Table of Contents Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Graphing Quadratic Functions Transforming Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic Equations by Factoring Solve Quadratic Equations Using Square Roots Solve Quadratic Equations by Completing the Square Solve Quadratic Equations by Using the Quadratic Formula The Discriminant Solving Application Problems
Key Terms Return to Table of Contents
Axis of symmetry: The vertical line that divides a parabola into two symmetrical halves
Maximum: The y-value of the vertex if a < 0 and the parabola opens downward Minimum: The y-value of the vertex if a > 0 and the parabola opens upward Parabola: The curve result of graphing a quadratic equation Max (+ a) (- a) Min
Quadratic Equation: An equation that can be written in the standard form ax2 + bx + c = 0. Where a, b and c are real numbers and a does not = 0. Quadratic Function: Any function that can be written in the form y = ax2 + bx + c. Where a, b and c are real numbers and a does not equal 0. Vertex: The highest or lowest point on a parabola. Zero of a Function: An x value that makes the function equal zero.
Identifying Quadratic Functions Return to Table of Contents
Any quadratic function that can be written in the form y = ax2 + bx + c (standard form) Where a, b, and c are real numbers and a ≠ 0 Examples Question: Is y = 7x + 9x2- 4 written in standard form? Answer: No Question: Is y = 0.5x2 + 7x written in standard form? Answer: Yes
Characteristics of Quadratic Equations Return to Table of Contents
A quadratic equation is an equation of the form ax2 + bx + c = 0 , where a is not equal to 0. The form ax2 + bx + c = 0 is called the standard form of the quadratic equation. The standard form is not unique. For example, x2 - x + 1 = 0 can be written as the equivalent equation -x2 + x - 1 = 0. Also, 4x2 - 2x + 2 = 0 can be written as the equivalent equation 2x2 - x + 1 = 0. Why is this equivalent?
Practice writing quadratic equations in standard form: (Reduce if possible.) Write 2x2 = x + 4 in standard form: 2x2 - x - 4 = 0
Write 3x = -x2 + 7 in standard form: x2 + 3x - 7 = 0
Write 6x2 - 6x = 12 in standard form: x2 - x - 2 = 0
Write 3x - 2 = 5x in standard form: Not a quadratic equation
Characteristics of Quadratic Functions 1. Standard form is y = ax2 + bx + c, where a ≠ 0.
2. The graph of a quadratic is a parabola, a u-shaped figure. 3. The parabola will open upward or downward. ← upward downward →
4. 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
6. 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 to
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 to 10
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 x=2
8. 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 function will have two, one or no real x-intercepts.
1 True or False: The vertex is the highest or lowest value on the parabola. True False
2 If a parabola opens upward then... A a>0 B a<0 C a=0
3 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
4 What is the equation of the axis of symmetry of the parabola shown in the diagram below? 20 18 16 14 12 10 8 6 4 2 A x = -0.5 B x = 2 C x = 4.5 8 9 10 5 6 7 1 2 3 4 5 D x = 13
5 The height, y, of a ball tossed into the air can be represented by the equation y = −x2 + 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
6 What are the vertex and axis of symmetry of the parabola shown in the diagram below? A vertex: (1,−4); axis of symmetry: x = 1 B vertex: (1,−4); axis of symmetry: x = −4 C vertex: (−4,1); axis of symmetry: x = 1 D vertex: (−4,1); axis of symmetry: x = −4
7 The equation y = x2 + 3x − 18 is graphed on the set of axes below. Based on this graph, what are the roots of the equation x2 + 3x − 18 = 0? A −3 and 6 B 0 and −18 C 3 and −6 D 3 and −18
8 The equation y = − x2 − 2x + 8 is graphed on the set of axes below. Based on this graph, what are the roots of the equation − x2 − 2x + 8 = 0? A 8 and 0 B 2 and -4 C 9 and -1 D 4 and -2
Graphing Quadratic Functions Return to Table of Contents
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
Step 1 - Find the Axis of Symmetry What is the Axis of Symmetry? The line that runs down the center of a parabola. This line divides the graph into two perfect halves. Pull Axis of Symmetry
Step 1 - Find the Axis of Symmetry Graph y = 3x2 – 6x + 1 Formula: a = 3 b = -6 x = - (- 6) = 6 = 1 2(3) 6 The axis of symmetry is x = 1.
Step 2 - Find the vertex by substituting the value of x (the axis of symmetry) into the equation to get y. y = 3x2 - 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)
Step 3 - Find y intercept. What is the y-intercept? The point where the line passes through the y-axis. This occurs when the x-value is 0. Pull y- intercept
Graph y = 3x2 – 6x + 1 Step 3 - Find y- intercept. The y- intercept is always the c value, because x = 0. y = ax2 + bx + c c = 1 y = 3x2 – 6x + 1 The y-intercept is 1 and the graph passes through (0,1).
Graph y = 3x2 – 6x + 1 Step 4 - Find two points Choose different values of x and plug in to find points. Let's pick x = -1 and x = -2 y = 3x2 – 6x + 1 y = 3(-1)2 – 6(-1) + 1 y = 3 +6 + 1 y = 10 (-1,10)
Step 4 - Find two points Graph y = 3x2 – 6x + 1 y = 3x2 - 6x + 1 y = 3(-2)2 - 6(-2) + 1 y = 3(4) + 12 + 1 y = 25 (-2,25)
Step 5 - Graph the axis of symmetry, the vertex, the point containing the y-intercept and two other points.
Step 6 - Reflect the points across the axis of symmetry. Connect the points with a smooth curve. (4,25)
9 What is the axis of symmetry for y = x2 + 2x - 3 (Step 1)? A 1 B -1
10 What is the vertex for y = x2 + 2x - 3 (Step 2)? A (-1,-4) B (1,-4) C (-1,4)
11 What is the y-intercept for y = x2 + 2x - 3 (Step 3)? A -3 B 3
12 What is an equation of the axis of symmetry of the parabola represented by y = −x2 + 6x − 4? A x = 3 B y = 3 C x = 6 D y = 6
On the set of axes below, solve the following system of equations graphically for all values of x and y . y = -x2 - 4x + 12 y = -2x + 4 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
On the set of axes below, solve the following system of equations graphically for all values of x and y. y = x2 − 6x + 1 y + 2x = 6
13 The graphs of the equations y = x2 + 4x - 1 and y + 3 = x are drawn on the same set of axes. At which point do the graphs intersect? A (1,4) B (1,–2) C (–2,1) D (–2,–5)