Objectives:
1. Be able to describe a quadratic function and all its parts.
2. Be able to convert from vertex form into standard form.
3. Be able to convert from standard form into vertex form.
Warm Up: Factor the polynomials
1. 10x3 + 6x2 - 28x
2. 48x4 - 27x2
3. 6x2 + 11xy – 10y2
I. Quadratic Function (parts)
Standard form: y = ax2 + bx + c
Vertex form: y = a(x - h)2 + k
*An equation needs to be in vertex form to graph it.
a: Determines the size and direction of the parabola
If “a” is positive that parabola opens up
If “a” is negative that parabola opens down
a =1
a >1
a <1
Parent Width Parabola
Narrow Parabola
Wide Parabola
h: Shifts the parabola left and right
k: Shifts the parabola up and down
(h,k) is the vertex of the parabola
II. Converting Quadratics: Vertex Form into Standard Form:
Standard form: y = ax2 + bx + c
Vertex form: y = a(x - h)2 + k
*To convert from VF into SF always think FOIL
Example 1: Put the equation y = 2(x + 3)2 – 5 into standard form
Y = 2 (x+3)(x+3) - 5
(FOIL)
Y = 2(x2 + 6x + 9) - 5 (Distribute)
Y = 2x2 + 12x + 18 - 5
(Combine Like Terms)
Y = 2x2 + 12x + 13
II. Converting Quadratics: Vertex Form into Standard Form:
Vertex form: y = a(x - h)2 + k
Standard form: y = ax2 + bx + c
*To convert from SF into VF you need some formulas
*h = -b/2a
*k = f(-b/2a)
Example 2: Put the equation f(x) = 3x2 - 6x + 11 into vertex form
What are the values of a, b, and c?
h = -b/2a
k = f(1)
h = --6/2(3)
a = 3, b = -6, and c = 11
h = 6/6
f(1) = 3(1)2 – 6(1) + 11
h=1
f(1) = 3 - 6 + 11
k=8
f(x) = 3(x - 1)2 + 8
Example 3:
What is the equation of the parabola in vertex form and
standard form?
Equation: f(x) = a(x - h)2 + k vertex (-1, 0)
1st: Substitute the vertex
Equation: f(x) = a(x + 1)2
2nd: Choose another point on the
graph and substitute for
“x” and “y” (0, 1)
Equation: 1 = a(0 + 1)2
3rd: Solve for “a”
1 = a(1)2
1=a
Vertex Form: f(x) = (x + 1)2
Standard Form: f(x) = x2 + 2x + 1
Homework:
Worksheet “Standard Form and Vertex Form of Quadratics”