College Prep Math Notes Quadratics Unit Quadratics. Math Background

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1 Quadratics Math Background Previousl, ou Identified and graphed quadratic functions in Algebra II Applied transformations to parent functions Solved quadratic functions in Algebra II Worked with comple number operations in Algebra II In this unit ou will Review graphing quadratic functions, with and without technolog Review solving quadratic functions using graphing, factoring, completing the square and the quadratic formula Review operations with comple numbers You can use the skills in this unit to Predict the number and nature of the roots using the discriminant. Manipulate comple numbers including finding the quotients of comple numbers. Solve quadratic equations that have comple solutions. Graph solutions of single variable and two variable quadratic inequalities. Vocabular Ais of smmetr A line that passes through a figure in such a wa that the part of the figure on one side of the line is a mirror reflection of the part on the other side of the line. Completing the square A wa of simplifing or solving a quadratic equation b adding an epression to both sides to make one part of the equation a perfect square. Comple number An number that can be written as a+bi, where a and b are real numbers and i 1. Comple solutions The -values that make the function equal to zero are comple numbers (have an imaginar part). Discriminant The epression b 4ac in a quadratic equation a b c 0. It can be used to determine the characteristics of the solution. Domain The input values of the function. For a quadratic function, the domain is all real numbers. Maima The largest value (highest point) of the function. Minima The smallest value (lowest point) of the function. Parabola The U-shaped graph of a quadratic function. Parent Quadratic function The simplest quadratic function,. Quadratic Equation A polnomial equation in which the highest power of the variable is two. The general form of such equations in the variable is a b c 0. Quadratic Formula The formula for determining the roots of a quadratic equation from its coefficients. Quadratic function A function where the highest eponent of the variable is a square. Range The output values of a function. Reflection A transformation in which ever point of a figure is mapped to a corresponding image across a line of smmetr. CPM Notes Unit Quadratics Page 1 of 8 4/15/015 f ( )

2 Transformation A change in the position, size, or shape of a figure or graph. Translation A figure is moved from one location to another on the coordinate plane without changing its size, shape or orientation. Verte The point of intersection of a parabola and its line of smmetr. The minima or maima of the quadratic function. Zeros of a Function The -value or -values that make the function equal to zero. A zero ma be a real number or a comple number. Essential Questions What are the ke features of the graphs of quadratic functions? What are the different was to solve quadratic equations and which was are more efficient? Do all quadratics have real solutions? What is a comple number? What is the purpose for a comple number? When do ou use an equation versus an inequalit? Overall Big Ideas A quadratic function is represented b a U-shaped curve, called a parabola. It intercepts one or both aes and has one maimum or minimum value. Solving quadratic functions can be done b different techniques like square rooting, factoring, completing the square, graphing and using the quadratic formula. Some techniques are more efficient than others based on the characteristics of the quadratic function. Some quadratics do not have real solutions. Changing the form of the epression b factoring reveals important attributes about the function and its graph. Comple numbers epand the number sstem to include square roots of negative numbers and allows applications of comple numbers to electronics. We use the properties of operations as it applies to comple numbers to simplif epressions and to build foundations to solve quadratic equations having comple solutions. Variable equations or inequalities model real-life situations and generalize applications, building a foundation for solving equations and inequalities with more than one variable. CPM Notes Unit Quadratics Page of 8 4/15/015

3 Skill To graph quadratic functions in both standard and verte form, with and without technolog. To solve quadratic equations b graphing, factoring, completing the square, and the quadratic formula. To predict and analze the nature of the roots of a quadratic using the discriminant. To identif, simplif, and perform operations with comple numbers. To solve quadratic equations with comple numbers. To solve and graph solutions of single variable and two variable quadratic inequalities. Related Standards A.APR.B. Know and appl the Remainder Theorem: For a polnomial p( ) and a number a, the remainder on division b a is p( a ), so pa ( ) 0if and onl if ( a) is a factor of p( ). A.APR.B.3 Identif zeros of polnomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined b the polnomial. F.IF.A.1 Understand that a function from one set (called the domain) to another set (called the range) assign to each element of the domain eactl one element of the range. If f is a function and is an element of its domain, then f() denotes the output of f corresponding to the input. The graph of f is the graph of the equation =f(). F.IF.A. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a contet. F.IF.C.7a Graph linear and quadratic functions and show intercepts, maima, and minima. *(Modeling Standard) A.SSE.B.3a Factor a quadratic epression to reveal the zeros of the function it defines. *(Modeling Standard) A.SSE.B.3b Complete the square in a quadratic epression to reveal the maimum or minimum value of the function it defines. CPM Notes Unit Quadratics Page 3 of 8 4/15/015

4 A.REI.B.4a Use the method of completing the square to transform an quadratic equation in into an equation of the form( - p) = q that has the same solutions. Derive the quadratic formula from this form. A.REI.B.4b Solve quadratic equations b inspection (e.g., for = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives comple solutions and write them as a bifor real numbers a and b. F.IF.C.8a Use the process of factoring and completing the square in a quadratic function to show zeros, etreme values, and smmetr of the graph, and interpret these in terms of a contet. N.CN.A.1 Know there is a comple number i such that i 1, and ever comple number has the form a+bi with a and b real. N.CN.A. Use the relationi 1and the commutative, associative, and distributive properties to add, subtract, and multipl comple numbers. N.CN.A.3 Find the conjugate of a comple number; use conjugates to find moduli and quotients of comple numbers. N.CN.C.7 Solve quadratic equations with real coefficients that have comple solutions. A.CED.A.1-1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic, and eponential functions. Limit eponentials to have integer inputs onl. *(Modeling Standard) A.CED.A.-1 Create linear, eponential, and quadratic equations in two or more variables to represent relationships between quantities and graph equations on coordinate aes with labels and scales. Limit eponentials to have integer inputs onl. *(Modeling Standard) CPM Notes Unit Quadratics Page 4 of 8 4/15/015

5 Notes, Eamples, and Eam Questions Parent Function: The most basic quadratic function is f ( ). This function is often called the parent quadratic function. Once ou understand the parent function ou can shift, reflect, and stretch the parent graph to get graphs of other quadratic functions. The graph of the function f( ) is the set of all points, f( ), is in the domain of f. We match domain values along the -ais with their range values along the -ais to get the ordered pairs that ield the graph of f( ). Unit 1.1 To graph quadratic functions in both standard and verte form, with and without technolog. NOTE: This unit is a review unit. Unit 3, Quadratic Functions, is a five week unit covered in the first quarter of Algebra II. Most of these topics are not new and should be treated as review. Division with comple numbers and solving and graphing two variable quadratic inequalities are the onl new topics in this unit. Graphing a Quadratic Function in Standard Form: Standard Form of a Quadratic Function a b c, when a 0 ; a, b, and c are real numbers b b Verte: the verte is the point, f a a Ais of Smmetr: b -intercept: c a Minimum Value: When a parabola opens upward, the -value of the verte is the minimum value. Maimum Value: When a parabola opens downward, the -value of the verte is the maimum value. Ais of Smmetr: It is the vertical line that passes through the verte of a quadratic function Verte Maimum Ais of Smmetr Verte Minimum Ais of Smmetr D:{ } R :{ k} The domain is all real numbers The range is all values greater than or equal to the minimum. D:{ } R :{ k} The domain is all real numbers The range is all values less than or equal to the maimum. CPM Notes Unit Quadratics Page 5 of 8 4/15/015

6 First, find the verte: - list a =, b =, c = If a is positive, it opens up, if negative, it opens down. b - find = a - plug this -value into the function - this point (, ) is the verte of the parabola Second, find the ais of smmetr: b It is the line =. a Third, graph the points: - put the verte ou found on the graph - graph the ais of smmetr - plot two points on one side of the ais of smmetr. - Use smmetr to plot two more points on the opposite side. - graph all 5 points Fourth: - Draw a parabola through the points E 1: Graph the quadratic function 6 1. State the verte and ais of smmetr. b Step One: a = 1, b = -6, c = -1. The parabola opens up. Find, the -coordinate of the verte. a (3) 110 verte: (3,-10) Step Two: Find the ais of smmetr. The ais of smmetr is the line 3 Step Three: Pick two points on one side of the verte. Let = 1. Substituting into the function, we get = -6, or (1, -6). Let = 0. Since c 1, the -intercept is 1, so we have the point (0, -1). Using smmetr, notice that (1, -6) is units left of the ais of smmetr. The point on the parabola smmetrical to (1, -6) is units to the right of the ais at (5, -6). Notice that (0, 1) is 3 units left of the ais of smmetr. The point on the parabola smmetrical to (0, 1) is 3 units to the right of the ais at (6, 1). Step Four: Connect points with a smooth curve to draw the parabola. 10 Verte: 3, 10 Ais of Smmetr: CPM Notes Unit Quadratics Page 6 of 8 4/15/015

7 E : Graph the quadratic function State the verte and ais of smmetr. b Step One: a = -3, b = 1, c = -6. The parabola opens down. Find, the -coordinate of the verte. a () 6 6 verte: (,6) Step Two: Find the ais of smmetr. The ais of smmetr is the line Step Three: Pick two points on one side of the verte. Let = 1. Substituting into the function, we get = 3, or (1, 3). Let = 0. Since c = -6, the -intercept is -6, so we have the point (0, -6). Using smmetr, notice that (1, -6) is 1 unit left of the ais of smmetr. The point on the parabola smmetrical to (1, -6) is 1 unit to the right of the ais at (3, 3). Notice that (0, 6) is units left of the ais of smmetr. The point on the parabola smmetrical to (0, 6) is units to the right of the ais at (4, 6). Step Four: Connect points with a smooth curve to draw the parabola. Verte:,6 Ais of Smmetr: Graphing a Quadratic Function in Verte Form: a( h) k a indicates a reflection across the -ais and/or a vertical stretch or compression. h indicates a horizontal translation k indicates a vertical translation Ais of Smmetr h Verte is ( hk, ) If a is positive the parabola opens up. If a is negative the parabola opens down. Vertical Stretch: a 1 Vertical Shrink: 0 a 1 Horizontal Translation: h 0 moves left h 0 moves right Vertical Translation: k 0 moves down k 0 moves up CPM Notes Unit Quadratics Page 7 of 8 4/15/015

8 State the verte and ais of smmetr. 1 Step One: Determine if the graph opens up or opens down. Because a is the graph opens down. Step Two: 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 Three: Plot the verte and sketch the graph. Notice the parent graph is reflected over the ais, is wider (vertical shrink), shifted to the left 3 (opposite of what ou think), and is translated up four units. E 3: Graph the quadratic function Optional: Make a table of values. When choosing -values for the T-table, 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.) Five to seven points will give a nice graph of the parabola E 4: Graph the quadratic function 1 3. State the verte and ais of smmetr. Step One: Determine if the graph opens up or opens down. Because a is the graph opens up. Step Two: Identif the verte and ais of smmetr. The verte is 1, 3 and the ais of smmetr is 1. Step Three: Plot the verte and sketch the graph. Notice the parent graph is narrower (vertical stretch), shifted to the right 1and is translated up three units. Optional: CPM Notes Unit Quadratics Page 8 of 8 4/15/015

9 E 5: Use technolog to graph quadratic functions. a) 6 5 Kestrokes: b) 3( ) 6 Kestrokes: SAMPLE EXAM QUESTIONS 1. Find the verte of the parabola and determine whether the parabola opens up or down. ( ) (A) (0, -), up (B) (, -), down (C) (, ), down (D) (-, 0), up Ans: B. Find the coordinates of the verte of (A) (-5, -4) (B) (-4, -5) (C) (5, 4) (D) (4, 5) Ans: D CPM Notes Unit Quadratics Page 9 of 8 4/15/015

10 3. Graph the quadratic function f() = a. 5 c b. 5 d Ans: C 4. Which of the following parabolas will have a maimum rather than a minimum? (A) ( ) (B) 3 (C) 5 10 (D) ( )(3 ) Ans: B 5. What is the equation of the ais of smmetr of the parabola with equation ( 3) 4? (A) = 3 (B) = (C) = 4 (D) = -3 Ans: D CPM Notes Unit Quadratics Page 10 of 8 4/15/015

11 6. Find the domain and range of the function: f ( ) ( ) 5 (A) Domain: (, ); range: 5, (B) Domain: ( 5, ); range: (, ) (C) Domain: (5, ); range: (, ) (D) Domain: (, ); range: 5, Ans: D 7. Identif the verte of. a. (, ) c. (14, ) b. (, 8) d. (14, 8) Ans: A 8. Use this description to write the quadratic function in verte form: The parent function is verticall stretched b a factor of 3 and translated 8 units right and 1 unit down. a. c. b. d. Ans: C 9. What is the maimum of the quadratic function f ( ) 4 6? A. f 1 C. f 3 B. f 6 D. f 8 Ans: D 10. Consider. What are its verte and -intercept? a. verte: (, ), -intercept: (0, ) c. verte: (1, 1), -intercept: (0, ) b. verte: (, ), -intercept: (0, ) d. verte: (, 1), -intercept: (0, ) Ans: A CPM Notes Unit Quadratics Page 11 of 8 4/15/015

12 11. Which graph represents f 1? Ans: C Unit 1. To solve quadratic equations b graphing, factoring, completing the square, and the quadratic formula. 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 quadratic equation. A quadratic equation can have one real solution, two real solutions, or no real solutions. Note that the three words - zeros, roots and -intercepts - all represent the same thing the -values that make the function equal to zero. Graph the function and find where the graph crosses the -ais to solve the quadratic equation. Using a Graphing Calculator to Solve Quadratic Equations E 6: Approimate the solution(s) of 1 4 using a graphing calculator. Step One: Write the equation in the form 0. a b c 41 0 Step Two: Graph the function a b c. 4 1 Press Y, plug in the equation and then hit ZOOM 6 for a nice window for the graph. Step Three: Find the zero(s) of the function. Press nd TRACE to find the zeros. To find each zero, make sure the cursors are to the left and right of the zero. 4.36,0.36 CPM Notes Unit Quadratics Page 1 of 8 4/15/015

13 Factoring Factoring Method: Write the equation in standard form: a b c 0 1. Factor the quadratic epression.. Use the Zero Product Propert. This propert states that if the product of two factors is zero, then one or both of the factors must equal zero. Set each factor equal to zero. 3. Solve each corresponding linear equation for the variable. 4. The solutions to a quadratic equation are roots. The roots of an equation are the value(s) of the variable that make the equation true. E 7: Solve the equation Step One: Write the equation in standard form. Step Two: Factor the quadratic using the ac method. Step Three: Set each factor equal to zero and solve ac4 b and The solutions can be written in set notation: 1 7, 3 E 8: Solve the equation w w. Step One: Write the equation in standard form. Step Two: Factor the quadratic. It is a perfect square trinomial. 4w 8w49 0 w 8w7 w Note: 7 8w w 7 0 Step Three: Set each factor equal to zero and solve. w CPM Notes Unit Quadratics Page 13 of 8 4/15/015

14 Completing the Square If the equation is not factorable, a method called completing the square can be used to rewrite the equation so that the trinomial is a perfect square and can be factored. E 9: 1. Make sure the leading coefficient is ONE. If it is not, DIVIDE the entire equation b the leading coefficient, a.. Isolate the variable terms a +b. Place the constant on a side b itself. 3. Find b/ and ADD its square to both sides. 4. Factor the perfect square. 5. Take the square root of both sides. 6. Solve the resulting equation. E 10: Solve Solve 14 0 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, b completing the square. Step One: The leading coefficient is 1, go to step two. 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. 3 Step Si: Solve for the variable CPM Notes Unit Quadratics Page 14 of 8 4/15/015

15 Quadratic Formula The Quadratic Formula can be used to solve an equation in the form a b c 0, where a 0. To determine the roots of the equation, substitute the coefficients a and b and the constant c into the quadratic formula and then simplif the resulting epression. b b 4ac a E 11: 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 ac 4 a Step Four: Simplif The solution set is 4 15,4 15 E 1: Solve the quadratic equation using the quadratic formula. Step One: Rewrite in standard form (if necessar) Step Two: Identif a, b, and c. a 3, b5, c 4 Step Three: Substitute the values into the quadratic formula. Step Four: Simplif b b 4ac a CPM Notes Unit Quadratics Page 15 of 8 4/15/015

16 Unit 1.3 To predict and analze the nature of the roots of a quadratic using the discriminant. Models: A quadratic equation can have one real solution, two real solutions, or no real solutions depending on the number of times the graph crosses the -ais Two real solutions One Real Solution No real solution 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 b 4ac 0, then the equation has two real solutions (two -intercepts) and no imaginar solutions. 4ac 0, then the equation has one real solution (one -intercept) and one imaginar solution. 4ac 0, then the equation has two imaginar solutions (no -intercept). E 13: Determine the number and tpe of roots for the quadratic equation Discriminant: b ac Since the discriminant is 0, there is one real solution and one imaginar. E 14: Determine the number and tpe of roots that the quadratic equation 18n 4n10n 6n 5 has. Put the equation in standard form: 8n 10n 5 0. Therefore, a 8,b10 and c 5. Discriminant: b ac Since the discriminant is a positive value, there are two real roots (solutions). CPM Notes Unit Quadratics Page 16 of 8 4/15/015

17 To summarize: Five methods for solving quadratics have been discussed. When to use each one? Here s a plan of attack: If there is no linear term (b), factor (if possible). If ou have a trinomial, tr to factor it first. If it is not factorable, then use the quadratic formula or the completing the square method. The graphing method can onl estimate roots if the are not integers unless technolog (TI-84) is hand. SAMPLE EXAM QUESTIONS 1. What are the solutions of the quadratic equation ? A. B. C. D., 4 9 4, 3 3 4, 3 3 4, 9 Ans: B. What is the solution set for the quadratic equation A. 3 3, 3 3 C. 3 6, 3 6 B. 3 3, 3 3 D. 3 6, ? Ans: A 3. Which is one of the appropriate steps in finding solutions for square? 43 0 when completing the A. 4 3 B. 3 C. 4 7 D. 7 Ans: D CPM Notes Unit Quadratics Page 17 of 8 4/15/015

18 4. How man real and imaginar solutions are there for the equation 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 ? Ans: A 5. Solve the equation b factoring: 15 A. 5 3, C. 5 3, B. 5 3, D. 5 3, Ans: A 6. Use the quadratic formula to solve the equation: n 10n 7 A , C , B , D , 4 4 Ans: B 7. Which equation below has onl one real root? A. 1 1 C. 1 0 B. 1 1 D. 1 Ans: C Unit 1.4 To identif, simplif, and perform operations with comple numbers. A Comple Number is a combination of a real number and an imaginar number. Imaginar numbers are special because when squared, the give a negative result. Normall this doesn t happen, because when we square a positive number we get a positive result, and when we square a negative number we also get a positive result. But just imagine there is such a number, because we need it! The unit imaginar number (like 1 is for Real numbers) is i, which is the square root of -1. So, a Comple Number has a real part and an imaginar part, but either part can be 0, so all Real numbers and Imaginar numbers are also Comple Numbers. 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. CPM Notes Unit Quadratics Page 18 of 8 4/15/015

19 Imaginar Unit: i 1 Powers of i: i i i i i i 4 i i i i i i i... Note: The pattern continues ever 4 th power of i. E 15: Simplifing Square Roots of Negative Numbers i i 4 4 i i i 1 ( 1)( 1) i 45 (45)( 1) i 5 Sum and Difference of Comple Numbers: Add or subtract the real parts and the imaginar parts separatel. E 16: Find the sum: 7i3 4i E 17: Find the difference: 95i4 3i 3 7i4i 53i 94 5 i ( 3 i) 13 i Product of Comple Numbers: Use the distributive propert or FOIL method to multipl two comple numbers. E 18: Find the product 5 i4 3i. Use FOIL: 5453i i4 i3i 0 15i4i3i 0 19i i 1 CPM Notes Unit Quadratics Page 19 of 8 4/15/015

20 E 19: Multipl 7(3 i 5). i Use the distributive propert. 7(3 i 5) i ( 1) i i i 1i 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). Wh do we do this? When we multipl a comple number b its conjugate, we get a real number: ( a bi)( a bi) a abi abi b i a b E 0: Find the quotient 3 5 i. 1 i Multipl b the conjugate: (1+i): 3 5i 1 i 1i 1i 36i5i10i 1 4i 311i i i Note: The final answer is written in standard form. E 1: Find the quotient 3 i. 7i Multipl b the conjugate: (-7i): 3 i 7i 7i 7i 6i1i7i 7 63i i i CPM Notes Unit Quadratics Page 0 of 8 4/15/015

21 Unit 1.5 To solve quadratic equations with comple numbers. E : Solve Solve using square roots: Write the answer(s) in comple form: i 8 3 E 3: Solve 3q 11 5 q. Use the quadratic formula: Simplif: 3q 5q11 0 a 3, b5, c Write the answer(s) in comple form: 5 i SAMPLE EXAM QUESTIONS 1. What are the solutions of the quadratic equation 3 5 4? A. B. 5i 3, 6 5i 73, 6 5i 3 6 5i 73 6 C. D. 5 i 3, 6 5 i 73, 6 5 i i 73 6 Ans: A CPM Notes Unit Quadratics Page 1 of 8 4/15/015

22 . Write the epression 7 3 i 3 9i as a comple number in standard form. A. B. C. D i i i 1 1 i Ans: B 3. What are the solutions of 85 0? A. 86i or 8 6i B. 43i or 4 3i C. 4i or 4 i D. 43i or 4 3i Ans: B 4. What is the product 3 i3 i? A. 5 1i C. 13 1i B. 5 D. 13 Ans: D 5. What is the product 83 3 i i? A. 30 7i C. 18 5i B. 6i 7i 4 D. 30 7i Ans: A CPM Notes Unit Quadratics Page of 8 4/15/015

23 Unit 1.6 To solve and graph solutions of single variable and two variable quadratic inequalities. Solving a Quadratic Inequalit in One Variable: Solving inequalities is ver similar to solving equations. We do most of the same things. When solving equations, we tr to find POINTS, such as the ones marked = 0. But when we solve inequalities, we tr to find INTERVAL(S), such as one marked < 0. So, this is what we do: Find the roots the = 0 points using the methods alread learned. Start with factoring and use quadratic formula if it is not factorable. In between the roots are intervals that are either o Greater than zero (> 0), or o Less than zero (< 0) Then, pick a test value to find out which it is **Note: In this diagram, the roots were - and 3. Using a number line, pick values in each interval to determine whether the function is positive or negative. The corresponding graph is shown above. E 4: Solve Step One: Solve the quadratic equation 6 0 using an method. 3 0 We will use factoring 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 4. Choose an -value between and 3 and substitute into the inequalit. We will tr 0. Choose an -value to the right of 3 and substitute into the inequalit. We will tr true true false Step Three: Write the solution as a compound inequalit or in set notation and graph. or 3, 3, CPM Notes Unit Quadratics Page 3 of 8 4/15/015

24 E 5: Solve 4 0. Step One: Solve the quadratic equation 4 0 using an method. a, b1, c ()( 4) 1 33 We will use the quadratic formula. () or 1.19 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 -. Choose an -value between and 1.19, substitute into the inequalit. We will tr 0. Choose an -value to the right of 1.19 and substitute into the inequalit. We will tr. ( ) ( ) false (0) true () false Step Three: Write the solution as a compound inequalit or in set notation and graph ,1.19 E 6: Solve 1 0. Step One: Solve the quadratic equation 1 0 using an method. A quick check of the discriminant tells us that there are no real solutions or no -intercepts. b 4 ac( 1) 4(1)(1) 3 There are no roots to put on the sign chart. But this makes things easier! Because the line does not cross the - ais, the function must be either: Alwas > 0, or Alwas < 0 So, all we have to do is test one value (sa = 0) to see if the parabola is above or below the -ais. (0) 0 11 This means the function is alwas greater than zero. Since we are tring to find when it is less 1 0 than zero, this never occurs and our answer is: No Solution If we were tring to solve: 1 0, then our solution would be: All Real Numbers CPM Notes Unit Quadratics Page 4 of 8 4/15/015

25 Solving a Quadratic Inequalit in Two Variables: First, graph the corresponding quadratic equation. If the inequalit smbols are < or >, graph a dashed parabola. If the smbols are or, graph a solid parabola. If the inequalit smbol is greater than or greater than or equal to, shade up and if the smbol is less than or less than or equal to, shade down. A test point can also be used. Pick a point (choose (0, 0) if possible it s eas!) either inside or outside the parabola. If the test point makes the inequalit true, shade that area (either inside or outside). If the test point makes the inequalit false, shade the other area not tested. E 7: Graph the quadratic inequalit, 1. Step One: Graph the parabola. Find the verte and ais of smmetr. Since a is negative, it opens down. 1 ( 1) verte: (1,); ais of smmetr: 1 Note: Since the smbol includes equalit, we draw a solid parabola Step Two: Choose a test point inside the parabola and substitute it into the inequalit. -4 We will choose 1 0, false 4 Step Three: If the test point makes the inequalit true, shade inside the parabola. If it does not, shade outside the parabola. Also, note from above, that since we had a greater than smbol, it states to shade up E 8: Graph the quadratic inequalit, ( ) 3. Step One: Graph the parabola. Find the verte and ais of smmetr. Since a is positive, it opens up. verte: (,-3); ais of smmetr: Note: Since the smbol does not include equalit, we draw a dashed parabola. Step Two: Choose an eas test point and substitute it into the inequalit. 15 We will choose 0,0. ( ) 3 0 (0 ) false 10 5 Step Three: If the test point makes the inequalit true, shade outside the parabola. If it does not, shade inside the parabola. Also, note from above, that since we had a greater than smbol, it states to shade up CPM Notes Unit Quadratics Page 5 of 8 4/15/015

26 Graphing Calculator Activit E 9: Graph the quadratic inequalit the Inequalz Application. 9 b hand and then check our graph on the graphing calculator using 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 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). CPM Notes Unit Quadratics Page 6 of 8 4/15/015

27 SAMPLE EXAM QUESTIONS 1. Which of the following graphs represents the quadratic inequalit 4? Ans: D. What is the solution of 7 80? A. 8 or 1 B. 8 or 1 C. 8 1 D. 1 or 8 Ans: B 3. What is the solution set of 3 14? A. 7 B. or 7 C. 7 D. 7 Ans: D CPM Notes Unit Quadratics Page 7 of 8 4/15/015

28 4. Which of the following screens from a graphing calculator represents scale on each graph is one unit per tick mark.)? (Assume the 4 A. B. C. D. Ans: D CPM Notes Unit Quadratics Page 8 of 8 4/15/015