Nonlinear Systems. No solution One solution Two solutions. Solve the system by graphing. Check your answer.


 Ursula Hicks
 8 months ago
 Views:
Transcription
1 810 Nonlinear Sstems CC.91.A.REI.7 Solve a simple sstem consisting of a linear equation and a quadratic equation in two variables algebraicall and graphicall. Objective Solve sstems of equations in two variables in which one equation is linear and the other is quadratic. Vocabular nonlinear sstem of equations Wh learn this? You can solve a nonlinear sstem to find how long it takes for two objects to reach the same height. (See Eample.) Recall that a sstem of linear equations is a set of two or more linear equations. A solution of a sstem is an ordered pair that satisfies each equation in the sstem. Points where the graphs of the equations intersect represent solutions of the sstem. A nonlinear sstem of equations is a sstem in which at least one of the equations is nonlinear. For eample, a sstem that contains one quadratic equation and one linear equation is a nonlinear sstem. A sstem made up of a linear equation and a quadratic equation can have no solution, one solution, or two solutions, as shown below. No solution One solution Two solutions EXAMPLE 1 Solving a Nonlinear Sstem b Graphing A quadratic function has the form = a + b + c. To graph a quadratic function, start b using =  b a to find the ais of smmetr and the verte. Solve the sstem b graphing. Check our answer. = = Step 1 Graph = The ais of smmetr is = 1. The verte is (1, ). The intercept is 3. Another point is (1, 0). Graph the points and reflect them across the ais of smmetr. Step Graph = The slope is 1. The intercept is 1. (1, 0) Step 3 Find the points where the two graphs intersect. The solutions appear to be (1, 0) and (, 3). = = (, 3) 590 Chapter 8 Quadratic Functions and Equations
2 Check Substitute (1, 0) into the sstem. Substitute (, 3) into the sstem. = = = = (1)  (1) (1) () The solutions are (1, 0) and (, 3). 1. Solve the sstem b graphing. Check our answer. = = + 1 EXAMPLE Solving a Nonlinear Sstem b Substitution The substitution method is a good choice when either equation is solved for a variable, both equations are solved for the same variable, or a variable in either equation has a coefficient of 1 or 1. Solve the sstem b substitution. = = + = Both equations are solved for. = + = = ( + ) ( + ) = + = 1 + = 3 0 = = (  + 1) 0 = (  1)(  1) The solution is (1, 3). 0;  1 = 0 Check Use a graphing calculator. = 1 The graph supports the result found above. This sstem has eactl one real solution. The graph of the sstem consists of a line and a parabola that meet in eactl one point. Substitute + for in the first equation. Subtract + from both sides. Factor out the GCF,. Factor the trinomial. Use the Zero Product Propert; cannot equal 0. Solve the remaining equation. Write one of the original equations. Substitute 1 for.. Solve the sstem b substitution. Check our answer. = = Nonlinear Sstems 591
3 EXAMPLE 3 Solving a Nonlinear Sstem b Elimination The elimination method is a good choice when both equations have the same variable term with the same or opposite coefficients or when a variable term in one equation is a multiple of the corresponding variable term in the other equation. Solve each sstem b elimination. A B  = = = + = + 1 = = = (  1)(  3)  1 = 0 or  3 = 0 = 1 or = 3 = + 1 = + 1 = = = = 10 The solutions are (1, ) and (3, 10). = = 63 = 6 = () = 3( +  1)  3 = = 6 3 = = = = 1 ± (1)  (3)(3) (3) 1 ± = ± = There are no real solutions. Check Use a graphing calculator. To graph  3 = 6, first solve for.  3 = 63 = = _ 3  The graph supports that there are no real solutions. Write the sstem to align the terms. Add the equations to eliminate. Subtract from both sides. Factor the trinomial. Use the Zero Product Propert. Solve the equations. Write one of the original equations. Substitute each value and solve for. Write the sstem to align the terms. Multipl each term in the first equation b 3. Add the second equation to the new first equation to eliminate. Subtract from both sides. Use the Quadratic Formula, = b ± b  ac. a Note the discriminant: b  ac = 35. Its value is negative, so there are no real solutions. 59 Chapter 8 Quadratic Functions and Equations
4 Solve each sstem b elimination. Check our answers.  = 3a. = b. = = 5 EXAMPLE Phsics Application When t = 0, the ball and elevator are at the same height because the are both at ground level. An elevator is rising at a constant rate of 0 feet per second. Its height in feet after t seconds is given b h = 0t. At the instant the elevator is at ground level, a ball is thrown upward with an initial velocit of 80 feet per second from ground level. The height in feet of the ball after t seconds is given b h = 16t + 80t. Find the time it takes for the ball and the elevator to reach the same height. h = 16t Solve the sstem + 80t b substitution. h = 0t 16t + 80t = 0t  0t 0t 16t + 60t = 0 t(t  15) = 0 t = 0 or t  15 = 0 t = 0 t = 15 Solve the remaining equations. t = 3.75 Substitute 16t + 80t for h in the second equation. Subtract 0t from both sides. Factor out the GCF, t. Use the Zero Product Propert. It takes 3.75 seconds for the ball and the elevator to reach the same height.. An elevator is rising at a constant rate of 8 feet per second. Its height in feet after t seconds is given b h = 8t. At the instant the elevator is at ground level, a ball is dropped from a height of 10 feet. The height in feet of the ball after t seconds is given b h =16t Find the time it takes for the ball and the elevator to reach the same height. THINK AND DISCUSS 1. How is solving the sstems in this lesson similar to solving sstems of linear equations? How is it different?. When using elimination to solve a linear/quadratic sstem, which variable will be eliminated? Wh? 3. A sstem of linear equations can have infinitel man solutions. Wh can t a linear/quadratic sstem have infinitel man solutions?. GET ORGANIZED Cop and complete the graphic organizer b sketching diagrams to show eamples. Write not possible for an cases that are not possible. Sstem of Equations Linear Linear/Quadratic Number of Solutions 0 1 infinite 810 Nonlinear Sstems 593
5 810 Eercises GUIDED PRACTICE Vocabular Appl the vocabular from this lesson to answer each question. 1. A sstem of equations that includes a linear equation and a quadratic equation is?. (linear, nonlinear, or quadratic). Sketch a nonlinear sstem of equations that has two solutions. The sstem should include one quadratic equation and one linear equation. SEE EXAMPLE 1 Solve each sstem b graphing. Check our answers. = = = =  8 SEE EXAMPLE Solve each sstem b substitution. Check our answers. = =  3 SEE EXAMPLE 3 Solve each sstem b elimination. Check our answers. = = 6 = = = = 5 SEE EXAMPLE 9. Phsics A bird is fling upwards such that its height in feet after t seconds is given b h = t. At the instant the bird passes the height of a ball being held out of a window, the ball is thrown upward with an initial velocit of 80 feet per second. The height in feet of the ball after t seconds is given b h = 16t + 80t. Find the time it takes for the ball and the bird to reach the same height. Independent Practice For See Eercises Eample Etra Practice See Etra Practice for more Skills Practice and Applications Practice eercises. PRACTICE AND PROBLEM SOLVING Solve each sstem b graphing. Check our answers. = = = = 18 = = = + 5 = 61 Solve each sstem b substitution. Check our answers. = = = = + 9 = = = = Solve each sstem b elimination. Check our answers. = = = 13 = 15 = = 6 = = 3 59 Chapter 8 Quadratic Functions and Equations
6 . Demographics The growing population of town A can be modeled b the equation P(t) = 8t + 000, where t represents number of ears after 010. The growing population of town B can be modeled b the equation P(t) = 100t In which ear will the populations of the towns be approimatel equal? 3. Finance The value of Danielle s investments is modeled b the equation V(t) = 3t + 70t + 100, where t represents the number of months after she made her initial investment. Jeffre has no mone invested in stocks, but he deposits the same amount ever month into a savings account that he opened at the same time as Danielle began investing. His savings account balance can be modeled b the equation V(t) = 50t After how man months will the value of Danielle s investments be equal to the balance of Jeffre s savings account?. Amusement Parks A ride at an amusement park consists of an observation deck that travels directl up into the air at a constant rate of 0 feet per second. Its height in feet after t seconds is given b h = 0t. At the instant the deck is at ground level, a ball is thrown up with initial velocit 60 feet per second from ground level. The height in feet of the ball after t seconds is given b h = 16t + 60t. Find the time it takes for the ball and the deck to reach the same height. Round our answer to the nearest hundredth. 5. Business A compan s weekl revenue can be modeled b the equation C(p) = 0.75p + 10p + 00, where p represents the number of products sold. The weekl cost of running the business is modeled b the equation C(p) = 80p How man products must the compan sell in a week to break even (when revenue equal the costs of running the business)? Determine whether the point is a solution of the sstem of equations. = = 16 ; (, 1) 7. = ; (3, ) = 8 = = 3 ; (7, 6) 9. = ; (1, ) = + 1 = = ; (3, 8) 31. ; (1, ) = = 5 = = ; (5, 5) 33. ; (, 18)  = = 8 3. Write About It Eplain in our own words when ou should use the substitution method to solve a nonlinear sstem, and when ou should use the elimination method. 35. Critical Thinking Describe a scenario in which ou might use the graphing method to solve a sstem of nonlinear equations, even if ou didn t epect the solution(s) to consist of integer coordinates. Wolfgang Deuter/Corbis 36. Estimation Estimate the solution(s) to the sstem b graphing. = = Nonlinear Sstems 595
7 A 37. /ERROR ANALYSIS / Below are two solutions to the sstem of equations = Which is incorrect? Eplain the error. + 3 = 11 = = = = = = = (3 + )(  ) 3 + = 0 or  = 03 =  = = = = 11 ( ) = 11 () + 3 = = = = 3 3 = 3 5 = 9 = 1, ( ) The solutions are and (, 1). B = = = = = = =  b ± b  ac a =  (8) ± (8)  (1)(5) (1) = 8 ± = 8 ± 11 = ± or (7.3) (0.68) The solutions are approimatel (7.3, 6.09) and (0.68,.76). 38. Which are the solutions to the sstem of equations below? = = (3, ) and (1, 6) (1, 8) and (, 10) (3, ) and (1, 8) (1,6) and (, 10) 39. For which sstem of equations is (, 6) a solution? = +  = = 3  = = = = = 50. Which sstem below has no real solutions? 3 + = 9 = (  3) =   = 5 =  =   =  7 = Chapter 8 Quadratic Functions and Equations
8 = Which is the graph of = +  5? CHALLENGE AND EXTEND Find the coordinate(s) of the solution(s) of each sstem. = = = = 8 = = 3 = = 76. A sstem of two equations contains one quadratic equation and one linear equation. The quadratic equation in the sstem is = The solutions of the sstem are (3, 15) and (1, 13). What is the linear equation in the sstem? 7. Phsics The formula for the height of an object in free fall (neglecting air resistance) is h(t) = 16t + v 0 t + h 0, where v 0 is the object s initial velocit in feet per second and h 0 is the object s initial height above the ground in feet. One ball is thrown with an initial velocit of 90 ft/s from a height of 0 ft. A second ball is thrown at the eact same instant with an initial velocit of 80 ft/s and a height of 30 ft. After how man seconds will the balls reach the same height? 810 Nonlinear Sstems 597
9 CHAPTER SECTION 8B Make sense of problems and persevere in solving them. Solving Quadratic Equations Seeing Green A golf plaer hits a golf ball from a tee with an initial vertical velocit of 80 feet per second. The height of the golf ball t seconds after it is hit is given b h =  16t + 80t. 1. How long is the golf ball in the air?. What is the maimum height of the golf ball? 3. How long after the golf ball is hit does it reach its maimum height?. What is the height of the golf ball after 3.5 seconds? 5. At what times is the golf ball 6 feet in the air? Eplain. (tl),photodisc/gett Images; (tc), COMSTOCK, Inc.; (tr),stuart Franklin/Gett Images; (b),robert Laberge/Gett Images 598 Chapter 8 Quadratic Function and Equations
10 CHAPTER Quiz for Lessons 85 Through 810 SECTION 8B 85 Solving Quadratic Equations b Graphing Solve each equation b graphing the related function = = = 3. The height of a fireworks rocket launched from a platform 35 meters above the ground can be approimated b h = 5 t + 30t + 35, where h is the height in meters and t is the time in seconds. Find the time it takes the rocket to reach the ground after it is launched. 86 Solving Quadratic Equations b Factoring Use the Zero Product Propert to solve each equation. 5. ( + 1) ( + 3) = 0 6. (  6) (  3) = 0 7. ( + 6) (  3) = 0 8. ( + 7) (  10) = 0 Solve each quadratic equation b factoring = = = = Solving Quadratic Equations b Using Square Roots Solve using square roots = = = Completing the Square Solve b completing the square = = = The width of a rectangle is feet shorter than its length. The area of the rectangle is square feet. Find the length and width. Round our answer to the nearest tenth of a foot. 89 Using the Quadratic Formula and the Discriminant Solve using the Quadratic Formula. Round our answer to the nearest hundredth = = = 3 Find the number of real solutions of each equation using the discriminant = = = Nonlinear Sstems 6. Solve the sstem b graphing. Check our answer. { =  7 = Solve the sstem b substitution. Check our answer. { = = + 5 Read to Go On? 599
11 EXTENSION Cubic Functions and Equations CC.91.F.IF.7c Graph polnomial functions, identifing zeros when suitable factorizations are available, and showing end behavior. Also CC.91.A.REI.10, CC.91.A.REI.11*, CC.91.A.APR.3 Objectives Recognize and graph cubic functions. Solve cubic equations. Vocabular cubic function cubic equation A cubic function is a function that can be written in the form f () = a 3 + b + c + d, where a 0. The parent cubic function is f () = 3. To graph this function, choose several values of and find ordered pairs. f() From the graph of f() = 3, ou can see the general shape of a cubic function. that the domain and the range are all real numbers. that the intercept and the intercept are both 0. A B The graph of f () = illustrates another characteristic of the graphs of cubic functions. Points A and B are called turning points. In general, the graph of a cubic function will have two turning points. EXAMPLE 1 Graphing Cubic Functions Graph f () = Identif the intercepts and give the domain and range. f() = f() 1 ( 1) ( 1) (0) (0) (1) (1) + 1 () () + 6 Choose positive, negative, and zero values for, and find ordered pairs. 600 Chapter 8 Quadratic Functions and Equations
12 Plot the ordered pairs and connect them with a smooth curve. Notice that, in general, this graph falls from left to right. This is because the value of a is negative. The intercept is 1. The intercept is. The domain and range are all real numbers. Graph each cubic function. Identif the intercepts and give the domain and range. 1a. f () = ( 1) 3 1b. f () = Previousl, ou saw that ever quadratic function has a related quadratic equation. Cubic functions also have related cubic equations. A cubic equation is an equation that can be written in the form a 3 + b + c + d = 0, where a 0. One wa to solve a cubic equation is b graphing the related function and finding its zeros. EXAMPLE Solving Cubic Equations b Graphing Solve 3 = b graphing. Check our answer. Step 1 Rewrite the equation in the form a 3 + b + c + d = 0. 3 = 3 + = 0 Add to both sides of the equation. Step Write and graph the related function: f () = 3 + f() = f() 1 ( 1) 3  ( 1)  (1) (0) 3  (0) (1) 3  (1) () 3  () (3) 3  (3) Step 3 Find the zeros. The zeros appear to be 1, 1, and. Check these values in the original equation. 3 = 3 = 3 = ( 1) 3 ( 1) ( 1) 1 3 (1) 1 3 () ( 1) Etension 601
13 Solve each equation b graphing. Check our answer. a. 3 5 = 50 b = Cubic equations can also be solved algebraicall. Man of the methods used to solve quadratic equations can be applied to cubic equations as well. EXAMPLE 3 Solving Cubic Equations Algebraicall Solve each equation. Check our answer. A ( + 5) 3 = 7 3 ( + 5) 3 = = 3 = Check ( + 5) 3 = 7 ( + 5) Take the cube root of both sides. Subtract 5 from both sides. Substitute for in the original equation. B = = 0 ( ) = 0 ( + 1)( + ) = 0 = 0 or + 1 = 0 or + = 0 = 1 or = The solutions are 0, 1, and. Add to both sides. Factor out on the left side. Factor the quadratic trinomial. Zero Product Propert Solve each equation. The factored epression must equal zero to use the Zero Product Propert. Check = = = (0) (0) (1) (1) (1) () () () (1) 8 + 3() C = = 0 ( ) = 0 = 0 or = 0 = 1.5 ± (1.5)  (1)(3.15) (1) 1.5 ± 3.75 = =.5 or = 1.5 Add 1.5 to both sides. Factor out on the left side. Zero Product Propert Quadratic Formula Simplif. The solutions are.5, 0, and Chapter 8 Quadratic Functions and Equations
14 Check Use a graphing calculator. Graph the related function and look for the zeros. The solutions look reasonable. Solve each equation. Check our answer. 3a. ( + ) 3 = 6 3b = 0 3c = 10 EXTENSION Eercises Graph each cubic function. Identif the intercepts and give the domain and range. 1. f() = g() = 3 + Solve each equation b graphing. Check our answer = = 8 Solve each equation. Check our answer. 5. ( 9) 3 = = = 8. The SendIt Store uses shipping labels that are in. tall and in. wide. Si labels fit on the front of the store s standard shipping bo with an area of 3 in left over. Three labels fit on the side of the bo. The volume of the bo is 108 in 3. What is the area of one label? 9. a. Graph the functions f() = 3, f() = 3 + 1, and f () = 3 + on the same coordinate plane. Describe an patterns ou observe. Predict the shape of the graph of f() = 3 + c. b. Graph the functions g() = 3, g() = ( 1) 3, and g() = ( ) 3 on the same coordinate plane. Describe an patterns ou observe. Predict the shape of the graph of g() = ( c) 3. 1 in. Use a graphing calculator to find the approimate solution(s) of each cubic equation. Round to the nearest hundredth = = = _ _ 1 = 9 1. Critical Thinking How man zeros can a cubic function have? What does this tell ou about the number of real solutions possible for a cubic equation? Etension 603
15 CHAPTER Vocabular ais of smmetr completing the square discriminant maimum minimum nonlinear sstem of equations parabola quadratic equation quadratic function verte zero of a function Complete the sentences below with vocabular words from the list above. 1. The? is the highest or lowest point on a parabola.. A? can also be called an intercept of the function. 81 Identifing Quadratic Functions EXAMPLE Use a table of values to graph = Step 1 Make a table of values. Choose values of and use them to find values of Step Plot the points and connect them. EXERCISES Tell whether each function is quadratic. Eplain. 3. = = =  _ 1 6. = Use a table of values to graph each quadratic function. 7. = 6 8. =  9. = 1 _ 10. = 3 Tell whether the graph of each function opens upward or downward. Eplain. 11. = = Characteristics of Quadratic Functions EXAMPLE EXERCISES Find the zeros of = from its graph.  0 Use the graph to find the zeros. The zeros are 1 and Find the zeros of each quadratic function from its graph. Check our answer. 13. = = Chapter 8 Quadratic Functions and Equations
16 83 Graphing Quadratic Functions EXAMPLE Graph = Step 1 Find the ais of smmetr. = _ b a = _ (8) () = 8 _ = The ais of smmetr is =. Step 3 Find the intercept. c = 10 Step Find the verte. = = ()  8 ()  10 = 18 The verte is (, 18). Step Find one more point on the graph. = (1)  8 (1) 10 = 0 Let = 1. Use (1, 0). Step 5 Graph the ais of smmetr and the points. Reflect the points and connect with a smooth curve. EXERCISES Graph each quadratic function. 15. = = = = = = Water that is spraed upward from a sprinkler with an initial velocit of 0 m/s can be approimated b the function = , where is the height of a drop of water seconds after it is released. Graph this function. Find the time it takes a drop of water to reach its maimum height, the water s maimum height, and the time it takes the water to reach the ground. 8 Transforming Quadratic Functions EXAMPLE Compare the graph of g () = 3  with the graph of f () =. Use the functions. Both graphs open upward because a > 0. The ais of smmetr is the same, = 0, because b = 0 in both functions. The graph of g ( ) is narrower than the graph of f () because 3 > 1. The verte of f () is (0, 0). The verte of g () is translated units down to (0, ). f () has one zero at the origin. g ( ) has two zeros because the verte is below the origin and the parabola opens upward. EXERCISES Compare the widths of the graphs of the given quadratic functions. Order functions with different widths from narrowest graph to widest.. f () =, g () = 3. f () = 6, g () = 6. f () =, g () = 1 _ 3, h () = 3 Compare the graph of each function with the graph of f () =. 5. g () = g () = g () = + 3 Stud Guide: Review 605
17 85 Solving Quadratic Equations b Graphing EXAMPLE Solve  =  8 b graphing the related function. Step 1 Write the equation in standard form. 0 = Step Graph the related function. = Step 3 Find the zeros. The onl zero is 1. The solution is = 1. EXERCISES Solve each equation b graphing the related function = = = = = = = Solving Quadratic Equations b Factoring EXAMPLE Solve 36 = b factoring. 36 = Write the equation in = 0 standard form. 3 (   8) = 0 Factor out 3. 3 ( + ) (  ) = 0 Factor the trinomial. Zero Product 3 0, + = 0 or  = 0 Propert =  or = Solve each equation. EXERCISES Solve each quadratic equation b factoring = = = = = = A rectangle is feet longer than it is wide. The area of the rectangle is 8 square feet. Write and solve an equation that can be used to find the width of the rectangle. 87 Solving Quadratic Equations b Using Square Roots EXAMPLE Solve = 98 using square roots. = 98 Divide both sides of the equation b to isolate. = 9 = ± 9 Take the square root of both sides. = ±7 Use ± to show both roots. The solutions are 7 and 7. EXERCISES Solve using square roots.. 5 = = 0. = = 7 6. = = 5 8. A rectangle is twice as long as it is wide. The area of the rectangle is 3 square feet. Find the rectangle s width. 606 Chapter 8 Quadratic Functions and Equations
18 88 Completing the Square EXAMPLE Solve  6 = 5 b completing the square. _ ( 6 ) = 9 Find ( b ) = = Complete the square. (  3) = Factor the trinomial. 3 =± Take the square root of both sides.  3 = ± Use the ± smbol.  3 = or  3 = Solve each = 5 or = 1 equation. The solutions are 5 and 1. EXERCISES Solve b completing the square = = = = A homeowner is planning an addition to her house. She wants the new famil room to be a rectangle with an area of 19 square feet. The contractor sas that the length needs to be more feet than the width. What will the dimensions of the new room be? 89 The Quadratic Formula and the Discriminant EXAMPLE Solve + + = 0 using the Quadratic Formula. = b ± b  ac a  ±  (1)() = (1) =  ± = _  ± 0 = _  =  The solution is = . Write the Quadratic Formula. Substitute for a, b, and c. Simplif. EXERCISES Solve using the Quadratic Formula = = = = 7 Find the number of real solutions of each equation using the discriminant = = = = Nonlinear Sstems EXAMPLE Solve { = b substitution. = + + =. 0 =   Substitute + for in the first equation. 0 = (  )( + 1) Factor the trinomial.  = 0 or + 1 = 0 Solve the equations. = or = 1 Write one of the = + = + original equations. = + =1 + Substitute each value = = 1 and solve for. The solutions are (, ) and (1, 1). EXERCISES Solve each sstem. 6. { =  = { = = { = =  6 Stud Guide: Review 607
19 CHAPTER Tell whether each function is quadratic. Eplain. 1. { (10, 50), (11, 71), (1, 9), (13, 119), (1, 16) }. 3 + = Tell whether the graph of = opens upward or downward and whether the parabola has a maimum or a minimum.. Estimate the zeros of the quadratic function. 5. Find the ais of smmetr of the parabola. 6. Find the verte of the graph of = Graph the quadratic function =  +. Compare the graph of each function with the graph of f () =. 8. g () = h () = 1 _ g () = A hammer is dropped from a 0foot scaffold. Another one is dropped from a 60foot scaffold. a. Write the two height functions and compare their graphs. Use h (t) = 16 t + c, where c is the height of the scaffold. b. Use the graphs to estimate when each hammer will reach the ground. 1. A rocket is launched with an initial vertical velocit of 110 m/s. The height of the rocket in meters is approimated b the quadratic equation h = 5 t + 110t where t is the time after launch in seconds. About how long does it take for the rocket to return to the ground? Solve each quadratic equation b factoring = = = 0 Solve b using square roots = = = 0 Solve b completing the square = = = 0 Solve each quadratic equation. Round to the nearest hundredth if necessar = = = 0 Find the number of real solutions of each equation using the discriminant = = _ + 8 = 0 8. Solve the sstem. { = = Chapter 8 Quadratic Functions and Equations
20 CHAPTER FOCUS ON SAT SUBJECT TESTS In addition to the SAT, some colleges require the SAT Subject Tests for admission. Colleges that don t require the SAT Subject Tests ma still use the scores to learn about our academic background and to place ou in the appropriate college math class. You ma want to time ourself as ou take this practice test. It should take ou about 6 minutes to complete. Take the SAT Subject Test in mathematics while the material is still fresh in our mind. You are not epected to be familiar with all of the test content, but ou should have completed at least three ears of collegeprep math. 1. The graph below corresponds to which of the following quadratic functions? (A) f () = (B) f () = (C) f () = (D) f () = (E) f () = What is the sum of the solutions to the equation 96 = 8? (A) _ 3 (B) _ 3 (C) 1 _ 3 (D)  _ 3 (E)  8 _ 3 3. If h () = a + b + c, where b  ac < 0 and a < 0, which of the following statements must be true? I. The graph of h () has no points in the first or second quadrants. II. The graph of h () has no points in the third or fourth quadrants. III. The graph of h () has points in all quadrants. (A) I onl (B) II onl (C) III onl (D) I and II onl (E) None of the statements are true.. What is the ais of smmetr for the graph of a quadratic function whose zeros are  and? (A) =  (B) = 0 (C) = 1 (D) = (E) = 6 5. How man realnumber solutions does 0 = have? (A) None (B) One (C) Two (D) All real numbers (E) It is impossible to determine. College Entrance Eam Practice 609
21 CHAPTER Etended Response: Eplain Your Reasoning Etended response test items often include multipart questions that evaluate our understanding of a math concept. To receive full credit, ou must answer the problem correctl, show all of our work, and eplain our reasoning. Use complete sentences and show our problemsolving method clearl. Etended Response Given 1 + =  3 and =  1, identif which is a quadratic function. Provide an eplanation for our decision. For the quadratic function, tell whether the graph of the function opens upward or downward and whether the parabola has a maimum or a minimum. Eplain our reasoning. Read the solutions provided b two different students. Student A Ecellent eplanation The response includes the correct answers along with a detailed eplanation for each part of the problem. The eplanation is written using complete sentences and is presented in an order that is eas to follow and to understand. It is obvious that this student knows how to determine and interpret a quadratic function. Student B Poor eplanation The response includes the correct answers, but the eplanation does not include details. The reason for defining the function as quadratic does not show knowledge of the concept. The student shows a lack of understanding of how to write and interpret a quadratic function in standard form. 610 Chapter 8 Quadratic Functions and Equations
22 Include as man details as possible to support our reasoning. This increases the chance of getting full credit for our response. Read each test item and answer the questions that follow. Item A The height in feet of a tennis ball seconds after it is ejected from a serving machine is given b the ordered pairs { (0, 10), (0.5, 9), (1, 7), (1.5, ), (, 0) }. Determine whether the function is quadratic. Find its domain and range. Eplain our answers. 1. What should a student include in the eplanation to receive full credit?. Read the two eplanations below. Which eplanation is better? Wh? Item C A science teacher set off a bottle rocket as part of a lab eperiment. The function h = 16 t + 96t represents the height in feet of a rocket that is shot out of a bottle with an initial vertical velocit of 96 feet per second. Find the time that the rocket is in the air. Eplain how ou found our answer.. Read the two responses below. a. Which student provided the better eplanation? Wh? b. What advice would ou give the other student to improve his or her eplanation? Student C Student A Student B Student D Item B The height of a golf ball can be approimated b the function = , where is the height in meters above the ground and is the time in seconds after the ball is hit. What is the maimum height of the ball? How long does it take for the ball to reach its maimum height? Eplain. 3. A student correctl found the following answers. Use this information to write a clear and concise eplanation. Item D The base of a parallelogram is 1 centimeters more than its height. The area of the parallelogram is 13 square centimeters. Eplain how to determine the height and base of the figure. What is the height? What is the base? 5. Read the following response. Identif an areas that need improvement. Rewrite the response so that it will receive full credit. Test Tackler 611
23 CHAPTER State Test Practice CUMULATIVE ASSESSMENT Multiple Choice 1. Which epression is NOT equal to the other three? (1) 0. Which function s graph is a translation of the graph of f () = 3 + seven units down? f () =  + f () = 10 + f () = 33 f () = The area of a circle in square units is π ( ). Which epression represents the circumference of the circle in units? π (3 + 7) π (3 + 7) π (3 + 7) CberCafe charges a computer station rental fee of $5, plus $0.0 for each quarterhour spent surfing. Which epression represents the total amount Carl will pa to use a computer station for three and a half hours? (3.5) (3.5) () _ _ 1 _ What is the numerical solution to the equation five less than three times a number equals four more than eight times the number?  9 _ 51 _ 5 1_ 11 1_ 5 6. Which is a possible situation for the graph? A car travels at a stead speed, slows down in a school zone, and then resumes its previous speed. A child climbs the ladder of a slide and then slides down. A person flies in an airplane for a while, parachutes out, and gets stuck in a tree. The number of visitors increases in the summer, declines in the fall, and levels off in the winter. 7. Which of the following is the graph of f () =  +? The value of varies directl with, and = 0 when = 5. Find when = What is the slope of the line that passes through the points (, 7) and (5, 3)? _ 1 1_ Chapter 8 Quadratic Functions and Equations
24 10. Putting Green Mini Golf charges a $ golf club rental fee plus $1.5 per game. Good Times Golf charges a $1.5 golf club rental fee plus $3.75 per game. Which sstem of equations could be solved to determine for how man games the cost is the same at both places? = = = = = = = = The graph of which quadratic function has an ais of smmetr of = ? = = = = Which polnomial is the product of  and  + 1? Gridded Response The problems on man standardized tests are ordered from least to most difficult, but all items are usuall worth the same amount of points. If ou are stuck on a question near the end of the test, our time ma be better spent rechecking our answers to earlier questions. 13. The length of a rectangle is units greater than the width. The area of the rectangle is square units. What is its width in units? 1. Find the value of the discriminant of the equation 0 = Short Response 16. The data in the table shows ordered pair solutions to a linear function. Find the missing value. Show our work. 17. Answer the following questions using the function f () = a. Make a table of values and give five points on the graph. b. Find the ais of smmetr and verte. Show all calculations. 18. a. Show how to solve = 0 b graphing the related function. Show all our work. b. Show another wa to solve the equation in part a. Show all our work. 19. What can ou sa about the value of a if the graph of = a  8 has no intercepts? Eplain. Etended Response 0. The graph shows the quadratic function f () = a + b + c. a. What are the solutions of the equation 0 = a + b + c? Eplain how ou know b. If the point (5, 1) lies on the graph of f (), the point (a, 1) also lies on the graph. Find the value of a. c. What do ou know about the relationship between the values of a and b? Use the coordinates of the verte in our eplanation. d. Use what ou know about solving quadratic equations b factoring to make a conjecture about the values of a, b, and c in the function f () = a + b + c What is the positive solution of = 10 +? Round our answer to the nearest hundredth if necessar. Standardized Test Prep 613
3.1 Graph Quadratic Functions
3. Graph Quadratic Functions in Standard Form Georgia Performance Standard(s) MMA3b, MMA3c Goal p Use intervals of increase and decrease to understand average rates of change of quadratic functions. Your
More informationSystems of Linear and Quadratic Equations. Check Skills You ll Need. y x. Solve by Graphing. Solve the following system by graphing.
NY Learning Standards for Mathematics A.A. Solve a sstem of one linear and one quadratic equation in two variables, where onl factoring is required. A.G.9 Solve sstems of linear and quadratic equations
More informationQuadratic Function. Parabola. Parent quadratic function. Vertex. Axis of Symmetry
Name: Chapter 10: Quadratic Equations and Functions Section 10.1: Graph = a + c Quadratic Function Parabola Parent quadratic function Verte Ais of Smmetr Parent Function =  1 0 1 1 Eample 1: Make a table,
More information9 (0, 3) and solve equations to earn full credit.
Math 0 Intermediate Algebra II Final Eam Review Page of Instructions: (6, ) Use our own paper for the review questions. For the final eam, show all work on the eam. (6, ) This is an algebra class do not
More information1Write and graph. 2Solve problems. Now. Then. Why? New Vocabulary
Direct Variation Then You found rates of change of linear functions. (Lesson ) Now Write and graph direct variation equations. Solve problems involving direct variation. Wh? Bianca is saving her mone
More information= x. Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions. Math Background
Algebra II Notes Quadratic Functions Unit 3.1 3. Graphing Quadratic Functions Math Background Previousl, ou Identified and graphed linear functions Applied transformations to parent functions Graphed quadratic
More informationObjectives To solve quadratic equations using the quadratic formula To find the number of solutions of a quadratic equation
96 The Quadratic Formula and the Discriminant Content Standards A.REI..a Use the method of completing the square to transform an quadratic equation in into an equation of the form ( p) 5 q... Derive the
More information2 variables. is the same value as the solution of. 1 variable. You can use similar reasoning to solve quadratic equations. Work with a partner.
9. b Graphing Essential Question How can ou use a graph to solve a quadratic equation in one variable? Based on what ou learned about the intercepts of a graph in Section., it follows that the intercept
More informationUsing Intercept Form
8.5 Using Intercept Form Essential Question What are some of the characteristics of the graph of f () = a( p)( q)? Using Zeros to Write Functions Work with a partner. Each graph represents a function of
More informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polnomial Degree and Finite Differences 1. Identif the degree of each polnomial. a. 1 b. 0. 1. 3. 3 c. 0 16 0. Determine which of the epressions are polnomials. For each polnomial, state its
More informationLesson 9.1 Using the Distance Formula
Lesson. Using the Distance Formula. Find the eact distance between each pair of points. a. (0, 0) and (, ) b. (0, 0) and (7, ) c. (, 8) and (, ) d. (, ) and (, 7) e. (, 7) and (8, ) f. (8, ) and (, 0)
More informationLESSON #11  FORMS OF A LINE COMMON CORE ALGEBRA II
LESSON #  FORMS OF A LINE COMMON CORE ALGEBRA II Linear functions come in a variet of forms. The two shown below have been introduced in Common Core Algebra I and Common Core Geometr. TWO COMMON FORMS
More informationLinear and Nonlinear Systems of Equations. The Method of Substitution. Equation 1 Equation 2. Check (2, 1) in Equation 1 and Equation 2: 2x y 5?
3330_070.qd 96 /5/05 Chapter 7 7. 9:39 AM Page 96 Sstems of Equations and Inequalities Linear and Nonlinear Sstems of Equations What ou should learn Use the method of substitution to solve sstems of linear
More informationOne of the most common applications of Calculus involves determining maximum or minimum values.
8 LESSON 5 MAX/MIN APPLICATIONS (OPTIMIZATION) One of the most common applications of Calculus involves determining maimum or minimum values. Procedure:. Choose variables and/or draw a labeled figure..
More informationChapters 8 & 9 Review for Final
Math 203  Intermediate Algebra Professor Valdez Chapters 8 & 9 Review for Final SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the formula for
More information3.1. Shape and Structure Forms of Quadratic Functions ESSENTIAL IDEAS TEXAS ESSENTIAL KNOWLEDGE AND SKILLS FOR MATHEMATICS 169A
Shape and Structure Forms of Quadratic Functions.1 LEARNING GOALS KEY TERMS In this lesson, ou will: Match a quadratic function with its corresponding graph. Identif ke characteristics of quadratic functions
More informationMAT 1033C  MartinGay Intermediate Algebra Chapter 8 (8.1, 8.2, 8.5, 8.6) Practice for the Exam
MAT 33C  MartinGa Intermediate Algebra Chapter 8 (8.1 8. 8. 8.6) Practice for the Eam Name Date Da/Time: SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
More informationAlgebra II Notes Unit Six: Polynomials Syllabus Objectives: 6.2 The student will simplify polynomial expressions.
Algebra II Notes Unit Si: Polnomials Sllabus Objectives: 6. The student will simplif polnomial epressions. Review: Properties of Eponents (Allow students to come up with these on their own.) Let a and
More informationChapter 6 Resource Masters
Chapter 6 Resource Masters Consumable Workbooks Man of the worksheets contained in the Chapter Resource Masters booklets are available as consumable workbooks. Stud Guide and Intervention Workbook 0078809X
More information2.3 Quadratic Functions
88 Linear and Quadratic Functions. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions:
More informationInstructor: Imelda Valencia Course: A3 Honors Pre Calculus
Student: Date: Instructor: Imelda Valencia Course: A3 Honors Pre Calculus 01 017 Assignment: Summer Homework for those who will be taking FOCA 017 01 onl available until Sept. 15 1. Write the epression
More informationAlgebra Notes Quadratic Functions and Equations Unit 08
Note: This Unit contains concepts that are separated for teacher use, but which must be integrated by the completion of the unit so students can make sense of choosing appropriate methods for solving quadratic
More informationLESSON #12  FORMS OF A LINE COMMON CORE ALGEBRA II
LESSON #  FORMS OF A LINE COMMON CORE ALGEBRA II Linear functions come in a variet of forms. The two shown below have been introduced in Common Core Algebra I and Common Core Geometr. TWO COMMON FORMS
More informationMath 0240 Final Exam Review Questions 11 ( 9) 6(10 4)
Math 040 Final Eam Review Questions 11 ( 9) 6(10 4) 1. Simplif: 4 8 3 + 8 ( 7). Simplif: 34 3. Simplif ( 5 7) 3( ) 8 6 4. Simplif: (4 3 ) 9 5. Simplif: 6. Evaluate 4 7 3 3 4 5 when and 3 Write each of
More informationSolving Quadratic Equations by Graphing 9.1. ACTIVITY: Solving a Quadratic Equation by Graphing. How can you use a graph to solve a quadratic
9. Solving Quadratic Equations b Graphing equation in one variable? How can ou use a graph to solve a quadratic Earlier in the book, ou learned that the intercept of the graph of = a + b variables is
More informationHonors Algebra 2 ~ Spring 2014 Name 1 Unit 3: Quadratic Functions and Equations
Honors Algebra ~ Spring Name Unit : Quadratic Functions and Equations NC Objectives Covered:. Define and compute with comple numbers. Operate with algebraic epressions (polnomial, rational, comple fractions)
More informationM122 College Algebra Review for Final Exam
M1 College Algebra Review for Final Eam Revised Fall 017 for College Algebra  Beecher All answers should include our work (this could be a written eplanation of the result, a graph with the relevant feature
More informationProperties of the Graph of a Quadratic Function. has a vertex with an xcoordinate of 2 b } 2a
0.2 Graph 5 a 2 b c Before You graphed simple quadratic functions. Now You will graph general quadratic functions. Wh? So ou can investigate a cable s height, as in Eample 4. Ke Vocabular minimum value
More informationSkills Practice Skills Practice for Lesson 1.1
Skills Practice Skills Practice for Lesson. Name Date Lots and Projectiles Introduction to Quadratic Functions Vocabular Give an eample of each term.. quadratic function 9 0. vertical motion equation s
More informationMATH 115: Final Exam Review. Can you find the distance between two points and the midpoint of a line segment? (1.1)
MATH : Final Eam Review Can ou find the distance between two points and the midpoint of a line segment? (.) () Consider the points A (,) and ( 6, ) B. (a) Find the distance between A and B. (b) Find the
More informationChapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs
Ch 5 Alg Note Sheet Ke Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs Definition: Standard Form of a Quadratic Function The
More informationReview Topics for MATH 1400 Elements of Calculus Table of Contents
Math 1400  Mano Table of Contents  Review  page 1 of 2 Review Topics for MATH 1400 Elements of Calculus Table of Contents MATH 1400 Elements of Calculus is one of the Marquette Core Courses for Mathematical
More informationSystems of Linear Equations: Solving by Graphing
8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From
More informationMATH 021 UNIT 1 HOMEWORK ASSIGNMENTS
MATH 01 UNIT 1 HOMEWORK ASSIGNMENTS General Instructions You will notice that most of the homework assignments for a section have more than one part. Usuall, the part (A) questions ask for eplanations,
More informationAnswer Explanations. The SAT Subject Tests. Mathematics Level 1 & 2 TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE
The SAT Subject Tests Answer Eplanations TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE Mathematics Level & Visit sat.org/stpractice to get more practice and stud tips for the Subject Test
More informationFactoring Polynomials
5. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS 2A.7.D 2A.7.E Factoring Polnomials Essential Question How can ou factor a polnomial? Factoring Polnomials Work with a partner. Match each polnomial equation with
More informationPRINCIPLES OF MATHEMATICS 11 Chapter 2 Quadratic Functions Lesson 1 Graphs of Quadratic Functions (2.1) where a, b, and c are constants and a 0
PRINCIPLES OF MATHEMATICS 11 Chapter Quadratic Functions Lesson 1 Graphs of Quadratic Functions (.1) Date A. QUADRATIC FUNCTIONS A quadratic function is an equation that can be written in the following
More informationName Class Date. Quadratic Functions and Transformations. 4 6 x
 Quadratic Functions and Transformations For Eercises, choose the correct letter.. What is the verte of the function 53()? D (, ) (, ) (, ) (, ). Which is the graph of the function f ()5(3) 5? F 6 6 O
More informationSample Problems For Grade 9 Mathematics. Grade. 1. If x 3
Sample roblems For 9 Mathematics DIRECTIONS: This section provides sample mathematics problems for the 9 test forms. These problems are based on material included in the New York Cit curriculum for 8.
More informationUnit 2 Notes Packet on Quadratic Functions and Factoring
Name: Period: Unit Notes Packet on Quadratic Functions and Factoring Notes #: Graphing quadratic equations in standard form, verte form, and intercept form. A. Intro to Graphs of Quadratic Equations: a
More informationSolving Systems Using Tables and Graphs
31 Solving Sstems Using Tables and Graphs Vocabular Review 1. Cross out the equation that is NOT in slopeintercept form. 1 5 7 r 5 s a 5!3b 1 5 3 1 7 5 13 Vocabular Builder linear sstem (noun) LIN ee
More informationPRACTICE FINAL EXAM. 3. Solve: 3x 8 < 7. Write your answer using interval notation. Graph your solution on the number line.
MAC 1105 PRACTICE FINAL EXAM College Algebra *Note: this eam is provided as practice onl. It was based on a book previousl used for this course. You should not onl stud these problems in preparing for
More information11.1 Solving Linear Systems by Graphing
Name Class Date 11.1 Solving Linear Sstems b Graphing Essential Question: How can ou find the solution of a sstem of linear equations b graphing? Resource Locker Eplore Tpes of Sstems of Linear Equations
More informationSummary and Vocabulary
Chapter 2 Chapter 2 Summar and Vocabular The functions studied in this chapter are all based on direct and inverse variation. When k and n >, formulas of the form = k n define directvariation functions,
More informationSolve Quadratic Equations
Skill: solve quadratic equations by factoring. Solve Quadratic Equations A.SSE.A. Interpret the structure of epressions. Use the structure of an epression to identify ways to rewrite it. For eample, see
More informationSolving Systems of Linear Equations
5 Solving Sstems of Linear Equations 5. Solving Sstems of Linear Equations b Graphing 5. Solving Sstems of Linear Equations b Substitution 5.3 Solving Sstems of Linear Equations b Elimination 5. Solving
More information4 B. 4 D. 4 F. 3. How can you use the graph of a quadratic equation to determine the number of real solutions of the equation?
3.1 Solving Quadratic Equations COMMON CORE Learning Standards HSASSE.A. HSAREI.B.b HSFIF.C.8a Essential Question Essential Question How can ou use the graph of a quadratic equation to determine the
More informationSolving Linear Systems
1.4 Solving Linear Sstems Essential Question How can ou determine the number of solutions of a linear sstem? A linear sstem is consistent when it has at least one solution. A linear sstem is inconsistent
More information2.1 Evaluate and Graph Polynomial
2. Evaluate and Graph Polnomial Functions Georgia Performance Standard(s) MM3Ab, MM3Ac, MM3Ad Your Notes Goal p Evaluate and graph polnomial functions. VOCABULARY Polnomial Polnomial function Degree of
More informationDiagnostic Tests Study Guide
California State Universit, Sacramento Department of Mathematics and Statistics Diagnostic Tests Stud Guide Descriptions Stud Guides Sample Tests & Answers Table of Contents: Introduction Elementar Algebra
More informationWrite Quadratic Functions and Models
4.0 A..B, A.6.B, A.6.C, A.8.A TEKS Write Quadratic Functions and Models Before You wrote linear functions and models. Now You will write quadratic functions and models. Wh? So ou can model the cross section
More information3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS
Section. Logarithmic Functions and Their Graphs 7. LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Ariel Skelle/Corbis What ou should learn Recognize and evaluate logarithmic functions with base a. Graph logarithmic
More informationFunctions. Essential Question What is a function? Work with a partner. Functions can be described in many ways.
. Functions Essential Question What is a function? A relation pairs inputs with outputs. When a relation is given as ordered pairs, the coordinates are inputs and the coordinates are outputs. A relation
More informationAnalytic Geometry 300 UNIT 9 ANALYTIC GEOMETRY. An air traffi c controller uses algebra and geometry to help airplanes get from one point to another.
UNIT 9 Analtic Geometr An air traffi c controller uses algebra and geometr to help airplanes get from one point to another. 00 UNIT 9 ANALYTIC GEOMETRY Copright 00, K Inc. All rights reserved. This material
More informationCHAPTER 3 Graphs and Functions
CHAPTER Graphs and Functions Section. The Rectangular Coordinate Sstem............ Section. Graphs of Equations..................... 7 Section. Slope and Graphs of Linear Equations........... 7 Section.
More informationEquations and Inequalities
Equations and Inequalities Figure 1 CHAPTER OUTLINE.1 The Rectangular Coordinate Systems and Graphs. Linear Equations in One Variable.3 Models and Applications. Comple Numbers.5 Quadratic Equations.6 Other
More informationSolving a LinearQuadratic System
CC18 Solving LinearQuadratic Systems Objective Content Standards A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables... A.REI.11 Explain why the xcoordinates
More informationQuadratic Functions and Equations
Quadratic Functions and Equations Quadratic Graphs and Their Properties Objective: To graph quadratic functions of the form y = ax 2 and y = ax 2 + c. Objectives I can identify a vertex. I can grapy y
More information8.4. If we let x denote the number of gallons pumped, then the price y in dollars can $ $1.70 $ $1.70 $ $1.70 $ $1.
8.4 An Introduction to Functions: Linear Functions, Applications, and Models We often describe one quantit in terms of another; for eample, the growth of a plant is related to the amount of light it receives,
More informationInverse Trigonometric Functions. inverse sine, inverse cosine, and inverse tangent are given below. where tan = a and º π 2 < < π 2 (or º90 < < 90 ).
Page 1 of 7 1. Inverse Trigonometric Functions What ou should learn GOAL 1 Evaluate inverse trigonometric functions. GOAL Use inverse trigonometric functions to solve reallife problems, such as finding
More informationSEE the Big Idea. Quonset Hut (p. 218) Zebra Mussels (p. 203) Ruins of Caesarea (p. 195) Basketball (p. 178) Electric Vehicles (p.
Polnomial Functions.1 Graphing Polnomial Functions. Adding, Subtracting, and Multipling Polnomials.3 Dividing Polnomials. Factoring Polnomials.5 Solving Polnomial Equations. The Fundamental Theorem of
More informationSection 5: Quadratic Equations and Functions Part 1
Section 5: Quadratic Equations and Functions Part 1 Topic 1: RealWorld Examples of Quadratic Functions... 121 Topic 2: Factoring Quadratic Expressions... 125 Topic 3: Solving Quadratic Equations by Factoring...
More informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 1 b. 0.2 1. 2 3.2 3 c. 20 16 2 20 2. Determine which of the epressions are polynomials. For each polynomial,
More informationEvaluate Logarithms and Graph Logarithmic Functions
TEKS 7.4 2A.4.C, 2A..A, 2A..B, 2A..C Before Now Evaluate Logarithms and Graph Logarithmic Functions You evaluated and graphed eponential functions. You will evaluate logarithms and graph logarithmic functions.
More informationEquations and Inequalities
Equations and Inequalities Figure 1 CHAPTER OUTLINE 1 The Rectangular Coordinate Systems and Graphs Linear Equations in One Variable Models and Applications Comple Numbers Quadratic Equations 6 Other Types
More informationSolving Systems of Linear Equations by Graphing
. Solving Sstems of Linear Equations b Graphing How can ou solve a sstem of linear equations? ACTIVITY: Writing a Sstem of Linear Equations Work with a partner. Your famil starts a bedandbreakfast. The
More informationInfinite Limits. Let f be the function given by. f x 3 x 2.
0_005.qd //0 :07 PM Page 8 SECTION.5 Infinite Limits 8, as Section.5, as + f() = f increases and decreases without bound as approaches. Figure.9 Infinite Limits Determine infinite its from the left and
More information6.3 Interpreting Vertex Form and Standard Form
Name Class Date 6.3 Interpreting Verte Form and Standard Form Essential Question: How can ou change the verte form of a quadratic function to standard form? Resource Locker Eplore Identifing Quadratic
More informationSummer Review For Students Entering Algebra 2
Summer Review For Students Entering Algebra Teachers and administrators at Tuscarora High School activel encourage parents and communit members to engage in children s learning. This Summer Review For
More informationDefine General Angles and Use Radian Measure
1.2 a.1, a.4, a.5; P..E TEKS Define General Angles and Use Radian Measure Before You used acute angles measured in degrees. Now You will use general angles that ma be measured in radians. Wh? So ou can
More information76. nth Roots. Vocabulary. Geometric Sequences in Music. Lesson. Mental Math
Lesson 76 nth Roots Vocabular cube root n th root BIG IDEA If is the nth power of, then is an nth root of. Real numbers ma have 0, 1, or 2 real nth roots. Geometric Sequences in Music A piano tuner adjusts
More informationLaw of Sines, Law of Cosines, Heron s Formula:
PreAP Math Analsis nd Semester Review Law of Sines, Law of Cosines, Heron s Formula:. Determine how man solutions the triangle has and eplain our reasoning. (FIND YOUR FLOW CHART) a. A = 4, a = 4 ards,
More informationRational Exponents and Radical Functions
.1..... Rational Eponents and Radical Functions nth Roots and Rational Eponents Properties of Rational Eponents and Radicals Graphing Radical Functions Solving Radical Equations and Inequalities Performing
More informationModeling with Exponential and Logarithmic Functions 6.7. Essential Question How can you recognize polynomial, exponential, and logarithmic models?
.7 Modeling with Eponential and Logarithmic Functions Essential Question How can ou recognize polnomial, eponential, and logarithmic models? Recognizing Different Tpes of Models Work with a partner. Match
More informationChapter 5: Systems of Equations
Chapter : Sstems of Equations Section.: Sstems in Two Variables... 0 Section. Eercises... 9 Section.: Sstems in Three Variables... Section. Eercises... Section.: Linear Inequalities... Section.: Eercises.
More informationLinear Functions. Essential Question How can you determine whether a function is linear or nonlinear?
. Linear Functions Essential Question How can ou determine whether a function is linear or nonlinear? Finding Patterns for Similar Figures Work with a partner. Cop and complete each table for the sequence
More informationAlgebra 1 Skills Needed to be Successful in Algebra 2
Algebra 1 Skills Needed to be Successful in Algebra A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed
More informationEssential Question How can you solve a system of linear equations? $15 per night. Cost, C (in dollars) $75 per Number of. Revenue, R (in dollars)
5.1 Solving Sstems of Linear Equations b Graphing Essential Question How can ou solve a sstem of linear equations? Writing a Sstem of Linear Equations Work with a partner. Your famil opens a bedandbreakfast.
More informationWriting Equations in PointSlope Form
. Writing Equations in PointSlope Form Essential Question How can ou write an equation of a line when ou are given the slope and a point on the line? Writing Equations of Lines Work with a partner. Sketch
More informationUse Properties of Exponents
4. Georgia Performance Standard(s) MMAa Your Notes Use Properties of Eponents Goal p Simplif epressions involving powers. VOCABULARY Scientific notation PROPERTIES OF EXPONENTS Let a and b be real numbers
More informationA calculator may be used on the exam.
The Algebra Semester A eamination has the following tpes of questions: Selected Response Student Produced Response (Gridin) Brief Constructed Response (BCR) Etended Constructed Response (ECR) Short Answer
More informationSection 3.1 Solving Linear Systems by Graphing
Section 3.1 Solving Linear Sstems b Graphing Name: Period: Objective(s): Solve a sstem of linear equations in two variables using graphing. Essential Question: Eplain how to tell from a graph of a sstem
More informationPractice Problem List II
Math 46 Practice Problem List II  Section 4.: 3, 3, 5, 9, 3, 9, 34, 39, 43, 53, 67 odd
More informationChapter 10 Answers. Practice (0,0); maximum 2. (0,0); maximum 3. (0,0); minimum y = x 2, y = 3x 2, y = 5x 2 8. y 1
Chapter 0 Answers Practice 0. (0,0); maimum. (0,0); maimum. (0,0); minimum. (0,0); minimum. (0,0); maimum. (0,0); minimum 7. =, =, =. =, =, =. =, =, = 0. =, =, =. =, =, =7. =, =, =........ 7. 0 7...
More informationStudy Guide and Intervention. The Quadratic Formula and the Discriminant. Quadratic Formula. Replace a with 1, b with 5, and c with 14.
Study Guide and Intervention Quadratic Formula The Quadratic Formula can be used to solve any quadratic equation once it is written in the form a 2 + b + c = 0. Quadratic Formula The solutions of a 2 +
More informationCopyrighted by Gabriel Tang B.Ed., B.Sc. Page 1.
Chapter : Linear and Quadratic Functions Chapter : Linear and Quadratic Functions : Points and Lines Sstem of Linear Equations:  two or more linear equations on the same coordinate grid. Solution of
More informationHow can you write an equation of a line when you are given the slope and a point on the line? ACTIVITY: Writing Equations of Lines
.7 Writing Equations in PointSlope Form How can ou write an equation of a line when ou are given the slope and a point on the line? ACTIVITY: Writing Equations of Lines Work with a partner. Sketch the
More informationLast modified Spring 2016
Math 00 Final Review Questions In problems 6, perform the indicated operations and simplif if necessar.. 8 6 8. 7 6. ( i) ( 4 i) 4. (8 i). ( 9 i)( 7 i) 6. ( i)( i) In problems 7, solve the following applications.
More informationChapter 9 BUILD YOUR VOCABULARY
C H A P T E R 9 BUILD YUR VCABULARY Chapter 9 This is an alphabetical list of new vocabular terms ou will learn in Chapter 9. As ou complete the stud notes for the chapter, ou will see Build Your Vocabular
More informationPRECALCULUS FINAL EXAM REVIEW
PRECALCULUS FINAL EXAM REVIEW Evaluate the function at the indicated value of. Round our result to three decimal places.. f () 4(5 ); 0.8. f () e ; 0.78 Use the graph of f to describe the transformation
More informationFinal Exam Review Part 2 #4
Final Eam Review Part # Intermediate Algebra / MAT 135 Fall 01 Master (Prof. Fleischner) Student Name/ID: 1. Solve for, where is a real number. + = 8. Solve for, where is a real number. 9 1 = 3. Solve
More informationMATH GRADE 8 UNIT 4 LINEAR RELATIONSHIPS EXERCISES
MATH GRADE 8 UNIT LINEAR RELATIONSHIPS Copright 01 Pearson Education, Inc., or its affiliate(s). All Rights Reserved. Printed in the United States of America. This publication is protected b copright,
More information26 Questions EOC Review #1 EOC REVIEW
Name Period 6 Questions EOC Review # EOC REVIEW Solve each: Give the BEST Answer. You may use a graphing calculator.. Which quadrant contains the verte of the following: f ( ) 8 st nd rd d. 4th. What type
More informationNAME DATE PERIOD. Study Guide and Intervention. Ax + By = C, where A 0, A and B are not both zero, and A, B, and C are integers with GCF of 1.
NAME DATE PERID 31 Stud Guide and Intervention Graphing Linear Equations Identif Linear Equations and Intercepts A linear equation is an equation that can be written in the form A + B = C. This is called
More informationINTRODUCTION GOOD LUCK!
INTRODUCTION The Summer Skills Assignment for has been developed to provide all learners of our St. Mar s Count Public Schools communit an opportunit to shore up their prerequisite mathematical skills
More information13.2 Exponential Decay Functions
6 6   Locker LESSON. Eponential Deca Functions Common Core Math Standards The student is epected to: F.BF. Identif the effect on the graph of replacing f() b f() + k, kf(), f(k), and f( + k) for specific
More information121D Practice Test #
D Practice Test # College Algebra / Math D GARAGE (Prof. Vasan) Student Name/ID:. Write the epression as a single logarithm. 5log 8 w + 5 log 8 3log 8 z. Solve for. log + 3 = log + 6 ALEKS D Practice Test
More informationLesson 10.1 Solving Quadratic Equations
Lesson 10.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with each set of conditions. a. One intercept and all nonnegative yvalues b. The verte in the third quadrant and no
More informationWhat Did You Learn? Key Terms. Key Concepts. 158 Chapter 1 Functions and Their Graphs
333371_010R.qxp 12/27/0 10:37 AM Page 158 158 Chapter 1 Functions and Their Graphs Ke Terms What Did You Learn? equation, p. 77 solution point, p. 77 intercepts, p. 78 slope, p. 88 pointslope form, p.
More informationQuadratic Equations and Complex Numbers
Quadratic Equations and Comple Numbers.1 Solving Quadratic Equations. Comple Numbers.3 Completing the Square. Using the Quadratic Formula.5 Solving Nonlinear Sstems. Quadratic Inequalities RobotBuilding
More information