# Quadratic Functions

### Bianca Joseph

## Definition

## 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

## 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 *ax*2 + *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 *ax*2 + *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

*ax*2 +

*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*(

*x*–

*h*)2 +

*k*" of the quadratic is the same as the "

*a*" in the common form of the quadratic equation, "

*y*=

*ax*2 +

*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*/2*a* = –(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 | ## shiftThis 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. | ## horizontal stretch example |

## 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.