15 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent function. 7 y = – – x 2 5 Answer up, wider A up, narrower B down, wider C down, narrower D 40
What Does "C" Do? What does "c" do in y = ax 2 + bx + c ? y = x 2 + 6 y = x 2 + 3 y = x 2 y = x 2 – 2 y = x 2 – 5 y = x 2 – 9 41
What Does "C" Do? What does "c" do in y = ax 2 + bx + c ? "c" moves the graph up or down the same value as "c." "c" is the y- intercept. 42
16 Without graphing, what is the y- intercept of the the given parabola? y = x 2 + 17 Answer 43
17 Without graphing, what is the y- intercept of the the given parabola? y = –x 2 –6 Answer 44
18 Without graphing, what is the y- intercept of the the given parabola? y = –3x 2 + 13x – 9 Answer 45
19 Without graphing, what is the y- intercept of the the given parabola? y = 2x 2 + 5x Answer 46
20 Choose all that apply to the following quadratic: f(x) = –.7x 2 –4 opens up y-intercept of y = –4 A A Answer opens down y-intercept of y = –2 B B wider than parent function y-intercept of y = 0 C C narrower than parent y-intercept of y = 2 D D function y-intercept of y = 4 E y-intercept of y = 6 F 47
21 Choose all that apply to the following quadratic: 4 f(x) = – – x 2 –6x 3 y-intercept of y = –4 E opens up A Answer opens down y-intercept of y = –2 B F wider than parent function y-intercept of y = 0 C G narrower than parent function D y-intercept of y = 2 H y-intercept of y = 4 I y-intercept of y = 6 J 48
22 The diagram below shows the graph of y = − x 2 − c. Answer Which diagram shows the graph of y = x 2 − c ? D C A B From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. 49
Graphing Quadratic Equations Return to Table of Contents 50
Graph by Following Six Steps: Step 1 - Find Axis of Symmetry Step 2 - Find Vertex Step 3 - Find Y intercept Step 4 - Find two more points Step 5 - Partially graph Step 6 - Reflect 51
Axis of Symmetry Step 1 - Find Axis of Symmetry What is the Axis of Symmetry? Teacher Notes Axis of Symmetry 52
Step 1 - Find Axis of Symmetry Graph y = 3x 2 – 6x + 1 –b x = –– Formula: 2a a = 3 b = –6 x = – (–6) = 6 = 1 2(3) 6 The axis of symmetry is x = 1. 53
Step 2 - Find Vertex Step 2 - Find the vertex by substituting the value of x (the axis of symmetry) into the equation to get y. y = 3x 2 – 6x + 1 a = 3, b = –6 and c = 1 y = 3(1) 2 + –6(1) + 1 y = 3 – 6 + 1 y = –2 Vertex = (1,–2) 54
Step 3 - Find y intercept What is the y-intercept? Teacher Notes y- intercept 55
Step 3 - Find y intercept Graph y = 3x 2 – 6x + 1 The y- intercept is always the c value, because x = 0. y = ax 2 + bx + c c = 1 y = 3x 2 – 6x + 1 The y-intercept is 1 and the graph passes through (0,1). 56
Step 4 - Find Two More Points Graph y = 3x 2 – 6x + 1 Find two more points on the parabola. Choose different values of x y = 3x 2 – 6x + 1 and plug in to find points. y = 3(–1) 2 – 6(–1) + 1 Let's pick x = –1 and x = –2 y = 3 + 6 + 1 y = 10 (–1,10) 57
Step 4 - Find Two More Points (continued) Graph y = 3x 2 – 6x + 1 y = 3x 2 – 6x + 1 y = 3(–2) 2 – 6(–2) + 1 y = 3(4) + 12 + 1 y = 25 (–2, 25) 58
Step 5 - Graph the Axis of Symmetry Graph the axis of symmetry, the vertex, the point containing the y-intercept and two other points. 59
Step 6 - Reflect the Points Reflect the points across the axis of symmetry. Connect the points with a smooth curve. (4,25) 60
23 What is the axis of symmetry for y = x 2 + 2x - 3 (Step 1)? Answer 61
24 What is the vertex for y = x 2 + 2x - 3 (Step 2)? A (-1, -4) Answer B (1, -4) C (-1, 4) 62
25 What is the y-intercept for y = x 2 + 2x - 3 (Step 3)? A -3 B 3 Answer 63
Graph y= x 2 + 2x – 3 axis of symmetry = –1 vertex = –1, –4 y intercept = –3 2 other points (step 4) (1,0) (2,5) Partially graph (step 5) Reflect (step 6) 64
Graph y = 2x 2 – 6x + 4 65
Graph y = –x 2 – 4x + 5 66
Graph y = 3x 2 – 7 67
Solve Quadratic Equations by Graphing Return to Table of Contents 68
Find the Zeros One way to solve a quadratic equation in standard form is find the zeros by graphing. A zero is the point at which the parabola intersects the x-axis. A quadratic may have one, two or no zeros. 69
Find the Zeros How many zeros do the parabolas have? What are the values of the zeros? No zeroes click click 2 zeroes; click 1 zero; (doesn't cross x = -1 and x=3 x=1 the "x" axis) 70
Review To solve a quadratic equation by graphing follow the 6 step process we already learned. Step 1 - Find Axis of Symmetry Step 2 - Find Vertex Step 3 - Find Y intercept Step 4 - Find two more points Step 5 - Partially graph Step 6 - Reflect 71
26 Solve the equation by graphing. –12x + 18 = –2x 2 Which of these is in standard form? Answer A y = –2x 2 – 12x + 18 B y = 2x 2 – 12x + 18 C y = –2x 2 + 12x – 18 72
27 What is the axis of symmetry? y = –2x 2 + 12x – 18 A –3 Answer B 3 C 4 D –5 73
28 y = –2x 2 + 12x – 18 What is the vertex? A (3,0) B (–3,0) Answer C (4,0) D (–5,0) 74
29 y = –2x 2 + 12x – 18 What is the y- intercept? (0, 0) A (0, 18) B Answer (0, –18) C (0, 12) D 75
If two other points are (5, –8) and (4 ,–2),what does 30 the graph of y = –2x 2 + 12x – 18 look like? B A Answer C D 76
31 y = –2x 2 + 12x – 18 What is(are) the zero(s)? A –18 click for graph of answer B 4 Answer C 3 D –8 77
Solve Quadratic Equations by Factoring Return to Table of Contents 78
Solving Quadratic Equations by Factoring Review of factoring - To factor a quadratic trinomial of the form x 2 + bx + c, find two factors of c whose sum is b. Example - To factor x 2 + 9x + 18, look for factors whose sum is 9. Factors of 18 Sum 1 and 18 19 2 and 9 11 x 2 + 9x + 18 = (x + 3)(x + 6) 3 and 6 9 79
Solving Quadratic Equations by Factoring When c is positive, it's factors have the same sign. The sign of b tells you whether the factors are positive or negative. When b is positive, the factors are positive. When b is negative, the factors are negative. 80
Solving Quadratic Equations by Factoring Remember the FOIL method for multiplying binomials 1. Multiply the First terms (x + 3)(x + 2) x x = x 2 2. Multiply the Outer terms (x + 3)(x + 2) x 2 = 2x 3. Multiply the Inner terms (x + 3)(x + 2) 3 x = 3x 4. Multiply the Last terms (x + 3)(x + 2) 3 2 = 6 (x + 3)(x + 2) = x 2 + 2x + 3x + 6 = x 2 + 5x + 6 I F O L 81
Zero Product Property For all real numbers a and b, if the product of two quantities equals zero, at least one of the quantities equals zero. Numbers Algebra 3(0) = 0 If ab = 0, 4(0) = 0 Then a = 0 or b = 0 82
Zero Product Property Example 1: Solve x 2 + 4x – 12 = 0 Use "FUSE" ! (x + 6) (x – 2) = 0 Factor the trinomial using the FOIL method. x + 6 = 0 or x – 2 = 0 Use the Zero property –6 –6 + 2 +2 x = –6 x = 2 –6 2 + 4(–6) – 12 = 0 Substitue found value into original –6 2 + (–24) – 12 = 0 equation 36 – 24 – 12 = 0 0 = 0 or Equal - problem 2 2 + 4(2) – 12 = 0 solved! The 4 + 8 – 12 = 0 solutions are -6 and 2. 0 = 0 83
Zero Product Property The equation has to be Example 2: Solve x 2 + 36 = 12x –12x –12x written in standard form (ax 2 + bx + c). So subtract 12x from both sides. x 2 – 12x + 36 = 0 (x – 6)(x – 6) = 0 Factor the trinomial using the FOIL x – 6 = 0 method. +6 +6 x = 6 Use the Zero property Substitue found 6 2 + 36 = 12(6) value into original equation 36 + 36 = 72 72 = 72 Equal - problem solved! 84
Zero Product Property Example 3: Solve x 2 – 16x + 48= 0 Factor the trinomial using the (x – 4)(x – 12) = 0 FOIL method. Use the Zero x – 4 = 0 x –12 = 0 property +4 +4 +12 +12 x = 4 x = 12 4 2 – 16(4) + 48 = 0 Substitue found 16 – 64 + 48 = 0 value into original –48+48 = 0 equation 0 = 0 12 2 – 16(12) + 48 = 0 144 –192 + 48 = 0 –48 + 48 = 0 0 = 0 –48 Equal - problem solved! 85
32 Solve x 2 – 5x + 6 = 0 –7 3 A F –5 5 B G Answer –3 6 C H –2 7 D I 2 15 E J 86
33 Solve m 2 + 10m + 25 = 0 –7 3 A F Answer –5 5 B G 6 –3 H C –2 7 D I 15 J 2 E 87
34 Solve h 2 – h = 12 3 –12 F A 4 Answer G –4 B –3 6 C H –2 D 8 I 2 12 E J 88
35 Solve d 2 – 35d = 2d 0 –7 F A 5 –5 G B Answer 6 –3 H C 35 7 D I 37 12 J E 89
36 Solve 8y 2 + 2y = 3 3 / 4 – 3 / 4 F A 1 / 2 G – 1 / 2 B Answer 4 / 3 H – 4 / 3 C –2 D –3 I 3 J 2 E 90
37 Which equation has roots of − 3 and 5? A x 2 + 2x − 15 = 0 B x 2 − 2x − 15 = 0 Answer C x 2 + 2x + 15 = 0 D x 2 − 2x + 15 = 0 91
Solve Quadratic Equations Using Square Roots Return to Table of Contents 92
Square Root Method You can solve a quadratic equation by the square root method if you can write it in the form: x ² = c If x and c are algebraic expressions, then: x = c or x = – c √ √ written as: x = ± c √ 93
Square Root Method Solve for z: z ² = 49 z = ± 49 √ z = ±7 The solution set is 7 and –7 94
Square Root Method A quadratic equation of the form x 2 = c can be solved using the Square Root Property. Example: Solve 4x 2 = 20 4x 2 = 20 Divide both sides 4 4 by 4 to isolate x ² x 2 = 5 √ 5 5 The solution set is and – √ 5 x = ± √ 95
Square Root Method Solve 5x ² = 20 using the square root method: 5x 2 = 20 5 5 x 2 = 4 x = or x = – √ 4 √ 4 x = ± 2 96
Square Root Method Solve (2x – 1) ² = 20 using the square root method. click click or 2x – 1 = 20 √ 2x – 1 = – 20 √ 2x – 1 = (4)(5) 2x – 1 = – (4)(5) √ √ 2x – 1 = 2 5 2x – 1 = –2 5 √ √ 2x = 1 + 2 5 √ 2x = 1 – 2 5 √ √ 1 + 2 5 1 – 2 5 √ x = 2 x = 2 solution: x = 1 ± 2 5 √ click 2 97
38 When you take the square root of a real number, your answer will always be positive. True Answer False 98
39 If x 2 = 16, then x = A 4 B 2 Answer C –2 D 26 E –4 99
40 If y 2 = 4, then y = A 4 B 2 Answer C –2 D 26 E –4 100
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