Quadratic Functions
Bianca Joseph
Definition
Finding Solutions
The Factoring Method applies the Zero Product Property which states that if the product of two or more factors equals zero, then at least one of the factors equals zero. Thus if B·C=0, then B=0 or C=0 or both.
STEPS:
Write the equation in standard form ax2 + bx + c = 0.
Factor the left side completely.
Apply the Zero Product Property to find the solution set.
Example:
x^2 - 7x =0
x(x-7)
0, 7
Using Square Root Method
The Square Root Property states that if an expression squared is equal to a constant , then the expression is equal to the positive or negative square root of the constant.
if x^2 = P, then x = +- the square root of P
Note:
1. The variable squared must be isolated first (coefficient equal to 1)
2. If P>0 is a real number, the equation x^2 =P has real 2 distinct real solutions, x = square root of P and x = - square root of P
3. If P=0, the equation x^2 =P has a double root of 0
4. IF p<0, the equation x^2=p has exactly 2 imaginary solutions
Example:
4(x-4)^2 +6=86
-6 -6
(4(x-4)^2)/4 = 80/4
(x-4)^2=20
x-4= +-square root of 20
x= 4+-square root of 5
Using Quadratic Formula
The roots of the quadratic equation ax2 + bx + c = 0, where a, b, and c are constants and a 0 are given by:
x=(-b+-squareroot of b^2-4ac)/(2a)
The quadratic equation must be in standard form ax^2 _bx + c =0 in order to identify a, b, and c.
Example:
4x^2-6x-7=0
-(-6)+-square root of ((-6)^2 -4 *4 * (-7))/2_4)
6+-square root of (36+112)/8
6+-square root of 148/8
3+-square root of 37/4
By Completing the Square
Steps:
1. Express the quadratic equation in the following form
x^2 +bx = c
2. Divide b by 2 and square the result, then add the square to both sides.
x^2 +bx+(b/2)^2 = c+(b/2)^2
3. Write the left side of the equation as a perfect square
(x+b/2)^2 = c+(b/2)^2
4. Solve using square root method
Example:
3x^2+12x+9=0
(3x^2+12x+9)/3
x^2+4x+3=0
x^2+4x+4=-3+4
x^2+4x+4=1
(x+2)^2=square root of 1
x+2 = +- 1
-1-2=3
1-2=-1
x=-3, -1
Discriminant
Types Of Answers
Example:
The roots of the equation x^2 - 5x - 24 = 0 are -3 and because (-3)^2 - 5(-3) -24=0 and 8^2 -5(8) -24=0
A number (r) is a zero of a function (f) if f(r)=0.
Y-intercepts are where a line or curve crosses the x-axis. Plug in 0 for x in your equation to find your y intercepts.
Suppose you have ax2 + bx + c = y, and you are told to plug zero in for y. The corresponding x-values are the x-intercepts of the graph. So solving ax2 + bx + c = 0 for x means, among other things, that you are trying to find x-intercepts.
Example:
Find all real solutions to the equation exp(-x2) = -x.
Note: exp(x) is the exponential function and is often written as ex. It is ex (2nd LN) on the calculator, but gets displayed as e^(x).
Begin by moving the -x to the left side where it becomes +x. The function to graph is y1 = x + e^(-x2).
Graph Zero Left Right Guess Solution
The calculator says the solution is x = -0.6529186. The y-value of 0 is ignored. There was no y in the original problem.
Vertex Form of a Quadratic Function
Example:
Find the vertex of y = 3x2 + x – 2 and graph the parabola.
To find the vertex, I look at the coefficients a, b, and c. The formula for the vertex gives me:
h = –b/2a = –(1)/2(3) = –1/6
Then I can find k by evaluating y at h = –1/6:
k = 3( –1/6 )2 + ( –1/6 ) – 2
= 3/36 – 1/6 – 2
= 1/12 – 2/12 – 24/12
= –25/12
So now I know that the vertex is at ( –1/6 , –25/12 ).
transformations
- –f(x) is f(x) flipped upside down ("reflected about the x-axis")
- f(x) + a is f(x) shifted upward a units
- f(x) – a is f(x) shifted downward a units
- f(x + a) is f(x) shifted left a units
- f(x – a) is f(x) shifted right a units
The "multiply" transformations are also called stretching.
2(x2 – 4)
stretch
shift
This is three units higher than the basic quadratic, f(x) = x2. That is, x2 + 3 is f(x) + 3. We added a "3" outside the basic squaring function f(x) = x2 and thereby went from the basic quadratic x2 to the transformed function x2 + 3. This is always true: To move a function up, you add outside the function: f(x) + b is f(x) moved up b units. Moving the function down works the same way; f(x) – b is f(x) moved down b units.