Module 2-2: Characteristics of Quadratic Equations

Module 2Characteristics of Quadratic Equations : 

Domain and Range Module 2Characteristics of Quadratic Equations

Characteristics of Quadratic Equations : 

Vertex Symmetry Domain Range Characteristics of Quadratic Equations

What is a Quadratic Equation?(review) : 

A quadratic equation is a polynomial equation of the second degree. The general form is ax2+bx+c=0, where x represents a variable, and a, b, and c, constants, with a ≠ 0. What is a Quadratic Equation?(review)

Graphs of Quadratic Equations : 

y=x2 Graphs of Quadratic Equations y=-x2

Domain : 

For a function f(x) defined by an expression with variable x, the domain of f(x) is the set of all real numbers variable x can take such that the expression defining the function is real or defined. The domain will not include numbers for which a function is undefined (i.e 1/0) Generally, all quadratic equations have a domain of (- ∞, + ∞) or all real numbers - ℝ Domain

Range : 

The range of f(x) is the set of all values that the function takes when x takes values in the domain. Once you find the domain, you can evaluate the minimum and maximum of the domain in the function to find the minimum and maximum of the range. The vertex can help with finding the range of a quadratic equation. Range

The VERTEX can HELP : 

Parabolas always have a lowest point (or a highest point, if the parabola is upside-down). This point, where the parabola changes direction, is called the "vertex". Since it is the lowest point or highest point, it will tell you the minimum or maximum value in the range of the quadratic equation The VERTEX can HELP

Example 1 : 

Example 1 y = (x–1)² + 1 Written in vertex form - y = a(x – h)2 + k a=1 , h=1, k=1 Vertex = (h,k) = (1,1) Domain: (- ∞, + ∞) Range: (1, + ∞) Vertex

Example 2 : 

Example 2 y = 3x2 + 12x – 12 Written in standard form - y = ax2 + bx + c a=3 , b=12, c=-12 Vertex = (h,k) h = –b/2a = –12/2(3) = –12/6 = -2 Substitute h for x in equation to find k 3(-2)2 + 12(-2) – 12 3(4)-24-12 12-24-12 -24 Vertex = (-2,-24) Domain: (- ∞, + ∞) Range: (-24, + ∞) Vertex

Example 3 : 

Example 3 y = -x2 + 6x - 8 Written in standard form - y = ax2 + bx + c a=-1 , b=6, c=-8 Vertex = (h,k) h = –b/2a = –6/2(-1) = –6/-2 = 3 Substitute h for x in equation to find k =-(3)2 + 6(3) – 8 -(9)+18-8 -9+18-8 1 Vertex = (3,1) Domain: (- ∞, + ∞) Range: (- ∞, 1] Vertex

Practice Problems : 

Try these problems for further practice. For each equation, identify the domain and range of each function  x2 + 1x - 20 = 0 x2 - 2x + 35 = 0 -x2 - 8x + 9 = 0 -x2 + 12x - 36 = 0 Practice Problems