Key Terms Solve Quadratic Equations by Factoring Application of - PDF document

Slide 1 / 222 Slide 2 / 222 Algebra II Quadratic Functions 2014-10-14 www.njctl.org Slide 3 / 222 Slide 4 / 222 Table of Contents Key Terms click on the topic to go to that section Explain Characteristics of Quadratic Functions Combining Transformations (review) Graph Quadratic Functions Solve Quadratic Equations by Graphing Key Terms Solve Quadratic Equations by Factoring Application of Zero Product Property Solve Quadratic Equations Using Square Roots Solve Quadratic Equations by Completing the Square Solve Quadratic Equations using the Quadratic Formula Return to The Discriminant Table of Contents Vertex Form More Application Problems using Quadratics Slide 5 / 222 Slide 6 / 222

Slide 7 / 222 Slide 8 / 222 Key Terms Key Terms Zero(s) of a Function: An x value that makes the function equal zero. Also called a "root," "solution" or "x-intercept" Vertex: The highest or lowest point on a parabola. Minimum Value: The y-value of the vertex if a > 0 and the parabola opens upward Maximum Value: The y-value of the vertex if a < 0 and the parabola opens downward Slide 9 / 222 Slide 10 / 222 Key Terms Explain Axis of symmetry: The vertical line that divides a parabola into two Characteristics symmetrical halves of Quadratic Functions Return to Table of Contents Slide 11 / 222 Slide 12 / 222 Characteristics of Quadratics Characteristics of Quadratics The form ax 2 + bx + c = 0 is called the standard form of a Remember: A quadratic equation is any equation that quadratic equation. can be written in the form ax 2 + bx + c =0 Where a, b, and c are real numbers and a ≠ 0 The standard form is not unique. For example, Question 1: Is a quadratic equation? x 2 - x + 1 = 0 can also be written -x 2 + x - 1 = 0. Also, 4x 2 - 2x + 2 = 0 can be written 2x 2 - x + 1 = 0. Question 2: Is a quadratic equation?

Slide 13 / 222 Slide 14 / 222 Standard Form Standard Form Practice writing quadratic equations in standard form: (Simplify if possible.) Write 3x = -x 2 + 7 in standard form, if possible: Write 2x 2 = x + 4 in standard form: Slide 15 / 222 Slide 16 / 222 Standard Form Standard Form Write 6x 2 - 6x = 12 in standard form and simplify, if possible: Write 3x - 2 = 5x in standard form: Slide 17 / 222 Slide 18 / 222 Standard Form Graph Similar to Quadratic Equations, the standard form of a Quadratic When graphed, a quadratic function will make the shape of a parabola. Function is y = ax 2 + bx + c, where a ≠ 0. Notice, a can be positive or negative. The parabola will open upward if a > 0 or downward if a < 0.

Slide 19 / 222 Slide 20 / 222 Domain The domain of a quadratic function is all real numbers. Slide 21 / 222 Slide 22 / 222 Range Axis of Symmetry If the vertex is a maximum, then the range is all real numbers less An axis of symmetry (also known as a line of symmetry) will than or equal to the y-value of the vertex. divide the parabola into mirror images. The range of this quadratic is The line of symmetry is always a vertical line of the form Slide 23 / 222 Slide 24 / 222 X-Intercepts 1 If a parabola opens downward, the vertex is the highest value on the parabola. The x-intercepts are the points at which a parabola intersects the x-axis. These points are also known as zeros, roots or solutions True and solution sets. Each quadratic function will have 0, 1, or 2 or real solutions. False 2 real solutions no real solutions 1 real solution

Slide 25 / 222 Slide 26 / 222 3 The vertical line that divides a parabola into 2 If a parabola opens upward then... two symmetrical halves is called... A a > 0 discriminant A B a < 0 B quadratic equation C a = 0 C axis of symmetry D vertex E maximum Slide 27 / 222 Slide 28 / 222 Slide 29 / 222 Slide 30 / 222 What is the range of the quadratic function below?

Slide 31 / 222 Slide 32 / 222 Combining Transformations (REVIEW) Return to Table of Contents Slide 33 / 222 Slide 34 / 222 Combining Transformations Combining Transformations Let the graph of f(x) be Let the graph of f(x) be Graph y = 2f(.5x+1) - 2 Graph y =(- 1 / 2 )f(2x + 1) + 2 Slide 35 / 222 Slide 36 / 222 Combining Transformations Combining Transformations Let the graph of f(x) be Let the graph of f(x) be Answer Graph y = 3f(-.5x - 2) + 1 Graph y = (- 1 / 2 )f(-x + 2) +1

Slide 37 / 222 Slide 38 / 222 7 Given the graph of h(x), which of the following Graph the Transformation graphs is y = 2h(-x+1) - 3? Consider the graph y = x 2 and the rules for stretches and shrinks, Graph A B D C Slide 39 / 222 Slide 40 / 222 8 Given the graph of h(x), which of the following graphs is y = -0.5h(2x - 1) + 2? A B Graph Quadratic Functions D C Return to Table of Contents Slide 41 / 222 Slide 42 / 222 Graph by Following Five Steps: Step 1 - Find Axis of Symmetry Step 2 - Find Vertex Step 3 - Find y-intercept Step 4 - Locate another point Step 5 - Reflect and Connect

Slide 43 / 222 Slide 44 / 222 Slide 45 / 222 Slide 46 / 222 Slide 47 / 222 Slide 48 / 222 9 What is the axis of symmetry for y = x 2 + 2x - 3 10 What is the vertex for y = x 2 + 2x - 3 (Step 2)? (Step 1)? A (-1, -4) A x = 1 B x = -1 B (1, -4) C x = 2 C (-1, -6) D x = -3 D (1, -6)

Slide 49 / 222 Slide 50 / 222 Graph 11 What is the y-intercept for y = x 2 + 2x - 3 (Step 3)? Practice: Graph A (0 , -3) B (0 , 3) Slide 51 / 222 Slide 52 / 222 Graph Graph Practice: Graph Practice: Graph Slide 53 / 222 Slide 54 / 222 Solve by Graphing When asked to solve a quadratic equation, there are several ways to do so. Solve Quadratic One way to solve a quadratic equation in standard form is to find the Equations by zeros of the related function by graphing. A zero is the point at which the parabola intersects the x-axis. Graphing A quadratic function may have one, two or no zeros. Return to Table of Contents

Slide 55 / 222 Slide 56 / 222 Solve by Graphing Vocabulary How many zeros do the parabolas have? What are the Every quadratic function has a related quadratic equation. values of the zeros? A quadratic equation is used to find the zeroes of a quadratic function. When a function intersects the x-axis its y-value is zero. When writing a quadratic function as its related quadratic equation, you replace y with 0. So y = 0. y = ax 2 + bx + c Quadratic Function 0 = ax 2 + bx + c No zeroes 2 zeroes; 1 zero; ax 2 + bx + c = 0 Quadratic Equation click click click x = -1 and x=3 x=1 Slide 57 / 222 Slide 58 / 222 Solve by Graphing Solve by Graphing Step 1 - Write the Related Function One way to solve a quadratic equation in standard form is to find the zeros or x-intercepts of the related function. 2x 2 - 18 = 0 Solve a quadratic equation by graphing: 2x 2 - 18 = y Step 1 - Write the related function. y = 2x 2 + 0x - 18 Step 2 - Graph the related function. Step 3 - Find the zeros (or x-intercepts) of the related function. Slide 59 / 222 Slide 60 / 222 Solve by Graphing Solve by Graphing Step 2 - Graph the Function Step 2 - Graph the Function y = 2x 2 + 0x – 18 y = 2x 2 + 0x – 18 Use the same five-step process for graphing The axis of symmetry is x = 0. x = 0 # # The vertex is (0, -18). (3,0) The y-intercept is (0, -18). Since the vertex is the y-intercept, locate two other points by substituting values for x. We'll # # use (2,-10) and (3,0) (2,-10) Graph these points and use reflection across the # axis of symmetry. Connect all points with a (0,-18) smooth curve.

Slide 61 / 222 Slide 62 / 222 Solve by Graphing Solve by Graphing Step 3 - Find the zeros Step 3 - Find the zeros y = 2x 2 + 0x – 18 y = 2x 2 + 0x – 18 Substitute 3 and -3 for x in the quadratic equation. Check 2x 2 – 18 = 0 x = 0 # # (3,0) 2(3) 2 – 18 = 0 2(-3) 2 – 18 = 0 The zeros appear to be 2(9) - 18 = 0 2(9) - 18 = 0 3 and -3. # # 18 - 18 = 0 18 - 18 = 0 (2,-10) # 0 = 0 # 0 = 0 # (0,-18) The zeros are 3 and -3. Slide 63 / 222 Slide 64 / 222 13 What is the axis of symmetry? 12 Solve the equation by graphing the related y = -2x 2 + 12x - 18 function and identifying the zeros. -12x + 18 = -2x 2 Formula: -b x = -3 A 2a Step 1: Which of these is the related function? B x = 3 x = 4 C y = -2x 2 - 12x + 18 A x = -5 D B y = 2x 2 - 12x - 18 y = -2x 2 + 12x - 18 C Slide 65 / 222 Slide 66 / 222 y = -2x 2 + 12x - 18 y = -2x 2 + 12x - 18 14 15 What is the vertex? What is the y-intercept? A (3,0) A (0,0) (-3,0) (0, 18) B B (4,0) C (0, -18) C (-5,0) D (0, 12) D