# Algebra II Notes Unit Five: Quadratic Functions. Syllabus Objectives: 5.1 The student will graph quadratic functions with and without technology.

Size: px
Start display at page:

Transcription

1 Sllabus Objectives:.1 The student will graph quadratic functions with and without technolog. Quadratic Function: a function that can be written in the form are real numbers Parabola: the U-shaped graph of a quadratic function a b c, when a 0 Verte: the highest or lowest point on a quadratic function (maimum or minimum) Ais of Smmetr: the vertical line that passes through the verte of a quadratic function ; a, b, and c Verte Ais of Smmetr Verte - - Ais of Smmetr Graphing a Parabola E: Graph the quadratic function. Step One: Make a table of values (t-chart) Step Two: Plot the points on a coordinate grid and connect to draw the parabola Note: The verte is 0,0, and the ais of smmetr is 0. Page 1 of 31 McDougal Littell.1.8

2 E: Graph the parabola Step One: Make a table of values (t-chart) Step Two: Plot the points on a coordinate grid and connect to draw the parabola Note: The verte is 0,0, and the ais of smmetr is 0. - Comparing and : The verte is 0,0, and the ais of smmetr is 0 for both graphs. When a is positive, the parabola opens up; when a is negative, the parabola opens down. Activit: Transformations with and. Use the graphing calculator to graph the quadratic functions. Describe the effect on the graphs of and dotted line.). (Note: In the calculator graphs shown, or is graphed as a 1. Compare to Verte: Same 0,0 Opens narrower than. 1 3 Compare to Verte: Same 0,0 Opens wider than 3. 3 Compare to Verte: Up 3 0,3 Opens the same as 4. 4 Compare to Verte: Down 4 0, 4 Opens the same as Page of 31 McDougal Littell.1.8

3 Conclusions (sample): For quadratic functions of the form a c a 1 Opens down Narrower than 1a 0 Opens down Wider than 0a 1 Opens up Wider than a 1 Opens up Narrower than c 0 Verte moves down c units c 0 Verte moves up c units Standard Form of a Quadratic Function: a b c Verte: the -coordinate of the verte is b Ais of Smmetr: a b -intercept: c a Graphing a Quadratic Function in Standard Form E: Graph the quadratic function 6 1. State the verte and ais of smmetr. Step One: Find the -coordinate of the verte. a 1, b Step Two: Make a table of values. When choosing -values, use the verte, a few values to the left of the verte, and a few values to the right of the verte coordinate of verte: Note: When calculating the -coordinate of points to the right and left of the verte, notice the smmetr. Step Three: Plot the points from the table and draw the parabola. Verte: 3, Ais of Smmetr: Page 3 of 31 McDougal Littell.1.8

4 Verte Form of a Quadratic Function: ah k Verte: hk, Graphing a Quadratic Function in Verte Form E: Graph the quadratic function State the verte and ais of smmetr. Step One: Identif the verte and ais of smmetr. Note: Another wa of writing the function is So the verte is 3, 4 and the ais of smmetr is 3. Step Two: Make a table of values. When choosing -values, use the verte, a few values to the left of the verte, and a few values to the right of the verte. (Note: Because of the fraction, ou ma want to choose values that will guarantee whole numbers for the -coordinates.) Step Three: Plot the points from the table and draw the parabola intercept: the -coordinate of the point where the curve intersects the -ais - Intercept Form of a Quadratic Function: a p q -Intercepts: p, q -Coordinate of Verte: p q (verte is halfwa between the -intercepts) Graphing a Quadratic Function in Intercept Form E: Graph the quadratic function 1. Step One: Identif the -intercepts. -intercepts are and 1 Note: It ma be helpful to write the equation as 1 Step Two: Identif the verte. -coordinate of verte: 1 -coordinate of verte: Page 4 of 31 McDougal Littell.1.8

5 Step Three: Plot the -intercepts and the verte and draw the parabola. 0 1 You Tr: Find the verte and ais of smmetr for the following quadratic functions. Determine if the parabola will open up or down. Then graph the parabola QOD: Describe the three forms of an equation of a quadratic function. Sample CCSD Common Eam Practice Question(s): 1. Which graph represents f 1?. What is the maimum of the quadratic function A. f 1 B. f 3 C. f 6 D. f 8 f 4 6? Page of 31 McDougal Littell.1.8

6 Sample SAT Question(s): Taken from College Board online practice problems. The figure above shows the graph of a quadratic function in the -plane. Of all the points, on the graph, for what value of is the value of greatest? Grid-In Page 6 of 31 McDougal Littell.1.8

7 Sllabus Objective:. The student will solve quadratic equations b factoring, graphing, completing the square, and the quadratic formula. Review: Factoring Quadratic Trinomials into Two Binomials (Using the ac method or splitting the middle term.) Factoring a b c a, 1 E: Factor 7 1. Find two integers such that their product is 1 and their sum is 7. 4 and 3 Write the two binomials as a product. 4 3 Factoring a b c a, 1 E: Factor 7 3. Step One: Multipl a c. 3 6 Step Two: Find two integers such that their product is a c 6 and their sum is b 7. 6 and 1 Step Three: Rewrite ( split ) the middle term as a sum of two terms using the numbers from Step Two (order does not matter when splitting the middle term) Step Four: Factor b grouping. Group the first terms and last terms and factor out the GCF from each pair Step Five: If Step Four was done correctl, there should be a common binomial factor. Factor this binomial out and write what remains from each term as the second binomial factor. 1 3 E: Factor 7. Step One: Multipl a c. Step Two: Find two integers such that their product is a c and their sum is b 7. and Step Three: Rewrite ( split ) the middle term as a sum of two terms using the numbers from Step Two. Page 7 of 31 McDougal Littell.1.8

8 Step Four: Factor b grouping. Group the first terms and last terms and factor out the GCF from each pair. 1 Step Five: If Step Four was done correctl, there should be a common binomial factor. Factor this binomial out and write what remains from each term as the second binomial factor. 1 Special Factoring Patterns: Memorize these! Difference of Two Squares: a b aba b Perfect Square Trinomial: a abb ab a abb ab E: Factor This appears to be a difference of two squares, since each term is a perfect square. Rewrite each term as a monomial squared then use the pattern to factor E: Factor 49m 14mn n. This appears to be a perfect square trinomial. Rewrite the first and last terms as a monomial squared and check to see if the middle term is twice the product of these monomials. Then use the pattern to factor. 7m 14mnn 7mn 7m n 14mn) (Check: Factoring a GCF Monomial E: Factor 7 0 completel. Step One: Factor out the GCF of. 36 Step Two: Factor the remaining polnomial. 6 6 Page 8 of 31 McDougal Littell.1.8

9 Standard Form of a Quadratic Equation: a b c 0 Zero Product Propert: If the product of two factors is 0, then one or both of the factors must equal 0. Solving a Quadratic Equation b Factoring E: Solve the equation 48. Step One: Write the equation in standard form. Step Two: Factor the quadratic using the ac method a c 4 b 8 and Step Three: Set each factor equal to zero and solve. The solutions can be written in set notation: , 3 E: Solve the equation Step One: Write the equation in standard form Step Two: Factor the quadratic Note: Step Three: Set each factor equal to zero and solve. The solution can be written in set notation: Zero(s) of Quadratic Functions: the -value(s) where the function intersects the -ais To find the zero(s), factor the quadratic and set each factor equal to 0. Reminder: We can graph quadratic functions b plotting the zeros. The verte is halfwa between the zeros. Page 9 of 31 McDougal Littell.1.8

10 E: Find the zero(s) of the quadratic function Step One: Factor the quadratic polnomial. Step Two: Set each factor equal to 0 and solve. Step Three: Find the coordinates of the verte. Step Four: Plot the points and sketch the parabola. 3and graph the parabola You Tr: Solve the quadratic equation t 4t 6 b factoring. QOD: What must be true about a quadratic equation before ou can solve it using the zero product propert? Sample CCSD Common Eam Practice Question(s): What are the solutions of the quadratic equation A., B., C., 3 3 D. 4, 9? Page of 31 McDougal Littell.1.8

11 Sample SAT Question(s): Taken from College Board online practice problems. If 6, which of the following must be true? (A) 6 (B) 3 (C) 0 (D) (E) Page 11 of 31 McDougal Littell.1.8

12 Sllabus Objective:. The student will solve quadratic equations b factoring, graphing, completing the square, and the quadratic formula. The student will solve quadratic equations b finding square roots. Square Root: Radical Sign: Radicand: the number beneath the radical sign Properties of Square Roots a 0, b 0 Product Propert: ab a b Quotient Propert: a b a b Review: Simplifing Square Roots A square root is simplified if the radicand has no perfect square factor (other than 1) and there is no radical in the denominator of a fraction. E: Simplif the square root 7. Method 1 Step One: Find the largest perfect square that is a factor of Step Two: Rewrite 7 as a product using 36 as a factor. 36 Step Three: Rewrite as the product of two radicals. 36 Step Four: Evaluate the square root of the perfect square. 6 Method Step One: Rewrite 7 as a product of prime factors Step Two: Find the square root of each pair of factors. 3 3 E: Simplif the epression We must rationalize the denominator b multipling b 1. 8 Now simplif the radical and the fraction Page 1 of 31 McDougal Littell.1.8

13 Solving a Quadratic Equation b Finding Square Roots E: Solve the equation Step One: Isolate the squared epression Step Two: Find the square root of both sides Step Three: Solve for the variable. 6 or 6 E: Solve the equation n 16. Step One: Isolate the squared epression. 81 n Step Two: Find the square root of both sides. Step Three: Solve for the variable. n 81 n 9 n 9 n 9 n 14 n 4 n 7 n E: Solve the equation a, Step One: Isolate the squared epression. a 8 8 Step Two: Find the square root of both sides. Step Three: Solve for the variable. a 8 8 a 8 7 a8 7 a8 7 a 8 7 a 8 7 a 8 7 Note: The ( plus or minus ) smbol is used to write both solutions in a shorter wa. In set notation, the solutions would be written 8 7,8 7. Page 13 of 31 McDougal Littell.1.8

14 Real-Life Application: Free Fall On Earth, the equation for the height (h) of an object for t seconds after it is dropped can be modeled b the function h 16t h0, where h0 is the initial height of the object. E: A ball is dropped from a height of 81 ft. How long will it take for the ball to hit the ground? Use the free-fall function. h 16t h0 h0 81, h 0 Initial height is 81 ft. The ball will hit the ground when its height is 0 ft. Solve for t. 016t 81 16t t 16 9 t t, 4 4 Solution: Since time is positive, the onl feasible answer is 9 4. seconds You Tr: Solve the equation 7 1. QOD: When is it necessar to simplif a square root? Sample CCSD Common Eam Practice Question(s): Which of the following shows the solutions of 3? A. 8, B., C. 4, 4 D.,8 Sample SAT Question(s): Taken from College Board online practice problems. If 3, then 6 must equal which of the following? 4 (A) (B) (C) (D) (E) 4 4 Page 14 of 31 McDougal Littell.1.8

15 Sllabus Objectives:.4 The student will solve quadratic equations with comple solutions.. The student will perform operations with comple numbers. Imaginar Unit: i 1 Comple Number (in Standard Form): a bi, where a is the real part of the comple number and bi is the imaginar part of the comple number Graphing Comple Numbers in the Comple Plane imaginar E: Plot the comple numbers in the comple plane: C 4 i, D 3i Note: The horizontal ais is the real ais, and the vertical ais is the imaginar ais. C D - - real - Absolute Value of a Comple Number: the distance a comple number is from the origin on the comple plane. imaginar - E: Find the absolute value 3 4i. Graph on the comple plane. Draw the right triangle formed b the point, the origin, and the real ais. - - real The length of the hpotenuse of this triangle is the distance from the point to the origin. So, b the Pthagorean Theorem: i Note: The formula for the absolute value of a comple number, z, is z a b Sum and Difference of Comple Numbers: Add or subtract the real parts and the imaginar parts separatel. E: Find the sum: 3i6 i E: Find the difference: 48i3 i 36ii 37i 43 8ii 1i Page 1 of 31 McDougal Littell.1.8

16 Powers of i: i i i ii i 4 i i i 1 4 i i i i... Note: The pattern continues ever 4 th power of i. E: Evaluate i 8. The eponent of 8 has a remainder of when divided b 4. Therefore, i 8 will be the same as i 1. Product of Comple Numbers: Use the distributive propert or FOIL method to multipl two comple numbers. E: Find the product 3 6i i. Use FOIL: 33i6i 6ii 1 6i30i1i 1 4i i Comple Conjugate: The comple conjugate of a bi is a bi. Quotient of Comple Numbers: To divide two comple numbers, multipl the numerator and denominator b the comple conjugate of the divisor (denominator). E: Find the quotient 3 i. 1 i 3i 1i 1i 1i 36iii 1 4i 311i i 7 11 i Note: The final answer is written in standard form. Page 16 of 31 McDougal Littell.1.8

17 Solving Quadratic Equations with Comple Solutions E: Solve 4 0. Solve b square roots: 4 4 Write the answer(s) in comple form: 4 1 i E: Solve the equation Solve b square roots: Write the answer(s) in comple form: i 8 3i 7 Note: The two solutions written in set notation are 3i 7, 3 i 7. You Tr: 1. Solve the equation Evaluate the following. Write all comple number answers in standard form. b. 3i4 i c. 4 8i 3 i a. 4 6i d. 3 i 4i QOD: Tell whether the statement is true or false, and justif our answer. Ever comple number is an imaginar number. Sample CCSD Common Eam Practice Question(s): Which is the product 8 i6 i A. 0 i B. 48 4i C. 46 i D. 48 0i in standard form? Page 17 of 31 McDougal Littell.1.8

18 Sllabus Objective:. The student will solve quadratic equations b factoring, graphing, completing the square, and the quadratic formula. Review: Factoring a Perfect Square Trinomial a abb ab a abb ab Completing the Square: writing an epression of the form to factor it as a binomial squared b as a perfect square trinomial in order To complete the square of b, we must add b. Teacher Note: Algebra Tiles work well to illustrate completing the square. See Page 81 for an activit. E: Find the value of c such that is a perfect square trinomial. c b, therefore we must add Note: c to complete the square. Solving a Quadratic Equation b Completing the Square E: Solve b completing the square. Step One: Rewrite to make the lead coefficient 1. Step Two: Take the constant term to the other side. b Step Three: Complete the square (add to both sides) Step Four: Factor the perfect square trinomial Step Five: Take the square roots of both sides. Step Si: Solve for the variable. The solution set is 3 11, 3 11 Page 18 of 31 McDougal Littell

19 E: Solve b completing the square. Step One: Rewrite to make the lead coefficient 1. Step Two: Take the constant term to the other side. b Step Three: Complete the square (add to both sides). Step Four: Factor the perfect square trinomial Step Five: Take the square roots of both sides. Step Si: Solve for the variable i 7i 7 i 7i 7i The solution set is 7 i,7 i Verte Form of a Quadratic Function: ah k Verte: hk, E: Write the quadratic function in verte form and identif the verte of 4 7. Step One: Factor out the lead coefficient from the variable terms (if other than one). 7 Step Two: Complete the square. b Note: We must add a to both sides b the distributive propert Step Three: Factor the perfect square trinomial. Step Four: Solve for The verte is 1, 9. Page 19 of 31 McDougal Littell.1.8

20 E: Write the quadratic function in verte form and identif the verte of. Step One: Factor out the lead coefficient (if other than one). Step Two: Complete the square. 4 4 Step Three: Factor the perfect square trinomial. 4 Step Four: Solve for The verte is 17, 4. You Tr: 1. A rectangle has sides and. The area of the rectangle is 0. Use completing the square to find the value of.. Write the quadratic function the function? in verte form. What is the maimum value of QOD: Wh is completing the square helpful when finding the maimum or minimum value of a quadratic function? Sample CCSD Common Eam Practice Question(s): 1. What are the solutions of 6 0? A. = or = 4 B. = or = 4 C. = 3 + i or = 3 i D. = 3 + i or = 3 i Page 0 of 31 McDougal Littell.1.8

21 . Which is one of the appropriate steps in finding solutions for completing the square? 43 0 when A. 4 3 B. 3 C. 4 7 D. 7 Page 1 of 31 McDougal Littell.1.8

22 Sllabus Objective:. The student will solve quadratic equations b factoring, graphing, completing the square, and the quadratic formula..3 The student will analze the nature of the roots of a quadratic equation. Deriving the Quadratic Formula b Completing the Square Solve the quadratic equation a b c 0 b completing the square. Step One: Rewrite so that the lead coefficient is 1. a b c 0 a a a a b c 0 a a Step Two: Take the constant term to the other side. b c a a b Step Three: Complete the square (add to both sides). b b c b a a a a b b 4acb a 4a 4a b b 4ac Step Four: Factor the perfect square trinomial. a 4a b b 4ac a 4a Step Five: Take the square roots of both sides. b b 4ac a 4a b b 4ac b b 4ac a a a a Step Si: Solve for the variable. b b 4ac b b 4ac a a The Quadratic Formula: To solve a quadratic equation in the form b b 4ac. a a b c 0, use the formula Note: To help memorize the quadratic formula, sing it to the tune of the song Pop Goes the Weasel. Page of 31 McDougal Littell.1.8

23 E: Solve the quadratic equation 8 1 using the quadratic formula. Step One: Rewrite in standard form (if necessar) Step Two: Identif a, b, and c. a 1, b8, c 1 Step Three: Substitute the values into the quadratic formula. b b 4ac a Step Four: Simplif. 4 1 The solution set is 4 1,4 1 E: Solve the quadratic equation 3 7 using the quadratic formula. Step One: Rewrite in standard form (if necessar) Step Two: Identif a, b, and c. a, b3, c 7 Step Three: Substitute the values into the quadratic formula. Step Four: Simplif. The solution set is i, i b b 4ac a i i Discriminant: The number under the square root in the quadratic formula. b 4ac The sign of the discriminant determines the number and tpe of solutions of a quadratic equation. If If If b b 4ac 0, then the equation has two real solutions (two -intercepts). 4ac 0, then the equation has one real solution (one -intercept). b 4ac 0, then the equation has two imaginar solutions (no -intercept). Page 3 of 31 McDougal Littell.1.8

24 E: What is the discriminant of the quadratic equation ? Give the number and tpe of solutions the quadratic equation has. Then graph the quadratic function to verif our answer. Discriminant: b ac Since the disciminant is 0, there is one real solution. The -coordinate of the verte of the function is 1 1 The -coordinate of the verte is b 4 1. a 4 Plot a couple of other points to graph the parabola. - - Note that the graph has one -intercept (the verte). - - You Tr: Determine the number and tpe of solutions the quadratic equation has. Then solve the equation using the quadratic formula. 0n 6n 6n 13n 3 QOD: Solve the equation a b c 0 b completing the square. Sample CCSD Common Eam Practice Question(s): 1. How man real and imaginar solutions are there for the equation? 7 0 A. no real solutions, imaginar solutions B. 1 real solution, no imaginar solutions C. 1 real solution, 1 imaginar solution D. real solutions, no imaginar solutions. What is the solution set for the quadratic equation A. 3 3, 3 3 B. 3 6, 3 6 C. 3 3,3 3 D. 3 6,3 6 Page 4 of 31 McDougal Littell ?

25 3. What are the solutions of? A. i 3 6 B. 3 6 C. i D Page of 31 McDougal Littell.1.8

26 Sllabus Objective:.6 The student will graph and solve quadratic inequalities with and without technolog. Graphing a Quadratic Inequalit in Two Variables E: Graph the quadratic inequalit 6. Step One: Graph the parabola. Make the parabola dashed if < or > and solid if or. We will write the inequalit in verte form using completing the square Note: We draw a solid parabola Step Two: Choose a test point inside the parabola and substitute it into the inequalit. We will choose 6 0, true Step Three: If the test point makes the inequalit true, shade inside the parabola. If it does not, shade outside the parabola Graphing Calculator Activit - E: Graph the quadratic inequalit 9 b hand and then check our graph on the graphing calculator using the Inequalz Application. Step One: Graph the parabola. Make the parabola dashed if < or > and solid if or. The verte of the parabola is the point 0,9. Note: We draw a dashed parabola that opens down. Step Two: Choose a test point inside (not on) the parabola and substitute it into the inequalit. We will choose 0, false Step Three: If the test point makes the inequalit true, shade inside the parabola. If it does not, shade outside the parabola. We will shade outside the parabola. Page 6 of 31 McDougal Littell.1.8

27 To check our graph, turn on the application b choosing Inequalz after pressing the APPS ke. Press an ke, and now our Y= screen should look like this: Enter the function 9 into Y1. Then use the command (function) buttons along the bottom of our calculator screen to choose >. Note In order to use the command buttons, ou must first tpe the ALPHA ke. So to choose >, we will press ALPHA TRACE. Graph the inequalit. (For the graph shown, we used ZOOM STANDARD). Solving a Quadratic Inequalit in One Variable E: Solve 6 0. Step One: Solve the quadratic equation We will use factoring. 6 0 using an method Step Two: Draw a sign chart on a number line to test which values for satisf the inequalit. Choose an -value to the left of and substitute into the inequalit. We will tr true Choose an -value between and 3 and substitute into the inequalit. We will tr false Choose an -value to the right of 3 and substitute into the inequalit. We will tr true Step Three: Write the solution as a compound inequalit. or 3 Page 7 of 31 McDougal Littell.1.8

28 You Tr: 1. Graph the quadratic inequalit 3.. Solve the quadratic inequalit QOD: What is the purpose of a sign chart when solving a quadratic inequalit in one variable? Sample CCSD Common Eam Practice Question(s): Which of the following graphs represents the quadratic inequalit 4? Page 8 of 31 McDougal Littell.1.8

29 Sllabus Objective:.7 The student will develop mathematical models involving quadratic equations to solve real-world problems. Writing the Equation of a Quadratic Function in Verte Form E: Write an equation for the parabola in verte form. The verte is at 1, 7 a So the verte form of the equation is To solve for a, we will choose a point on the parabola and substitute it into the equation for,. a 41 7 Choose 4,. 9a 7 1 a Writing the Equation of a Quadratic Function in Standard Form So the verte form of the equation is 1 7. E: Write and equation for the parabola in standard form. - - The -intercepts (zeros) of the parabola are at and 3. So the a 3. intercept form of the quadratic equation is - - To solve for a, we will choose a point on the parabola and substitute it into the equation for,. Choose,. a 3 4a 1 a 1. So the intercept form of the equation is 3 To rewrite in standard form, multipl the binomials and distribute the constant Page 9 of 31 McDougal Littell.1.8

30 Writing the Equation of a Quadratic Function Given Three Points E: Write a quadratic function in standard form for the parabola whose graph passes through the points,,3,4, and 0,. Use the standard form a b c sstem of three equations for a, b, and c.. Substitute each point in for, and solve the remaining a b c 4abc 4 a 3 b 3 c 4 9a3bc a 0 b 0 c c Since c, we can substitute this value into the first two equations. 4ab 04ab Solve the sstem of the remaining two equations. 4 9a3b 69a3b We will use the substitution method. 0 4a b 4a b a b 69a3 a 6 3a a 04 b 8b 4 b Substitute the values for a, b, and c into the standard form equation Graphing Calculator Activit: Using a quadratic model to represent data. a b c. E: The table shows the average sale price p of a house for various ears t since Use a quadratic regression on the graphing calculator to write a quadratic model for the data. Years Since 1988, t Average Sale Price (thousands of dollars), p Enter the data from the table into the Lists. Enter t values into L1 and p values into L. (Use STAT Edit to enter data into the Lists.) On the home screen, use the QuadReg to find the quadratic regression. (Use STAT Calc to find QuadReg.) Note: To store this into Y1, ou can tpe in Y1 after QuadReg on the home screen before pressing enter. Kestrokes for entering Y1: Page 30 of 31 McDougal Littell.1.8

31 Take a look at the graph of the quadratic model with the scatter plot of the data. You Tr: 1. Write the verte form, intercept form, and standard form of the parabola shown in the graph. -. Write the equation of the quadratic function that passes through the points, 1, 1,11, and,7. - QOD: Give three was to find a quadratic model for a set of data points. Sample CCSD Common Eam Practice Question(s): Use the formula below, where h is the height (in feet) of a falling object after t seconds and h 0 is the object s initial height (in feet). h16t h0 A coote is standing on a cliff 64 feet above a roadrunner. The coote drops a boulder from the cliff. How much time does the roadrunner have to move out of its wa? A. 1 4 second B. 1 second C. seconds D. 4 seconds Sample SAT Question(s): Taken from College Board online practice problems. If and 8, what is the value of? (A) 1 (B) (C) 4 (D) 8 (E) 16 Page 31 of 31 McDougal Littell.1.8

Sllabus Objective: 9.4 The student will solve quadratic equations using graphic and algebraic techniques to include the quadratic formula, square roots, factoring, completing the square, and graphing.

### Algebra 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

### Unit 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

### x (vertex is halfway between the x-intercepts)

Big Idea: A quadratic equation in the form a b c 0 has a related function f ( ) a b c. The zeros of the function are the -intercepts of its graph. These -values are the solutions or roots of the related

### Algebra 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

### Chapter 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

### Algebra 2 Semester Exam Review

Algebra Semester Eam Review 7 Graph the numbers,,,, and 0 on a number line Identif the propert shown rs rs r when r and s Evaluate What is the value of k k when k? Simplif the epression 7 7 Solve the equation

### Glossary. Also available at BigIdeasMath.com: multi-language glossary vocabulary flash cards. An equation that contains an absolute value expression

Glossar This student friendl glossar is designed to be a reference for ke vocabular, properties, and mathematical terms. Several of the entries include a short eample to aid our understanding of important

### Learning Goals. College of Charleston Department of Mathematics Math 101: College Algebra Final Exam Review Problems 1

College of Charleston Department of Mathematics Math 0: College Algebra Final Eam Review Problems Learning Goals (AL-) Arithmetic of Real and Comple Numbers: I can classif numbers as natural, integer,

### Ch 5 Alg 2 L2 Note Sheet Key Do Activity 1 on your Ch 5 Activity Sheet.

Ch Alg L Note Sheet Ke Do Activit 1 on our Ch Activit Sheet. Chapter : Quadratic Equations and Functions.1 Modeling Data With Quadratic Functions You had three forms for linear equations, ou will have

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

### f(x) = 2x 2 + 2x - 4

4-1 Graphing Quadratic Functions What You ll Learn Scan the tet under the Now heading. List two things ou will learn about in the lesson. 1. Active Vocabular 2. New Vocabular Label each bo with the terms

### Algebra 2 Honors Summer Packet 2018

Algebra Honors Summer Packet 018 Solving Linear Equations with Fractional Coefficients For these problems, ou should be able to: A) determine the LCD when given two or more fractions B) solve a linear

### Section 5.5 Complex Numbers

Name: Period: Section 5.5 Comple Numbers Objective(s): Perform operations with comple numbers. Essential Question: Tell whether the statement is true or false, and justify your answer. Every comple number

### Ready To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions

Read To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions Find these vocabular words in Lesson 5-1 and the Multilingual Glossar. Vocabular quadratic function parabola verte

### MATH 0312 FINAL EXAM REVIEW ITEMS

MATH 012 FINAL EXAM REVIEW ITEMS Name The items on this review are representative of the items that ou might see on our course final eam. No formul sheets are allowed and calculators are not allowed on

### math FALL developmental mathematics sullivan 1e

TSIpractice eam review 1 131 180 plus 34 TSI questions for elementar and intermediate algebra m0300004301 aaa Name www.alvarezmathhelp.com math0300004301 FALL 01 100 interactmath developmental mathematics

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

### review math0410 (1-174) and math 0320 ( ) aafinm mg

Eam Name review math04 (1-174) and math 0320 (17-243) 03201700aafinm0424300 mg MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Simplif. 1) 7 2-3 A)

### Math 030 Review for Final Exam Revised Fall 2010 RH/ DM 1

Math 00 Review for Final Eam Revised Fall 010 RH/ DM 1 1. Solve the equations: (-1) (7) (-) (-1) () 1 1 1 1 f. 1 g. h. 1 11 i. 9. Solve the following equations for the given variable: 1 Solve for. D ab

### Lesson 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

Eam Name algebra final eam review147 aam032020181t4highschool www.alvarezmathhelp.com MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the equation.

### math0320 FALL interactmath sections developmental mathematics sullivan 1e

Eam final eam review 180 plus 234 TSI questions for intermediate algebra m032000 013014 NEW Name www.alvarezmathhelp.com math0320 FALL 201 1400 interactmath sections developmental mathematics sullivan

### Algebra 2 CPA Summer Assignment 2018

Algebra CPA Summer Assignment 018 This assignment is designed for ou to practice topics learned in Algebra 1 that will be relevant in the Algebra CPA curriculum. This review is especiall important as ou

### APPLIED ALGEBRA II SEMESTER 1 EXAM ITEM SPECIFICATION SHEET & KEY

APPLIED ALGEBRA II SEMESTER 1 EXAM ITEM SPECIFICATION SHEET & KEY Constructed Response # Objective Sllabus Objective NV State Standard 1 Graph a polnomial function. 1.1.7.1 Analze graphs of polnomial functions

### FINAL EXAM REVIEW ITEMS Math 0312: Intermediate Algebra Name

FINAL EXAM REVIEW ITEMS Math 0312: Intermediate Algebra Name 1) Find the SUM of the solutions of the equation. 82 + 0 = 16 Use the quadratic formula to solve the equation. (All solutions are real numbers.)

### MATH College Algebra Review for Test 2

MATH 4 - College Algebra Review for Test Sections. and.. For f (x) = x + 4x + 5, give (a) the x-intercept(s), (b) the -intercept, (c) both coordinates of the vertex, and (d) the equation of the axis of

### REVIEW KEY VOCABULARY REVIEW EXAMPLES AND EXERCISES

Etra Eample. Graph.. 6. 7. (, ) (, ) REVIEW KEY VOCABULARY quadratic function, p. 6 standard form of a quadratic function, p. 6 parabola, p. 6 verte, p. 6 ais of smmetr, p. 6 minimum, maimum value, p.

### Math Intermediate Algebra

Math 095 - Intermediate Algebra Final Eam Review Objective 1: Determine whether a relation is a function. Given a graphical, tabular, or algebraic representation for a function, evaluate the function and

### H.Algebra 2 Summer Review Packet

H.Algebra Summer Review Packet 1 Correlation of Algebra Summer Packet with Algebra 1 Objectives A. Simplifing Polnomial Epressions Objectives: The student will be able to: Use the commutative, associative,

### INTRODUCTION 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

### Study Guide and Intervention

6- NAME DATE PERID Stud Guide and Intervention Graphing Quadratic Functions Graph Quadratic Functions Quadratic Function A function defined b an equation of the form f () a b c, where a 0 b Graph of a

### The American School of Marrakesh. Algebra 2 Algebra 2 Summer Preparation Packet

The American School of Marrakesh Algebra Algebra Summer Preparation Packet Summer 016 Algebra Summer Preparation Packet This summer packet contains eciting math problems designed to ensure our readiness

### 3 Polynomial and Rational Functions

3 Polnomial and Rational Functions 3.1 Quadratic Functions and Models 3.2 Polnomial Functions and Their Graphs 3.3 Dividing Polnomials 3.4 Real Zeros of Polnomials 3.5 Comple Zeros and the Fundamental

### Algebra 2 Unit 2 Practice

Algebra Unit Practice LESSON 7-1 1. Consider a rectangle that has a perimeter of 80 cm. a. Write a function A(l) that represents the area of the rectangle with length l.. A rectangle has a perimeter of

### Polynomial and Rational Functions

Name Date Chapter Polnomial and Rational Functions Section.1 Quadratic Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Important Vocabular Define

### LESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II

1 LESSON #8 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The

### d. 2x 3 7x 2 5x 2 2x 2 3x 1 x 2x 3 3x 2 1x 2 4x 2 6x 2 3. a. x 5 x x 2 5x 5 5x 25 b. x 4 2x 2x 2 8x 3 3x 12 c. x 6 x x 2 6x 6 6x 36

Vertices: (.8, 5.), (.37, 3.563), (.6, 0.980), (5.373, 6.66), (.8, 7.88), (.95,.) Graph the equation for an value of P (the second graph shows the circle with P 5) and imagine increasing the value of P,

### First Semester Final Review NON-Graphing Calculator

Algebra First Semester Final Review NON-Graphing Calculator Name:. 1. Find the slope of the line passing through the points ( 5, ) and ( 3, 7).. Find the slope-intercept equation of the line passing through

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

(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

### BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE. MTH 05 Review Sheet

BRONX COMMUNITY COLLEGE of the Cit Universit of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE MTH 05 Review Sheet Go to http://www.cun.edu/testing for more information on the CUNY Elementar Algebra

### MAT 1033C -- Martin-Gay Intermediate Algebra Chapter 8 (8.1, 8.2, 8.5, 8.6) Practice for the Exam

MAT 33C -- Martin-Ga 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.

### Course 15 Numbers and Their Properties

Course Numbers and Their Properties KEY Module: Objective: Rules for Eponents and Radicals To practice appling rules for eponents when the eponents are rational numbers Name: Date: Fill in the blanks.

### Graphing Calculator Computations 2

Graphing Calculator Computations A) Write the graphing calculator notation and B) Evaluate each epression. 4 1) 15 43 8 e) 15 - -4 * 3^ + 8 ^ 4/ - 1) ) 5 ) 8 3 3) 3 4 1 8 3) 7 9 4) 1 3 5 4) 5) 5 5) 6)

### Name Period Date. Practice FINAL EXAM Intro to Calculus (50 points) Show all work on separate sheet of paper for full credit!

Name Period Date Practice FINAL EXAM Intro to Calculus (0 points) Show all work on separate sheet of paper for full credit! ) Evaluate the algebraic epression for the given value or values of the variable(s).

### Review: Properties of Exponents (Allow students to come up with these on their own.) m n m n. a a a. n n n m. a a a. a b a

Algebra II Notes Unit Si: Polynomials Syllabus Objectives: 6. The student will simplify polynomial epressions. Review: Properties of Eponents (Allow students to come up with these on their own.) Let a

### Chapter 8 Vocabulary Check

28 CHAPTER 8 Quadratic Equations and Functions d. What is the level of methane emissions for that ear? (Use our rounded answer from part (c).) (Round this answer to 2 decimals places.) Use a graphing calculator

### Reteaching (continued)

Quadratic Functions and Transformations If a, the graph is a stretch or compression of the parent function b a factor of 0 a 0. 0 0 0 0 0 a a 7 The graph is a vertical The graph is a vertical compression

### 2.1 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

### MATH College Algebra Review for Test 2

MATH 34 - College Algebra Review for Test 2 Sections 3. and 3.2. For f (x) = x 2 + 4x + 5, give (a) the x-intercept(s), (b) the -intercept, (c) both coordinates of the vertex, and (d) the equation of the

### MATH 60 Review Problems for Final Exam

MATH 60 Review Problems for Final Eam Scientific Calculators Onl - Graphing Calculators Not Allowed NO CLASS NOTES PERMITTED Evaluate the epression for the given values. m 1) m + 3 for m = 3 2) m 2 - n2

### Unit 10 - Graphing Quadratic Functions

Unit - Graphing Quadratic Functions PREREQUISITE SKILLS: students should be able to add, subtract and multipl polnomials students should be able to factor polnomials students should be able to identif

### review for math TSI 55 practice aafm m

Eam TSI Name review for math TSI practice 01704041700aafm042430m www.alvarezmathhelp.com MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the

### + = + + = x = + = + = 36x

Ch 5 Alg L Homework Worksheets Computation Worksheet #1: You should be able to do these without a calculator! A) Addition (Subtraction = add the opposite of) B) Multiplication (Division = multipl b the

### review for math TSI 182 practice aafm m

Eam TSI 182 Name review for math TSI 182 practice 01704041700aafm042430m www.alvarezmathhelp.com MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Simplif.

### Ready To Go On? Skills Intervention 6-1 Polynomials

6A Read To Go On? Skills Intervention 6- Polnomials Find these vocabular words in Lesson 6- and the Multilingual Glossar. Vocabular monomial polnomial degree of a monomial degree of a polnomial leading

### C) x m A) 260 sq. m B) 26 sq. m C) 40 sq. m D) 364 sq. m. 7) x x - (6x + 24) = -4 A) 0 B) all real numbers C) 4 D) no solution

Sample Departmental Final - Math 46 Perform the indicated operation. Simplif if possible. 1) 7 - - 2-2 + 3 2 - A) + - 2 B) - + 4-2 C) + 4-2 D) - + - 2 Solve the problem. 2) The sum of a number and its

### Algebra 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

Name: Date: QUADRATIC FUNCTION REVIEW Linear and eponential functions are used throughout mathematics and science due to their simplicit and applicabilit. Quadratic functions comprise another ver important

### SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Spring 0 Math 08 Eam Preparation Ch Dressler Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the quadratic equation b the square root propert.

### LESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II

1 LESSON #4 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The

### Algebra I Notes Unit Eleven: Polynomials

Syllabus Objective: 9.1 The student will add, subtract, multiply, and factor polynomials connecting the arithmetic and algebraic processes. Teacher Note: A nice way to illustrate operations with polynomials

### PreCalculus. Ocean Township High School Mathematics Department

PreCalculus Summer Assignment Name Period Date Ocean Township High School Mathematics Department These are important topics from previous courses that ou must be comfortable doing before ou can be successful

### Algebra Final Exam Review Packet

Algebra 1 00 Final Eam Review Packet UNIT 1 EXPONENTS / RADICALS Eponents Degree of a monomial: Add the degrees of all the in the monomial together. o Eample - Find the degree of 5 7 yz Degree of a polynomial:

### = 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

### Unit 11 - Solving Quadratic Functions PART TWO

Unit 11 - Solving Quadratic Functions PART TWO PREREQUISITE SKILLS: students should be able to add, subtract and multiply polynomials students should be able to factor polynomials students should be able

### COUNCIL ROCK HIGH SCHOOL MATHEMATICS. A Note Guideline of Algebraic Concepts. Designed to assist students in A Summer Review of Algebra

COUNCIL ROCK HIGH SCHOOL MATHEMATICS A Note Guideline of Algebraic Concepts Designed to assist students in A Summer Review of Algebra [A teacher prepared compilation of the 7 Algebraic concepts deemed

### Math 100 Final Exam Review

Math 0 Final Eam Review Name The problems included in this review involve the important concepts covered this semester. Work in groups of 4. If our group gets stuck on a problem, let our instructor know.

### Algebra II Notes Unit Nine: Rational Equations and Functions

Syllabus Objectives: 9. The student will solve a problem by applying inverse and joint variation. 9.6 The student will develop mathematical models involving rational epressions to solve realworld problems.

### Algebra 1 Skills Needed for Success in Math

Algebra 1 Skills Needed for Success in Math A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed to simplif

### Mini-Lecture 8.1 Solving Quadratic Equations by Completing the Square

Mini-Lecture 8.1 Solving Quadratic Equations b Completing the Square Learning Objectives: 1. Use the square root propert to solve quadratic equations.. Solve quadratic equations b completing the square.

### Writing Quadratic Functions in Standard Form

Chapter Summar Ke Terms standard form (general form) of a quadratic function (.1) parabola (.1) leading coefficient (.) second differences (.) vertical motion model (.3) zeros (.3) interval (.3) open interval

### Review for Intermediate Algebra (MATD 0390) Final Exam Oct 2009

Review for Intermediate Algebra (MATD 090) Final Eam Oct 009 Students are epected to know all relevant formulas, including: All special factoring formulas Equation of a circle All formulas for linear equations

### Answer 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

### a 2 x y 1 y SOL AII.1a

SOL AII.a The student, given rational, radical, or polnomial epressions, will a) add, subtract, multipl, divide, and simplif rational algebraic epressions; Hints and Notes Rules for fractions: ) Alwas

### Final Exam Review Part 2 #1 Page 1 / 21

Final Eam Review Part #1 Intermediate Algebra / MAT 135 Spring 017 Master ( Master Templates) Student Name/ID: v 1. Solve for, where is a real number. v v + 1 + =. Solve for, where is a real number. +

### Keira Godwin. Time Allotment: 13 days. Unit Objectives: Upon completion of this unit, students will be able to:

Keira Godwin Time Allotment: 3 das Unit Objectives: Upon completion of this unit, students will be able to: o Simplif comple rational fractions. o Solve comple rational fractional equations. o Solve quadratic

### Copyrighted 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

### Northwest High School s Algebra 2/Honors Algebra 2

Northwest High School s Algebra /Honors Algebra Summer Review Packet 0 DUE Frida, September, 0 Student Name This packet has been designed to help ou review various mathematical topics that will be necessar

### Find the distance between the pair of points. 2) (7, -7) and (3, -5) A) 12 3 units B) 2 5 units C) 6 units D) 12 units B) 8 C) 63 2

Sample Departmental Final - Math 9 Write the first five terms of the sequence whose general term is given. 1) a n = n 2 - n 0, 2,, 12, 20 B) 2,, 12, 20, 30 C) 0, 3, 8, 1, 2 D) 1,, 9, 1, 2 Find the distance

### QUADRATIC FUNCTIONS AND COMPLEX NUMBERS

CHAPTER 86 5 CHAPTER TABLE F CNTENTS 5- Real Roots of a Quadratic Equation 5-2 The Quadratic Formula 5-3 The Discriminant 5-4 The Comple Numbers 5-5 perations with Comple Numbers 5-6 Comple Roots of a

### a 2 x y 1 x 1 y SOL AII.1a

SOL AII.a The student, given rational, radical, or polnomial epressions, will a) add, subtract, multipl, divide, and simplif rational algebraic epressions; Hints and Notes Rules for fractions: ) Alwas

### Write each expression in terms of i : Add: (3 4i) (5 7i) (3 5) ( 4 7)i. 8 3i. Subtract: (3 4i) (5 7i) (3 4i) ( 5 7i) Find each product:

7_Ch09_online 7// 0:7 AM Page 9-0 9-0 CHAPTER 9 Quadratic Equations SECTION 9. Comple Numbers DEFINITIONS AND CONCEPTS EXAMPLES The imaginar number i is defined as Write each epression in terms of i :

Secondar Mathematics Table of Contents Unit 1: Etending the Number Sstem Cluster 1: Etending Properties of Eponents (N.RN.1 and N.RN.)... 3 Cluster : Using Properties of Rational and Irrational Numbers

### UNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives

Chapter 3 3Quadratics Objectives To recognise and sketch the graphs of quadratic polnomials. To find the ke features of the graph of a quadratic polnomial: ais intercepts, turning point and ais of smmetr.

### Mth 95 Module 4 Chapter 8 Spring Review - Solving quadratic equations using the quadratic formula

Mth 95 Module 4 Chapter 8 Spring 04 Review - Solving quadratic equations using the quadratic formula Write the quadratic formula. The NUMBER of REAL and COMPLEX SOLUTIONS to a quadratic equation ( a b

### 2 nd Semester Final Exam Review Block Date

Algebra 1B Name nd Semester Final Eam Review Block Date Calculator NOT Allowed Graph each function. Identif the verte and ais of smmetr. 1 (10-1) 1. (10-1). 3 (10-) 3. 4 7 (10-) 4. 3 6 4 (10-1) 5. Predict

### Review of Essential Skills and Knowledge

Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope

### Chapter 9 Notes Alg. 1H 9-A1 (Lesson 9-3) Solving Quadratic Equations by Finding the Square Root and Completing the Square

Chapter Notes Alg. H -A (Lesson -) Solving Quadratic Equations b Finding the Square Root and Completing the Square p. *Calculator Find the Square Root: take the square root of. E: Solve b finding square

### California State University, Northridge

California State Universit, Northridge MATH 09 HYBRID WORKBOOKS Spring 00 Chapter Equations, Inequalities and Applications. The Addition Propert of Equalit Learning Objectives:. Use the Addition Propert

### MATH 91 Final Study Package Name

MATH 91 Final Stud Package Name Solve the sstem b the substitution method. If there is no solution or an infinite number of solutions, so state. Use set notation to epress the solution set. 1) - = 1 1)

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Math 2 Stud Guide-Chapters 8 and 9 Name Date: Time: MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find all square roots of the number. ) 600 9,

### Answers. Chapter Warm Up. Sample answer: The graph of h is a translation. 3 units right of the parent linear function.

Chapter. Start Thinking As the string V gets wider, the points on the string move closer to the -ais. This activit mimics a vertical shrink of a parabola... Warm Up.. Sample answer: The graph of f is a

Honors Algebra -3 Solving Quadratic Equations by Graphing and Factoring Learning Targets 1. I can solve quadratic equations by graphing. I can solve quadratic equations by factoring 3. I can write a quadratic

### 2 nd Semester Final Exam Review Block Date

Algebra 1B Name nd Semester Final Eam Review Block Date Calculator NOT Allowed Graph each function. 1 (10-1) 1. (10-1). (10-1) 3. (10-1) 4. 3 Graph each function. Identif the verte, ais of smmetr, and

### 12x y (4) 2x y (4) 5x y is the same as

Name: Unit #6 Review Quadratic Algebra Date: 1. When 6 is multiplied b the result is 0 1 () 9 1 () 9 1 () 1 0. When is multiplied b the result is 10 6 1 () 7 1 () 7 () 10 6. Written without negative eponents

### Maintaining Mathematical Proficiency

Chapter Maintaining Mathematical Proficienc Find the -intercept of the graph of the linear equation. 1. = + 3. = 3 + 5 3. = 10 75. = ( 9) 5. 7( 10) = +. 5 + 15 = 0 Find the distance between the two points.

### Name Please print your name as it appears on the class roster.

Berkele Cit College Practice Problems Math 1 Precalculus - Final Eam Preparation Name Please print our name as it appears on the class roster. SHORT ANSWER. Write the word or phrase that best completes