Quadratics

As Easy As 1-2-3

Basics of Quadratics (Review)

Basics of Parabolas

· May open upwards or downwards

· The zero is where the graph crosses the x-axis

· Zeroes are also called x-intercepts

· The axis of symmetry (AOS) divides the parabola into two equal halves

· Vertex is where the AOS and parabola meet. It is the parabola's maximum or minimum value

· Optimal value is y co-ordinate of the vertex

· Y-intercept is where the graph crosses y-axis

Graphing Quadratics Basics

The 3 Quadratic Equations

Vertex Form

y=a(x-h)+k


Factored Form

y=a(x-r)(x-s)


Standard Form

y=ax^2+bx+c

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VERTEX FORM

An equation in vertex form can be used to solve for:

- The optimal value (The y value of the vertex)

-The axis of symmetry (The x value of the vertex)

- Stretch or Compression of parabola

- Direction of opening

- x intercepts/zeroes


In this form:

x=h is the axis of symmetry

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y=k is the optimal value
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If "a" is positive then the parabola opens upwards like a regular "U".

If "a" is negative, then the graph opens downwards like an upside down "U".

If "a" < 1, the graph of the parabola widens. This just means that the "U" shape of parabola stretches out sideways.

If |a| > 1, the graph of the graph becomes narrower (The effect is the opposite of |a| < 1).

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Changing a Quadratic Function into Vertex Form
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The Step Pattern

The first step in graphing a parabola is to understand the step pattern:

With the equation y=x^2, the pattern is (1,1),(2,4),(3,9).


In the step pattern the x value increases at a constant rate of 1. In the equation y=x^2, the x value is squared to get the y value, which gives you the pattern 1,4,9 etc.


If the a value in the equation is more than 1, for example y=4x^2, the y values in the step pattern are multiplied by the a value. In the case, the points would become (1,4),(2,16),(3,36).

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Finding the Vertex

To find the vertex in an equation in vertex form, all you need to do is find the h and k values in the equation.


If there is no h or k value in the equation, then the value is 0.


In the equation y=x^2, there is no h or k value, meaning the vertex is at (0,0).

Finding x intercepts

To find the x intercepts with an equation in vertex form put y=0 and solve the equation.

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FACTORED FORM

An equation in factored form can be used to solve for:

-The x intercepts (r and s values)

-The axis of symmetry (x value of vertex)

-The optimal value (y value of vertex)

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Finding the x intercepts

The x intercepts are the r and s values in a factored form equation. Just like the h value in vertex form, the r and s values are negative, making their values opposite. For example if the equation is y=(x+5)(x-6), the x intercepts would be (-5,0) and (6,0).

Quadratic Relations of the Form y = a(x r)(x s)

The optimal value

To find the optimal value, you to find the axis of symmetry. Once you have that, you put it into to your equation as the value of x and solve.

The axis of symmetry

To find the axis of symmetry, all you need to do is add the r and s values and divide by 2 as shown in the image above.

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STANDARD FORM

An equation in standard form can be used to solve for:

-The x intercepts

-The axis of symmetry

-The optimal value

A standard form equation can also be converted to vertex form by completing the square.

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And from Vertex to Standard by expanding.
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Factoring to turn to Factored Form include:

Common Factoring

Used when the numbers have a common factor.

Simple Trinomial

The factors of c have a sum of b

Complex Trinomial

Factors of a and c expand to get b. Used with trial and error.
How to Factor Trinomials - Step By Step Tutorial

Perfect Square

a and b can both be square rooted.
factor perfect square trinomial

Difference of Squares

The variables a and b can be square rooted, b value must be negative, meaning one factor of b will be positive and one will be negative
Factoring the Difference of Two Squares - Ex 1

Questions & Answers

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Assessment & Reflection

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This is a question from the Quadratics mini test 1. I did this question right it was just a simple mistake shown above in the pink pen that was my mistake. After this test I went over question similar to it to make sure the mistake was never to be made twice.

Connection

In quadratics I realized that all the topics somehow related to one another. The information taught on the first day was needed to understand every other lesson that followed. So you would need to understand it form the start in order to understand everything.