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


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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 Ushaped 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 (tchart) 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 (tchart) 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? GridIn 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 RealLife 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 freefall 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 realworld 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
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