Module 2 Lesson 1 - Characteristics of Quadratic Equations

Module 2Characteristics of Quadratic Equations : 

Axis of Symmetry & Vertex 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

Vertex : 

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". If the quadratic equation is written in the vertex form y = a(x – h)2 + k, then the vertex is the point (h, k). If the quadratic equation is written in the standard form y = ax2 + bx + c, the vertex (h, k) is found by computing h = –b/2a, and then evaluating y at h to find k. Vertex

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)

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)

Axis of Symmetry : 

If you look at a parabola, you'll notice that you could draw a vertical line right up through the middle which would split the parabola into two mirrored halves. This vertical line, right through the vertex, is called the axis of symmetry. If you're asked for the axis, write down the line "x = h", where h is the x-coordinate of the vertex. Axis of Symmetry

Example 1 : 

Example 1 y = (x–1)² + 1 Vertex = (h,k) = (1,1) Axis of Symmetry: x=1 Axis of Symmetry

Example 2 : 

Example 2 y = 3x2 + 12x – 12 Vertex = (-2,-24) Axis of Symmetry : x=-2 Vertex

Practice Problems : 

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