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

Slide 1 / 175 Slide 2 / 175 Algebra I Quadratics 2015-11-04 www.njctl.org Slide 3 / 175 Slide 4 / 175 Table of Contents · Key Terms Click on the topic to go to that section · Characteristics of Quadratic Equations · Transforming Quadratic Equations · Graphing Quadratic Equations · Solve Quadratic Equations by Graphing Key Terms · Solve Quadratic Equations by Factoring · Solve Quadratic Equations Using Square Roots · Solve Quadratic Equations by Completing the Square · The Discriminant Return to Table · Solve Quadratic Equations by Using the Quadratic Formula of Contents · Solving Application Problems Slide 5 / 175 Slide 6 / 175 Axis of Symmetry Parabolas Maximum: The y-value of the Axis of symmetry: vertex if a < 0 and the The vertical line that parabola opens downward divides a parabola into two symmetrical halves Minimum: The y-value of the Max vertex if a > 0 and the (+ a) parabola opens upward (- a) Min Parabola: The curve result of graphing a quadratic equation

Slide 7 / 175 Slide 8 / 175 Quadratics Quadratic Equation: An equation that can be written in the Characteristics standard form ax 2 + bx + c = 0. Where a, b and c are real numbers of Quadratic and a does not = 0. Equations Vertex: The highest or lowest point on a parabola. Zero of a Function: An x value that makes the function equal zero. Return to Table of Contents Slide 9 / 175 Slide 10 / 175 Quadratics Writing Quadratic Equations A quadratic equation is an equation of the form Practice writing quadratic equations in standard form: ax 2 + bx + c = 0 , where a is not equal to 0. (Simplify if possible.) The form ax 2 + bx + c = 0 is called the standard form of the Write 2x 2 = x + 4 in standard form: quadratic equation. The standard form is not unique. For example, x 2 - x + 1 = 0 can be written as the equivalent equation -x 2 + x - 1 = 0. Also, 4x 2 - 2x + 2 = 0 can be written as the equivalent equation 2x 2 - x + 1 = 0. Why is this equivalent? Slide 11 / 175 Slide 12 / 175 1 Write 3x = -x 2 + 7 in standard form: 2 Write 6x 2 - 6x = 12 in standard form: A. 6x 2 - 6x -12 = 0 A. x 2 + 3x-7= 0 B. x 2 - x - 2 = 0 B. x 2 -3x +7=0 C. -x 2 + x + 2 = 0 C. -x 2 -3x -7= 0

Slide 13 / 175 Slide 14 / 175 Characteristics of Quadratic Functions 3 Write 3x - 2 = 5x in standard form: The graph of a quadratic is a parabola, a u-shaped figure. A. 2x + 2 = 0 The parabola will open upward or downward. B. -2x - 2 = 0 C. not a quadratic equation ← upward downward → Slide 15 / 175 Slide 16 / 175 Characteristics of Quadratic Functions Characteristics of Quadratic Functions The domain of a quadratic function is all real numbers. A parabola that opens upward contains a vertex that is a minimum point. A parabola that opens downward contains a vertex that is a maximum point. D = Reals vertex vertex Slide 17 / 175 Slide 18 / 175 Characteristics of Quadratic Functions Characteristics of Quadratic Functions To determine the range of a quadratic function, If the vertex is a maximum, then the range is all real numbers less ask yourself two questions: than or equal to the y-value. > Is the vertex a minimum or maximum? The range of this quadratic is (–∞,10] > What is the y-value of the vertex? If the vertex is a minimum, then the range is all real numbers greater than or equal to the y-value. The range of this quadratic is [–6,∞)

Slide 19 / 175 Slide 20 / 175 Characteristics of Quadratic Functions Characteristics of Quadratic Functions To find the axis of symmetry simply plug the values of a and b 7. An axis of symmetry (also known as a line of symmetry) will into the equation: Remember the form ax 2 + bx + c. In this divide the parabola into mirror images. The line of symmetry is example a = 2, b = -8 and c =2 always a vertical line of the form –b –b x = x = x=2 2a 2a –(–8) a b c x = = 2 x=2 y = 2x 2 – 8x + 2 2(2) –(–8) x = = 2 y = 2x 2 – 8x + 2 2(2) Slide 21 / 175 Slide 22 / 175 Characteristics of Quadratic Functions 4 The vertical line that divides a parabola into two symmetrical halves is called... The x-intercepts are the points at which a parabola intersects the x-axis. These points are also known as zeroes, roots or solutions and solution sets. Each quadratic equation will have two, one or no real A discriminant x-intercepts. B perfect square C axis of symmetry D vertex E slice Slide 23 / 175 Slide 24 / 175 6 The equation y = x 2 + 3x − 18 is graphed on the set of axes below. Based on this graph, what are the roots of the equation x 2 + 3x − 18 = 0? A −3 and 6 B 0 and −18 C 3 and −6 D 3 and −18 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.

Slide 25 / 175 Slide 26 / 175 7 The equation y = − x 2 − 2x + 8 is 8 What is an equation of the axis of symmetry of the graphed on the set of axes below. parabola represented by y = −x 2 + 6x − 4? Based on this graph, what are the roots of the A x = 3 equation − x 2 − 2 x + 8 = 0? B y = 3 C x = 6 A 8 and 0 D y = 6 B 2 and –4 C 9 and –1 D 4 and –2 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. 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. Slide 27 / 175 Slide 28 / 175 9 The height, y, of a ball tossed into the air can be represented by the equation y = –x 2 + 10x + 3, where x is the elapsed time. What is the equation of the axis of symmetry of this parabola? A y = 5 B y = –5 C x = 5 D x = –5 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. Slide 29 / 175 Slide 30 / 175 Quadratic Parent Equation 2 y = – – x 2 3 The quadratic parent equation is y = x 2 . The graph of all other quadratic equations are transformations of the graph of y= x 2 . Transforming x x 2 y = x 2 -3 9 Quadratic Equations -2 4 -1 1 0 0 1 1 2 4 Return to Table 3 9 of Contents

Slide 31 / 175 Slide 32 / 175 Quadratic Parent Equation Quadratic Parent Equation The quadratic parent equation is y = x 2 . How is y = x 2 changed into y = .5x 2 ? The quadratic parent equation is y = x 2 . How is y = x 2 changed into y = 2x 2 ? x 0.5 y = 2x 2 y = x 2 x 2 -3 4.5 -2 2 -3 18 y = x 2 -1 0.5 -2 8 0 0 -1 2 1 0.5 0 0 1 y = – x 2 2 2 1 2 2 3 4.5 2 8 3 18 Slide 33 / 175 Slide 34 / 175 What Does "A" Do? What Does "A" Do? What does "a" do in y = ax 2 + bx + c ? What does "a" also do in y =ax 2 + bx +c ? How does a > 0 affect the parabola? How does a < 0 affect the parabola? How does your conclusion about "a" change as "a" changes? y = x 2 y = –x 2 1 y = – x 2 y = 3x 2 y = x 2 2 1 y = –1x 2 y = – – x 2 y = –3x 2 2 Slide 35 / 175 Slide 36 / 175 What Does "A" Do? 11 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent equation. What does "a" do in y = ax 2 + bx + c ? y = .3x 2 If a > 0, the graph opens up. up, wider A B up, narrower If a < 0, the graph opens down. down, wider C If the absolute value of a is > 1, then the graph of the equation is narrower than the graph of the parent equation. down, narrower D If the absolute value of a is < 1, then the graph of the equation is wider than the graph of the parent equation.

Slide 37 / 175 Slide 38 / 175 12 Without graphing determine which direction does the 13 Without graphing determine which direction does the parabola open and if the graph is wider or narrower parabola open and if the graph is wider or narrower than the parent function. than the parent equation. y = –2x 2 + 100x + 45 y = –4x 2 up, wider up, wider A A up, narrower up, narrower B B C down, wider C down, wider down, narrower down, narrower D D Slide 39 / 175 Slide 40 / 175 14 Without graphing determine which direction does the 15 Without graphing determine which direction does the parabola open and if the graph is wider or narrower parabola open and if the graph is wider or narrower than the parent function. than the parent function. 2 7 y = – – x 2 y = – – x 2 3 5 up, wider up, wider A A up, narrower up, narrower B B down, wider down, wider C C down, narrower down, narrower D D Slide 41 / 175 Slide 42 / 175 What Does "C" Do? What Does "C" Do? What does "c" do in y = ax 2 + bx + c ? What does "c" do in y = ax 2 + bx + c ? y = x 2 + 6 "c" moves the graph up or down the same value as "c ." y = x 2 + 3 "c" is the y- intercept. y = x 2 y = x 2 – 2 y = x 2 – 5 y = x 2 – 9