Quadratic Functions

Bianca Joseph

Definition

A quadratic function (f) is a function that has the form as f(x) = ax2 + bx + c where a, b and c are real numbers and a not equal to zero (or a ≠ 0). • The graph of the quadratic function is called a parabola. It is a "U" or “n” shaped curve that may open up or down depending on the sign of coefficient a. Any equation that has 2 as the largest exponent of x is a quadratic function.

Finding Solutions

Using Factoring

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

The term inside the radical, b2 -4ac, is called the discriminant.The discriminant gives important information about the corresponding solutions or roots of where a, b, and c are real numbers and a can not equal 0 .
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Types Of Answers

The "solution" or "roots" or "zeroes" of a quadratic are usually required to be in the "exact" form of the answer. Zeros or roots of functions and solutions to equations can either be real or complex numbers. x-intercepts can only be real numbers. A root is any value that satifies the equation.


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

You can complete the square to convert ax2 + bx + c to vertex form, but, for finding the vertex, it's simpler to just use a formula. (The vertex formula is derived from the completing-the-square process, just as is the Quadratic Formula.The "a" in the vertex form "y = a(xh)2 + k" of the quadratic is the same as the "a" in the common form of the quadratic equation, "y = ax2 + bx + c".

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(xa) is f(x) shifted right a units

The "multiply" transformations are also called stretching.