F) 4 ( ) 60 ( ) Method : Using square roots To find the -intercepts, we solve for when f() = 0. Page 9 of 13 M@WUSD 01/15/16

10 nswer the following questions about the graph of the function g() shown below: 3 Match the function with its graph. 1. f () = D 3. 4 E a) What are the y-intercept(s)? The y-intercept is where the graph crosses the y-ais, in this case at the point (0, 4) or y = 4. b) What are the -intercept(s)? The -intercept is where the graph crosses the -ais, in this case at the points (, 0) and (, 0) or = or. D c) How has this graph transformed from the mother function, f() =? g() is shifted 4 units down from f() = (which means the constant term is 4). d) Is the average rate of change from = 0 to = negative, positive, zero, or undefined? If we look only at the portion of the graph between = 0 and =, we can see that the graph is increasing. This means that the average rate of change on this interval is positive. e) What is the approimate value of g( 3)? We are looking at the graph where = 3 and trying to determine what the y-value is at that point. We can approimate that g( 3) 5. f) Is g() increasing or decreasing on 0? If we look only at the portion of the graph between = and = 0, we can see that the graph is decreasing. E 4a 1) Given the function, 4 1 find the average rate of change from = 1 to = 5. To do this, I need to evaluate the function at = 1 and = 5. f (1) (1) 4(1) 1 f (5) (5) 4(5) 1 f (1) f (5) f (1) f (5) 6 Now, I can substitute those values into the formula. f ( b) f ( a) b a ( 6) () 5 1 The average rate of change of f() 8 from = 1 to = 5 is. 4 Page 10 of 13 M@WUSD 01/15/16

11 4a cont d ) Given the function, g() = 3 find the average rate of change from = 1 to = 3. To do this, I need to evaluate the function at = 1 and = 3. g(1) = 3 1 g(1) = 3 g(3) = 3 3 = g(3) = 7 5 Two functions, f() and g() are represented below. Function 1: 3 3 Function : g() Now, I can substitute those values into the formula. g( b) g( a) b a The average rate of change of g() from = 1 to = 3 is b 3 1 g( ) Select all of the following statements that are TRUE about the functions above. Given the functions above, select all of the following intervals on which the average rate of change of f() is greater than the average rate of change of g(). Select all that apply. Select all that apply. ) f() and g() have the same -intercept(s). ) f() and g() have the same y-intercept(s). ) 1 0 ) 0 3 ) 1 3 D) 3 ) f() and g() have the same average rate of change between = 3 and = 1. D) oth functions are decreasing everywhere. E) oth functions have a maimum at (0, ) Page 11 of 13 M@WUSD 01/15/16

12 6a Match each situation to the type of function that would be used to model it. Linear model Eponential model Neither 1) The amount of grain in a silo is 15 lbs and gains 10 lbs every 3 minutes. 7a Determine whether each sequence below is arithmetic, geometric, or neither. If possible, identify the common difference or ratio. 1. 7, 10, 13, 16, This sequence is arithmetic with a common difference of 3.. 1, 4, 9,16, This sequence is neither arithmetic nor geometric. Therefore, there is no common difference or ratio. ) soccer player kicks a ball off a cliff and it hits the ground 5 seconds later. 3) The amount of fish in a lake starts at 13 and triples every years. 3. 4, 40, 400, This sequence is geometric with a common ratio of , 87, 99, This sequence is arithmetic with a common difference of 1. 6b Match each eample with the function that could be used to model its behavior , 3, 3, 6, 1,... This sequence is geometric with a common ratio of. D 7b Match each sequence to its eplicit formula. 1) The number of people in line for a rollercoaster increases by 5 people every minute. D ) car s value increases by 5% every year. D 1) 3, 6, 9, 1, D 3) The number of cells in petri dish multiplies five times every hour. 4) teacher s salary increases $1,000 for every year they teach. ) 3, 6, 1, 4, 3) 1, 3, 9, 7, 4) 1, 18, 15, 1, Page 1 of 13 M@WUSD 01/15/16

13 8 Graph the following piecewise-defined function., for p ( ), for a) What is p( )? Why? This is the boundary point so you have to pay close attention to which piece of the function includes =. I can see from the graph that p() has a closed circle at the point (, 4), therefore, p( ) = 4. I could also evaluate the function at to show that it would be 4. b) Is p() increasing, decreasing, or neither between = and = 0? Eplain your We can answer. look at the piece of the graph from = to = 0 and see that the graph is decreasing. Page 13 of 13 M@WUSD 01/15/16