# Learning Standard Form

## Learning Goals

1. Learn About The Standard Form Equation
2. Learn How To Complete The Squares (Convert Standard to Vertex)
5. Learn How To Graph The Equation
6. Learn How To Solve Word Problems

## Standard Form Equation

The standard form equation is y=ax^2+bx+c.

Standard form is the expansion of a factored or vertex form equation.

The value of "a" gives you the shape and direction of opening.

The value of "c" is the y-intercept.

Solve using the quadratic formula, to get the x-intercepts.

Solve using completing the squares to get vertex.

MAX or MIN

In order to determine whether a quadratic equation has a max or min you must first check to see if the "a" value is positive or negative.

If the "a" value is positive you have a min.

If the "a" value is negative you have a max.

In order to determine the max or min coordinates of a standard form equation you must use completing the squares (convert to vertex form) and then determine the vertex.

## Completing The Square

y=ax^2+bx+c--->y=a(x-h)^2+k
When completing the square, you are converting a standard form equation to a vertex form equation.
This makes graphing a parabola easier.

1. To complete the square, you must factor the first 2 terms of a standard form equation.
2. Then you must divide the 'b' value by 2, then square that value.
3. Make sure that the value that you get is positive and negative.
4. Take the negative value out of the brackets, then multiply it with the 'a' value that we factored the first 2 terms of a standard form equation with.
5. Add or subtract the terms that are outside of the brackets.
6. Square root the first and the last values that are inside that bracket.
7. Put the numbers that are square rooted in a bracket without the squares, make sure that the bracket is squared.
8. Also make sure you take the 'b' values operation is in middle of the squared bracket of your final answer.
Ex:

y=2x^2-24x-5
y=2(x^2-12x)-5
y=2(x^2-12x+36-36)-5

y=2(x^2-12x+36)-36*2-5
y=2(x^2-12x+36)-72-5
y=2(x-6)^2-77

*Remember in order to convert it back into standard form all you have to do is follow BEDMAS*

3.14 Completing the square

The image above shows the quadratic formula.

The formula solves for the x-intercepts of a quadratic equation.

In order to use it all you have to do is get a quadratic equation in standard form and then get the a,b & c values.

Once you have the values, you have to input them into the equation and then solve in BEDMAS order.

*Remember ± means that you have to split the equation into 2; one which adds and another which subtracts (replacing the ±)*

## Discriminant

The discriminant is the final value in the square root part of the quadratic formula.

Discriminant formula: D=b^2-4ac

If the value of the discriminant is positive and greater than 0, the parabola will have 2 x-intercepts.
If the value of the discriminant is negative, the parabola will have no x-intercepts.
If the discriminant has a value of 0, the parabola will have 1 x-intercept.

In the image to the right the discriminant is 9, which is above 0, so there are 2 x-intercepts.

The Quadratic Formula: How to Use the Discriminant to Determine Roots

## Graphing

In order to graph a standard form equation, you must use the quadratic form to solve for the x-intercepts of the parabola, then you must complete the square of the standard form equation to find the vertex of the parabola. Finally, you must plot the points to show the parabola.

## Word Problem

Math WP Standard Form