Graph the linear system and estimate the solution. Then check the solution algebraically.

Transcription

1 (Chapters and ) A. Linear Sstems (pp. 6 0). Solve a Sstem b Graphing Vocabular Solution For a sstem of linear equations in two variables, an ordered pair (x, ) that satisfies each equation. Consistent A sstem that has at least one solution. Inconsistent A sstem that has no solution. Independent A consistent sstem that has exactl one solution. Dependent A consistent sstem that has infinitel man solutions. A. Linear Sstems Two distinct lines intersect in at most one point. Graph the linear sstem and estimate the solution. Then check the solution algebraicall. x 5 8 x 5 6 Begin b graphing both equations, as shown at the right. From the graph, the lines appear to intersect at (, ). You can check this algebraicall as follows. Equation Equation x 5 8 x 5 6 () () 5 8 () () 5 6 The solution is (, ). x (, ) Graph the linear sstem and estimate the solution. Then check the solution algebraicall.. 5 x. 5x 5 0. x x 9 x 5 5 x Solve the sstem. Then classif the sstem as consistent and independent, consistent and dependent, or inconsistent. x 5 x 5 The graphs of the equations are the same line. Each point on the line is a solution, and the sstem has infinitel man solutions. Therefore, the sstem is consistent and dependent. x 6 Algebra Benchmark Chapters and

2 (Chapters and ) The graph of an inconsistent sstem of linear equations consists of parallel lines. Solve the sstem. Then classif the sstem as consistent and independent, consistent and dependent, or inconsistent. x 5 0 x 5 The graphs of the equations are parallel. There are no solutions, and the sstem is inconsistent. x Use substitution to check our work. Solve the sstem. Then classif the sstem as consistent and independent, consistent and dependent, or inconsistent.. x x x 5 6x x 5 x 5. Use the Substitution Method Solve the sstem using the substitution method. x 5 5 x Step : Solve Equation for x. x 5 5 Revised Equation Step : Substitute the expression for x into Equation and solve for. (5 ) Step : Substitutethe value of into revised Equation and solve for x. x 5 5 () 5 The solution is (, ). Check the solution b substituting into the original equations. Solve the sstem using the substitution method. 7. 5x 5 8. x 5 7 9x 5 x 5 5 BENCHMARK 6 A. Linear Sstems Algebra Benchmark Chapters and 7

3 (Chapters and ) A. Linear Sstems Look for similarities between equations that ou can exploit to eliminate a variable.. Use the Elimination Method Solve the sstem using the elimination method. x 5 7 x 5 Step : Multipl Equation b so that the coefficient of x differ onl in sign. x 56 Step : Add the revised equations and solve for. x 5 7 x Step : Substitute the value of into one of the original equations. Solve for x. x x Check our solution b choosing a point on the coordinate plane and substituting into both inequalities. This helps to determine appropriate shading. x x 5 7 The solution is 7, 7. Check the solution b substituting into the original equations. Solve the sstem using the elimination method. 9. x 5 0. x 5 x 5 5 x 5 6. Graph a Sstem of Two Inequalities Graph the sstem of inequalities. x x Step : Graph each inequalit in the sstem. Use red for the first inequalit and blue for the second inequalit. Step : Identif the region that is common to both graphs. It is the region that is shaded purple. x 8 Algebra Benchmark Chapters and

4 (Chapters and ) Graph the sstem of inequalities x x x 5 5. Solve a Sstem of Three Linear Equations The process used in solving a sstem of three equations is no different from what ou used to solve a sstem of two equations. Solve the sstem. x z 5 8 x z 5 x 6 z 5 Step : Rewrite the sstem as a linear sstem in two variables. x z 5 8 Add times Equation x 6 z 5 to Equation. z 5 New Equation x z 5 Add times Equation x 8z 5 8 to Equation. 5 0z 5 0 New E quation Step : Solve the new linear sstem for both of its variables. BENCHMARK 6 A. Linear Sstems 0 0z 5 0 Add 0 times new Equation 5 0z 5 0 to times new Equation Solve for. z 50 Substitute into new Equation or to find z. Step : Substitute 5 and z 50 into an original equation and solve for x. x z 5 8 x () (0) 5 8 x 5 0 The solution is x 5 0, 5, and z 50 or the ordered triple (0,, 0). Check this solution in each of the original equations. Solve the sstem.. x z 5. x z 5 7 x z 5 x z 5 x z 5 x z 5 7 Algebra Benchmark Chapters and 9

5 (Chapters and ) Quiz. Graph the linear sstem and estimate the solution. Then check the solution algebraicall. 5 5x x 5 5. Solve the sstem. Then classif the sstem as consistent and independent, consistent and dependent, or inconsistent. x 5 8 x 5 0 A. Linear Sstems. Solve the sstem using the substitution method. x 5 7 x 5. Solve the sstem using the elimination method. x 5 5 x Graph the sstem of inequalities. x 6 x 6 6. Solve the sstem. x z 5 x z 5 x z Algebra Benchmark Chapters and

6 (Chapters and ) B. Matrices (pp. 5). Perform Basic Matrix Operations Vocabular Matrix A rectangular arrangement of numbers in rows and columns. Elements The numbers in a matrix. Matrix addition is not alwas defined. Matrix multiplication is not commutative. Perform the indicated operation, if possible. a. F G F b. 5G F a. F F b. G F G F 5 5 6G F G F () 5 5G F () G F () 5 5() (5) G F 5 5 G (5)G 5 F () () Perform the indicated operation, if possible F G. F 7. F G Perform the indicated operation, if possible. F G F 5 G F () () () () G 5 F 6 9 G F 5 () G Perform the indicated operation, if possible F 5 F 5G. Multipl Matrices a. Find AB if A 5 F and B 5 G F b. Find CD if C 5 F G and D 5 F G. G. 7 5 G 5G F G 7 6G G B. Matrices Algebra Benchmark Chapters and

7 (Chapters and ) a. Step : Multipl the numbers in the first row of A b the numbers in the first column of B, add the products, and put the result in the first row, first column of AB. B. Matrices Step : Multipl the numbers in the first row of A b the numbers in the second column of B, add the products, and put the results in the first row, second column of AB. Step : Multipl the numbers in the second row of A b the numbers in the first column of B, add the products, and put the result in the second row, second column of AB. Step : Multipl the numbers in the second row of A b the numbers in the second column of B, add the products, and put the results in the second row, second column of AB. Step 5: Simplif the product matrix. F G F 5 G F (() ()) (() ()) 5 F 0 5 G ( ) ( ) G b. Step : Multipl the numbers in the first row of C b the numbers in the first column of D, add the products, and put the result in the first row, first column of CD. Step : Multipl the numbers in the first row of C b the numbers in the second column of D, add the products, and put the results in the first row, second column of CD. Step : Multipl the numbers in the second row of C b the numbers in the first column of D, add the products, and put the result in the second row, first column of CD. Step : Multipl the numbers in the second row of C b the numbers in the second column of D, add the products, and put the results in the second row, second column of CD. Step 5: Multipl the numbers in the third row of C b the numbers in the first column of D, add the products, and put the results in the third row, first column of CD. Step 6: Multipl the numbers in the third row of C b the numbers in the second column of D, add the products, and put the results in the third row, second column of CD. F Step 7: Simplif the product matrix. G F G 5 F ( ()) ( ()) ( ()) (() (() (() )G F 5 0 8G 8 9 Algebra Benchmark Chapters and

8 (Chapters and ) Vocabular 6. Find AB if A 5 F 5 7. Find CD if C 5 F G and B 5 F. Evaluate Determinants G. and D 5 G F G. Determinant The determinant of a matrix is written ) a c difference ad bc. b d) and is defined as the Onl square matrices have determinants. Evaluate the determinants. a. ) a. ) b. ) ) ) ) 5 (() ()) 5 6 b. ) ) 5 ( 8 9) ( ) 5 B. Matrices Cramer s Rule provides an alternative approach to solving sstems of linear equations. Evaluate the determinants. 8. ) 7 a 6 8) 9. ) 0 5. Use Cramer s Rule Use Cramer s rule to solve this sstem: x 5 9 x ) ) ) Step : Evaluate the determinant of the coefficient matrix. ) 5 ) 5 (5) ()() 5 Step : Appl Cramer s rule because the determinant is not zero. 9 ) 5 ) x ) 9 ) 5 5 The solution is (, ). Check this solution in the original equations. Algebra Benchmark Chapters and

9 (Chapters and ) Use Cramer s rule to solve each sstem.. 5x 5. x 5 7 x 5 5 x Find the Inverse of a Matrix Onl square matrices have inverses, but not all square matrices have inverses. Find the inverse of A 5 A 5 F 5 G F 7 7 G. B. Matrices Multipling a matrix b its inverse is analogous to multipling a real number b its reciprocal. Find the inverse of the matrix. 8. F 5G. F 0 6. Solve a Matrix Equation 5. G F Solve the matrix equation AX 5 B for the matrix. F 5 G X 5 F 5 8 F G G Begin b finding the inverse of A. 7 F G 7G 5 7 To solve the equation for X, multipl both sides of the equation b A on the left. F 7 7 F 5 X G 7 7 F G F 0 G F 0 X 5 G F 5 X 5 F 5 G G G Solve the matrix equation AX 5 B for the matrix X. 6. F 7. G F G X 5 F G X 5 F G Algebra Benchmark Chapters and

10 (Chapters and ) A sstem of linear equations can be translated into a matrix problem which can be solved using inverses. 7. Solve a Linear Sstem Using Inverse Matrices Use an inverse matrix to solve the linear sstem. x x 58 Step : Write the linear sstem as a matrix equation AX 5 B. F 5 7 X 5 6 G F 8G Step : Find the inverse matrix A. F 5 6 G Step : Multipl the matrix of constant b on the left. X 5 A B 5 F G F 5 8G F 5 G F x G The solution of the sstem is (, ). Check this in the original equations. Use an inverse matrix to solve the linear sstem. 8. x x 5 x 5 5x 5 B. Matrices Quiz Perform the indicated operation, if possible.. F 5 G F. 5G F G F 0 G.. Find AB if A 5 F G and B 5 F G. 5. Find the determinant and the inverse for matrix A 5 F 6. Use an inverse matrix to solve the linear sstem. x 5 5 x 5 F G. G Algebra Benchmark Chapters and 5

11 (Chapters and ) C. Graphing Quadratics Vocabular A quadratic function has either a maximum or a minimum. It cannot have both. C. Graphing Quadratic Functions (pp. 6 ). Find the Minimum or Maximum Value Quadratic function A function that can be written in the standard form ax bx c where a ± 0. Parabola The graph of a quadratic function. Vertex The lowest or highest point on a parabola. Axis of smmetr A line that divides the parabola into mirror images and passes through the vertex. Tell whether the given function has a minimum value or a maximum value. Then find the minimum or maximum value. a. 5 x 8x 0 b. 5x x a. Because a 0, the function has a minimum value. To find it, calculate the coordinates of the vertex. x 5 b a Now substitute this value of x into the function. 5 () 8() 0 56 The minimum value is (, 6). b. Because a 0, the function has a maximum value. To find it, calculate the coordinates of the vertex. b x 5 a 5 5 Then substitute this value of x into the function. 5 ()() () 5 The maximum value is (, ). Tell whether each function has a minimum value or a maximum value. Then find the minimum or maximum value.. 5 x. 5x x 6. 5x x 6 6 Algebra Benchmark Chapters and

12 BENCHMARK 6 (Chapters and ). Graph a Quadratic in Standard Form Just as two points determine a line, three non-collinear points determine a parabola. So alwas plot and label three points when graphing parabolas. Vocabular Vertex form is the easiest form of the equation of a parabola to work with when graphing. Graph 5 x x 5 Step : Identif the coefficients of the function. The coefficients are a 5, b 5, c 55. Because a, the parabola opens up. Step : Find the vertex. Calculate the x-coordinate. x 5 b a 5 5 Then find the -coordinate of the vertex. 5 () () 5 57 So the vertex is (, 7). Plot this point. Step : Draw the axis of smmetr x x Step : Identif the -intercept c, which is 5. Plot the point (0, 5). Then reflect this point in the axis of smmetr to plot another point (, 5). Step 5: Evaluate the function for another value of x, such as x 5. 5 () () 5 5 Plot the point (, ) and its reflection (, ) in the axis of smmetr. Step 6: Draw a parabola through the plotted points. Graph the function. Label the vertex and axis of smmetr.. 5 x x 5. 5x 6x 6. 5x 0. Graph a Quadratic Function in Vertex Form Vertex form A form of a quadratic function written 5 a(x h) k. Graph 5 (x ) Step : Identif the constants a 5, h 5, and k 5. Because a 0, the parabola opens up. Step : Plot the vertex (h, k) 5 (, ) and draw the axis of smmetr x 5. C. Graphing Quadratics Algebra Benchmark Chapters and 7

13 (Chapters and ) Step : Evaluate the function for two values of x. x 5 0: 5 (0 ()) 5 x 5 : 5 ( ()) 5.5 Plot the points (0, ) and (,.5) and their reflections in the axis of smmetr. Step : Draw a parabola through the plotted points. x 5 5 x Graph each. Label the vertex and axis of smmetr. C. Graphing Quadratics Vocabular Now is a good time to review factoring trinomials, since these are quadratic functions (x ) 8. 5(x ) 9. 5(x ). Graph a Quadratic in Intercept Form Intercept Form A form of a quadratic function written 5 a(x p)(x q). Graph 5 (x )(x ) Step : Identif the x-intercepts. Because p 5 and q5, the x-intercepts occur at the points (, 0) and (, 0). Step : Find the coordinates of the vertex. x 5 p q 5 () 5 5 ( )( ) 58 So, the vertex is (, 8). Step : Draw the axis of smmetr x 5. x Step : Draw a parabola through the vertex and the points where the x-intercepts occur Graph each. Label the vertex, axis of smmetr, and x-intercepts. 0. 5(x )(x ). 5 (x )(x ). 5(x )(x ) 8 Algebra Benchmark Chapters and

14 BENCHMARK 6 (Chapters and ) 5. Rewrite Quadratic Functions Write in standard form. 5(x )(x ) Each of the different forms for quadratics has advantages depending on the situation. Therefore, ou should be able to convert from one form to another easil. 5(x x x ) 5(x x ) 5x x 6 Write f(x) 5 (x ) in standard form. ƒ(x) 5 (x )(x ) ƒ(x) 5 (x x ) ƒ(x) 5 x x ƒ(x) 5 x x Write the quadratic function in standard form.. 5 (x )(x ). 5(x )(x ) 5. 5(x ) 6. ƒ(x) 5 (x ) C. Graphing Quadratics A solution to an inequalit is a value that makes the inequalit true. 6. Solve a Quadratic Inequalit b Graphing Solve x 6x 0 b graphing. The solution consists of the x-values for which the graph of x 6x 0 lies on or above the x-axis. Find the graph s x-intercepts b using the quadratic formula to solve for x. 0 5 x 6x x 5 (6) 6 Ï (6) ()() () x Ï 6 x ø 0.6 and x ø.577 Sketch a parabola that opens up and has 0.6 and.577 as x-intercepts. The graph lies on or above the x-axis to the left of (and including) 0.6 and to the right of (and including).577. The solution of the inequalit is approximatel x 0.6 and x.577 Solve each inequalit using a graph. 7. x x 0 8. x x 0 Algebra Benchmark Chapters and 9

15 (Chapters and ) 7. Write a Quadratic Model Write a quadratic function for the parabola shown. Use the appropriate form of the equation of a parabola depending on the information ou are given. Use intercept form because the x-intercepts are given. 5 a(x p)(x q) 5 a(x )(x ) Use the other given point, (, ), to find a. 5 a( )( ) (, 0) (, 0) x (, ) C. Graphing Quadratics a 5 A quadratic function for the parabola is 5 (x )(x ). Write a quadratic function for the parabola shown. Use vertex form because the vertex is given. 5 a(x h) k 5 a(x ) Use the other given point (, ) to find a. 5 a( ) a 5 A quadratic function for the parabola is 5(x ). Write a quadratic function for the parabola shown. 9. (, ) (, 0) (0, 0) x 0. (, 0) (, 0) x (, ) (, ) (, ) x. (, ) x (, ) 0 Algebra Benchmark Chapters and

16 BENCHMARK 6 (Chapters and ) Quiz. Tell whether the function 5x 9x has a minimum value or a maximum value. Then find the minimum or maximum value.. Graph 5(x ). Label the vertex and axis of smmetr.. Graph 5 (x )(x ). Label the vertex, axis of smmetr, and x-intercepts.. Write 5(x ) in standard form. 5. Write a quadratic function for the parabola shown. (, ) (, ) x C. Graphing Quadratics Algebra Benchmark Chapters and

17 (Chapters and ) Vocabular Man trinomials cannot be factored. But ou need to tr all possibilities before ou can conclude that that is the case. D. Solving Quadratic Equations (pp. ). Factor Quadratic Trinomials Monomial An expression that is either a number, a variable, or the product of a number and one or more variables. Binomial The sum of two monomials. Trinomial Sum of three monomials. Factor the expression. a. x x b. x 0x a. You want x x 5 (x m)(x n) where mn 5 and m n 5 m 56 and n 5. So x x 5 (x 6)(x ). b. You want x 0x 5 (x m)(x n) where mn 5 and m n 50 m 57 and n 5. So x 0x 5 (x 7)(x ). Factor the quadratic trinomial.. 5 x x 9. 5 x x 8. 5 x x 6. Use Special Factoring Patterns WARNING: a b Þ (a b) When factoring ax bx c there can be man different combinations to consider if neither a nor c is prime. Factor the expression using special patterns. a. 5 x 6 b. 5 x x 9 c. 5 x 0x 5 a. 5 x 6 5 (x 8)(x 8) Difference of two squares b. 5 x x 9 5 (x 7) Perfect square trinomial c. 5 x 0x 5 5 (x 5) Perfect square trinomial Factor the expression using special patterns.. 5 x x 5. 5 x x 6x 9. Solve Quadratic Equations b Factoring a. Factor 6x x. b. Factor 0x x. a. The correct factorization is (x )(x ). b. The correct factorization is (x )(5x ). Algebra Benchmark Chapters and

18 (Chapters and ) Vocabular When simplifing radicals, find the greatest perfect square that is a factor of the value under the radical. Factor each expression. 7. x 0x 8. x 7x x x. Use Properties of Square Roots Radical The expression Ï s. Simplif the expression. a. Ï 0 b. Ï Ï 0 c. Ï 9 6 d. Ï 8 5 a. Ï 0 5 Ï Ï 0 5 Ï 0 b. Ï Ï 0 5 Ï 0 5 Ï Ï 5 5 Ï 5 c. Ï iï 9 Ï 6 5 i d. Ï Ï 8 Ï 5 5 Ï Ï 9 Ï 5 5 Ï 5 Simplif the expression. 0. Ï 5. Ï Ï. Ï 5 9. Ï 9 Simplif the expression. a. Ï 7 b. Ï 5 D. Solving Quadratics Vocabular Adding and subtracting complex numbers is similar to adding and subtracting binomials. But remember, i is not a variable. a. Ï 7 5 Ï 7 Ï Ï Ï 5 Ï b. Ï 5 5 Ï 5 5 Ï 5 Ï 5 Simplif the expression. 6 Ï 5 9 Ï 5 Ï Ï. Ï 5 5. Ï 5 6. Ï 7. 5 Ï Ï 5 5. Add, Subtract, and Multipl Complex Numbers Complex number A number a bi where a and b are real numbers. Imaginar number If b Þ 0, then a bi is an imaginar number. Write the expression as a complex number in standard form. a. ( i) ( i) b. (6 i) ( i) c. 8 ( i) 6i a. ( i) ( i) 5 ( ) ( )i 5 7 i b. (6 i) ( i) 5 (6 ) ( )i 5 c. 8 ( i) 6i 5 (8 ) ( 6)i 5 5 0i Algebra Benchmark Chapters and

19 (Chapters and ) 8. ( 5i) ( 6i) 9. ( i) (6 i) 0. ( i) 6i 6. Solve Quadratic Equations b Finding Square Roots a. Solve x b. Solve x 5. a. x x 5 8 x 5 D. Solving Quadratics x 56Ï 56Ï Ï 56Ï The solutions are Ï and Ï. b. x 5 x 58 x 5 x56ï 56iÏ 56i. x x 8 5. x 5 Quiz Factor each.. x 7x 0. x 6x 6. 6x 7x. x Simplif each expression. 5. Ï Ï Ï 8. ( i) (7 i) Solve for x. 9. x 5 Algebra Benchmark Chapters and

20 (Chapters and ) This process is known as completing the square because ou are building a perfect square trinomial from an existing trinomial. E. Quadratic Formula (pp. 5 7). Make a Perfect Square Trinomial Find the value of c that makes x x c a perfect square trinomial. Then write the expression as the square of a binomial. Step : Find half the coefficient of x. 5 7 Step : Square the result of Step : Step : Replace c with the result from Step : x x 9. Written as the square of a binomial, it looks like this: (x 7). Find the value of c that makes each a perfect square trinomial. Then write the expression as the square of a binomial.. x 6x c. x 8x c. x 0x c. Solve a Quadratic Equation b Completing the Square a. Solve x x b completing the square. E. Quadratic Formula Although completing the square requires some work, it makes solving quadratic equations simple. x x 5 8 x x 5 8 (x ) 5 x 56Ï x 56Ï x 5 6 Ï The solutions are Ï and Ï. b. Solve x 6x b completing the square. x 6x 5 9 x x 5 x x 5 (x ) 5 x 56Ï x 56 The solutions are and. Algebra Benchmark Chapters and 5

21 (Chapters and ) Solve each b completing the square.. x 8x x 0x x x x x Use the Discriminant Find the discriminant of the quadratic equation and give the number and tpe of solutions of the equation. Find all solutions. a. x x 0 b. x 0x 0 c. x x E. Quadratic Formula If the discriminant is greater than zero, there are two real solutions. If the discriminant is less than zero, there are two imaginar solutions. If the discriminant is equal to zero there is one real solution. a. Step : Find the discriminant. b ac 5 ()(0) 5 Because the discriminant is less than zero, conclude that there are exactl two imaginar solutions. Step : Find the solutions b using the quadratic formula. b 6 Ï b ac a 5 6 Ï 5 6 i Ï b. Step : Find the discriminant. b ac 5 (0) ()(0) 5 0 Because the discriminant is greater than zero, conclude that there are exactl two real solutions. Step : Find the solutions b using the quadratic formula. b 6 Ï b ac a Ï Ï Ï 5 c. Step : Find the discriminant. b ac 5 () ()() 5 0 Because the discriminant is zero, conclude that there is exactl one real solution. Step : Find the solutions b using the quadratic formula. b 6 Ï b ac a 5 6 Ï 0 5 Find the discriminant of the quadratic equation and give the number and tpe of solutions of the equation. Find all solutions. 8. x x x x x x Algebra Benchmark Chapters and

22 (Chapters and ) Quiz Find the value of c that makes each a perfect square trinomial. Then write the expression as the square of a binomial.. x x c. x x c Solve each quadratic b completing the square.. x 8x x x 5 0 Find the discriminant of the quadratic equation and give the number and tpe of solutions of the equation. Find all solutions. 5. x x x 6x E. Quadratic Formula Algebra Benchmark Chapters and 7