Solve the following system of equations. " 2x + 4y = 8 # $ x 3y = 1. 1 cont d. You try:

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1 1 Solve the following system of equations. " 2x + 4y = 8 # $ x 3y = 1 Method 1: Substitution 1. Solve for x in the second equation. 1 cont d Method 3: Eliminate y 1. Multiply first equation by 3 and second equation by 4 to create inverse y-terms. (3) (4) 2. Substitute ( 1 + 3y) for x in the first equation and solve for y. 2. dd both equations together and solve for x Substitute 2 for x in any equation (here, we chose the first equation) and solve for y. 3. Substitute 1 for y in any equation (here, we chose the second equation) and solve for x. The solution for this system is (2, 1)..REI.6 The solution for this system is (2, 1). 1 Method 2: Eliminate x 1. Multiply second equation by 2 to create inverse x-terms. State whether or not each of these statements could be the first step to solve the system above. ( 2) ) dd the equations together. 2. dd both equations together and solve for y. + B) Multiply both sides of one equation by 2. C) Multiply both sides of one equation by 3 and both sides of the other equation by Substitute 1 for y in any equation (here, we chose the first equation) and solve for x. ) Subtract 8x from both sides of one equation. E) Subtract 2y from both sides of one equation. F) Multiply both sides of one equation by 4. G) ivide both sides of one equation by 2. The solution for this system is (2, 1)..REI.6 Page 1 of 9 MCC@WCCUS 02/07/14

2 2 Graph the solution to this system of inequalities. Inequality 1 2 Inequality 2 1. Graph Inequality 1! B C (0, 0) E So, shade the half-plane with the point (0, 0). 2. Graph Inequality 2 (Change to slope-intercept form first)! Choose or to indicate which of the points on this coordinate plane are solutions to the system below. ) Point ( 1, 4) B) Point B So, shade the half-plane with the point ( 1, 4). C) Point C 3. The graph of the system is the intersection of half-planes. Check a point in that intersection.! ) Point (2, 1) E) Point E.REI.12 Page 2 of 9 MCC@WCCUS 02/07/14

3 3 The function f (x) = 2 x is graphed below. Using this graph of the parent function f(x), sketch a graph of each of the following functions. ) g(x) = 2 x cont d ) g(x) = 2 x + 3 g(x) is of the form f(x + 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. To check this, I could also create a table. x g(x) F.BF.3 g(x) is of the form f(x) + 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. To check this, I could also create a table. x g(x) B) g(x) = 2 x g(x) is of the form f( x). The domain of x is being transformed to x. This means the parent function will reflect over the y-axis. To check this, I could also create a table. x g(x) C Match the function with its graph B C) g(x) = 2 x g(x) is of the form f(x). The range of the function is being transformed to its negative. This means the parent function will reflect over the x-axis. To check this, I could also create a table. x g(x) E F.BF.3 Page 3 of 9 MCC@WCCUS 02/07/14

4 4 Solve 9 x = 27 Our goal is to get a common base. If we have a common base, we can use the property of equality in exponents to solve this equation. 4 Which of the following equations are equivalent to 16 = 8 x? ) B) C) ) E).SSE.3c F) G) H) Page 4 of 9 MCC@WCCUS 02/07/14

5 5 nswer the following questions about the graph of the function g(x) shown below: 5 nswer the following questions about the graph of the function h(x) below: a) What are the y-intercept(s)? The y-intercept is where the graph crosses the y-axis, in this case at the point (0, 1) or y = 1. b) What are the x-intercept(s)? The x-intercept is where the graph crosses the x-axis, in this case at the points ( 7, 0) and ( 1, 0) or x = 7, 1. c) Is the average rate of change between x = 4 and x = 0 negative, positive, zero, or undefined? If we look only at the portion of the graph between x = 4 and x = 0, we can see that the graph is increasing. This means that the average rate of change on this interval is positive. d) What is g( 4)? The y-value of the graph when x = 4 is 3. Therefore, g( 4) = 3. e) How has this graph transformed from its parent function? The parent function of g(x) is the function f(x) = x. g(x) has shifted 4 units to the left and 3 units down from this function. f) Write an equation for g(x). g(x) is shifted 3 units down from the parent function f(x) = x. This is a change to the range so I know I must subtract 3 from the parent function. lso, the function is shifted 4 units to the left. This is a change to the domain so I know I must add 4 to the domain of the function: g(x) = x F.IF.4 a) What are the y-intercept(s)? b) What are the x-intercept(s)? c) Is the average rate of change between x = 2 and x = 4 negative, positive, zero, or undefined? d) What is h( 2)? e) How has this graph transformed from its parent function? f) Write an equation for h(x). Page 5 of 9 MCC@WCCUS 02/07/14

6 6 Graph the following piecewise-defined function. # f (x) = x2, for x < 1 $ % 3x 1, for x 1 First, we will sketch the first piece of the graph, f(x) = x 2. We only need the part of this graph when x < 1, so we erase the other part and put an open circle at x = 1. 6 Graph the following piecewisedefined function. # x + 2, for x < 2 p(x) = $ % x 2, for x 2 Then, we will sketch the second piece of the graph, f(x) = 3x 1. We only need the part of this graph when x 1, so we erase the other part and put a closed circle at x = 1. a) What is p( 2)? 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 x = 1. I can see from the graph that f(x) has a closed circle at the point ( 1, 4), therefore, f( 1) = 4. I could also evaluate the function at 1 to show that it would be 4. b) Is f(x) increasing, decreasing, or neither between x = 1 and x = 0? Explain your answer. We can look at the piece of the graph from x = 1 to x = 0 and see that the graph is increasing. F.IF.7a b) Is p(x) increasing, decreasing, or neither between x = 2 and x = 0? Explain your answer. End of Study Guide Page 6 of 9 MCC@WCCUS 02/07/14

7 You Try Solutions: 2 1 B State whether or not each of these statements could be the first step to solve the system above. C ) dd the equations together. B) Multiply both sides of one equation by 2. E C) Multiply both sides of one equation by 3 and both sides of the other equation by 2. ) Subtract 8x from both sides of one equation. E) Subtract 2y from both sides of one equation. Choose or to indicate which of the points on this coordinate plane are solutions to the system below. F) Multiply both sides of one equation by 4. G) ivide both sides of one equation by 2. ) Point B) Point B C) Point C ) Point E) Point E Page 7 of 9 MCC@WCCUS 02/07/14

8 3 Match the function with its graph Which of the following equations are equivalent to 16 = 8 x? B C E ) B) C) B ) E) F) C G) H) E Page 8 of 9 MCC@WCCUS 02/07/14

9 5 nswer the following questions about the graph of the function h(x) below: 6 Graph the following piecewisedefined function. # x + 2, for x < 2 p(x) = $ % x 2, for x 2 a) What are the y-intercept(s)? The y-intercept is where the graph crosses the y-axis, in this case at the point (0, 2) or y = 2. b) What are the x-intercept(s)? The x-intercept is where the graph crosses the x-axis, in this case at the point (2, 0) or x = 2. c) Is the average rate of change between x = 2 and x = 4 negative, positive, zero, or undefined? If we look only at the portion of the graph between x = 2 and x = 4, we can see that the graph is decreasing. This means that the average rate of change on this interval is negative. d) What is h( 2)? The y-value of the graph when x = 2 is 4. Therefore, h( 2) = 4. e) How has this graph transformed from its parent function? The parent function of h(x) is the function f(x) = x. h(x) has shifted 2 units to the right and has reflected over the x-axis. f) Write an equation for h(x). h(x) has reflected over the x-axis from its parent function, so it must have a negative coefficient outside the absolute value. The function has also shifted 2 units to the right. This is a change to the domain so I know I must subtract 2 from the domain of the function. h(x) = x 2 a) What is p( 2)? Why? This is the boundary point so you have to pay close attention to which piece of the function includes x = 2. I can see from the graph that p(x) has a closed circle at the point ( 2, 4), therefore, p( 2) = 4. I could also evaluate the function at 2 to show that it would be 4. b) Is p(x) increasing, decreasing, or neither between x = 2 and x = 0? Explain your answer. We can look at the piece of the graph from x = 2 to x = 0 and see that the graph is decreasing. Page 9 of 9 MCC@WCCUS 02/07/14