Slide 1 / 175 Slide 2 / 175 Algebra I Quadratics 2015-11-04 www.njctl.org Slide 3 / 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 · Solve Quadratic Equations by Factoring · Solve Quadratic Equations Using Square Roots · Solve Quadratic Equations by Completing the Square · The Discriminant · Solve Quadratic Equations by Using the Quadratic Formula · Solving Application Problems
Slide 4 / 175 Key Terms Return to Table of Contents Slide 5 / 175 Axis of Symmetry Axis of symmetry: The vertical line that divides a parabola into two symmetrical halves Slide 6 / 175 Parabolas Maximum: The y-value of the vertex if a < 0 and the parabola opens downward 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 Quadratics Quadratic Equation: An equation that can be written in the standard form ax 2 + bx + c = 0. Where a, b and c are real numbers and a does not = 0. Vertex: The highest or lowest point on a parabola. Zero of a Function: An x value that makes the function equal zero. Slide 8 / 175 Characteristics of Quadratic Equations Return to Table of Contents Slide 9 / 175 Quadratics A quadratic equation is an equation of the form ax 2 + bx + c = 0 , where a is not equal to 0. The form ax 2 + bx + c = 0 is called the standard form of the 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 10 / 175 Writing Quadratic Equations Practice writing quadratic equations in standard form: (Simplify if possible.) Write 2x 2 = x + 4 in standard form: Slide 11 / 175 1 Write 3x = -x 2 + 7 in standard form: A. x 2 + 3x-7= 0 B. x 2 -3x +7=0 C. -x 2 -3x -7= 0 Slide 12 / 175 2 Write 6x 2 - 6x = 12 in standard form: A. 6x 2 - 6x -12 = 0 B. x 2 - x - 2 = 0 C. -x 2 + x + 2 = 0
Slide 13 / 175 3 Write 3x - 2 = 5x in standard form: A. 2x + 2 = 0 B. -2x - 2 = 0 C. not a quadratic equation Slide 14 / 175 Characteristics of Quadratic Functions The graph of a quadratic is a parabola, a u-shaped figure. The parabola will open upward or downward. ← upward downward → Slide 15 / 175 Characteristics of Quadratic Functions 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. vertex vertex
Slide 16 / 175 Characteristics of Quadratic Functions The domain of a quadratic function is all real numbers. D = Reals Slide 17 / 175 Characteristics of Quadratic Functions To determine the range of a quadratic function, ask yourself two questions: > Is the vertex a minimum or maximum? > 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 18 / 175 Characteristics of Quadratic Functions If the vertex is a maximum, then the range is all real numbers less than or equal to the y-value. The range of this quadratic is (–∞,10]
Slide 19 / 175 Characteristics of Quadratic Functions 7. An axis of symmetry (also known as a line of symmetry) will divide the parabola into mirror images. The line of symmetry is always a vertical line of the form –b x = 2a –(–8) x = = 2 x=2 y = 2x 2 – 8x + 2 2(2) Slide 20 / 175 Characteristics of Quadratic Functions To find the axis of symmetry simply plug the values of a and b into the equation: Remember the form ax 2 + bx + c. In this example a = 2, b = -8 and c =2 –b x = x=2 2a a b c –(–8) x = = 2 y = 2x 2 – 8x + 2 2(2) Slide 21 / 175 Characteristics of Quadratic Functions 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 x-intercepts.
Slide 22 / 175 4 The vertical line that divides a parabola into two symmetrical halves is called... A discriminant 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 7 The equation y = − x 2 − 2x + 8 is graphed on the set of axes below. Based on this graph, what are the roots of the equation − x 2 − 2 x + 8 = 0? A 8 and 0 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. Slide 26 / 175 8 What is an equation of the axis of symmetry of the parabola represented by y = −x 2 + 6x − 4? A x = 3 B y = 3 C x = 6 D y = 6 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 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 28 / 175 Slide 29 / 175 Transforming Quadratic Equations Return to Table of Contents 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 . x x 2 y = x 2 -3 9 -2 4 -1 1 0 0 1 1 2 4 3 9
Slide 31 / 175 Quadratic Parent Equation The quadratic parent equation is y = x 2 . How is y = x 2 changed into y = 2x 2 ? y = 2x 2 x 2 -3 18 y = x 2 -2 8 -1 2 0 0 1 2 2 8 3 18 Slide 32 / 175 Quadratic Parent Equation The quadratic parent equation is y = x 2 . How is y = x 2 changed into y = .5x 2 ? x 0.5 y = x 2 -3 4.5 -2 2 -1 0.5 0 0 1 0.5 1 y = – x 2 2 2 2 3 4.5 Slide 33 / 175 What Does "A" Do? What does "a" do in y = ax 2 + bx + c ? How does a > 0 affect the parabola? How does a < 0 affect the parabola? y = x 2 y = –x 2
Slide 34 / 175 What Does "A" Do? What does "a" also do in y =ax 2 + bx +c ? How does your conclusion about "a" change as "a" changes? 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 What Does "A" Do? What does "a" do in y = ax 2 + bx + c ? If a > 0, the graph opens up. If a < 0, the graph opens down. If the absolute value of a is > 1, then the graph of the equation is narrower than the graph of the parent equation. If the absolute value of a is < 1, then the graph of the equation is wider than the graph of the parent equation. Slide 36 / 175 11 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent equation. y = .3x 2 A up, wider up, narrower B down, wider C down, narrower D
Slide 37 / 175 12 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent function. y = –4x 2 up, wider A up, narrower B down, wider C down, narrower D Slide 38 / 175 13 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent equation. y = –2x 2 + 100x + 45 up, wider A up, narrower B C down, wider down, narrower D Slide 39 / 175 14 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent function. 2 y = – – x 2 3 A up, wider up, narrower B down, wider C down, narrower D
Slide 40 / 175 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 up, wider A up, narrower B down, wider C down, narrower D Slide 41 / 175 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 Slide 42 / 175 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.
Slide 43 / 175 16 Without graphing, what is the y- intercept of the the given parabola? y = x 2 + 17 Slide 44 / 175 17 Without graphing, what is the y- intercept of the the given parabola? y = –x 2 –6 Slide 45 / 175 18 Without graphing, what is the y- intercept of the the given parabola? y = –3x 2 + 13x – 9
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