Quadratics

101

Table of Contents

1. First and Second Differences

2. Parabolas

3. Vertex form

4. Graphing from factored form

5. Expanding

6. Multiplying Binomials

7. Common Factors

8. Factoring Simple Trinomials

9. Factoring Complex Trinomials

10. Perfect Squares

11. Differences of Squares

12. Quadratic formula

13. Discriminant

14. Standard to Vertex Form

15. Reflection on Unit

First & Second Differences

To tell if a relation is linear or not, you must look at the first differences. If the first differences are the same throughout the relation then that means the the relation is linear.


Second differences are used to tell if a relation is quadratic or not. In a quadratic relation the first differences don't have to be the same, but the second differences must.

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Parabolas

A very basic quadratic relation is y=x^2

A parabola is a symmetrical curve (both sides of the parabola are the same)


Properties of Parabolas:

Vertex- The vertex is the lowest point of the parabola and is located in the center parabola

Axis of Symmetry- A vertical line that divides the parabola in half perfectly

Y intercept- The point where the parabola crosses through the Y axis

X Intercept- The point where the parabola crosses through the X axis

Min/Max Value- The Y value of the vertex


A quadratic relationship can be graphed by using a chart to organize the x and y as shown above

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Vertex Form y=a(x-h)^2+k

There are many things that you can find with vertex form such as


- Axis of symmetry

-Optimal Value

-X-intercepts or zeros

-Transformations

-Step pattern

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Axis of Symmetry

To find the axis of symmetry from vertex form you should look at the H value. In the equation y=4(x+3)^2+9 the H value (3) will turn negative and is your axis of symmetry. You would write the axis of symmetry as x=-3.

Optimal Value

The K value represents the optimal value. In the equation y=4(x+3)^2+9 the optimal value would be 9. The optimal value is the Y coordinate of the vertex.

X Intercepts/Zeroes

To find the X intercepts of a parabola using vertex form you need to sub in the Y value as 0 and solve for X. Here is an example.

y=3(x+3)^2-12

0=3(x+3)^2-12 Sub Y in as 0

12=3(x+3)^2 Take 12 to the other side of the equation

12/3=(x+3)^2 Divide both sides by 3

4=(x+6)^2 Square booth sides by - & +

-+2=x+6 Now calculate for the zeroes


-2=x+6

-2-6=x

-8=x


2=x+6

2-6=x

-4=x

Transformations

All variables in the vertex form equation y=a(x-h)^2+k are used in transformations

  • a controls if the parabola opens up or down and how stretched/ compressed the parabola is
  • h controls the horizontal shift
  • k controls the vertical shift


If the A value is positive then the parabola will open upwards, if it is negative then it will open downwards.

If the A value is greater than one then the parabola will be stretched. If it is smaller than one but greater than zero it means the parabola is compressed.

The H value represents the horizontal position of the vertex in relation to the origin. If the h value is negative then the vertex shifts to the right, if it is positive the vertex shifts to the left.

The K value represents the vertex of the parabola on the Y axis.

Step Patterns

Step patterns are very important for graphing. The step pattern is a simple rule that will help you create a parabola. The most basic step pattern is 1:1, 2:4, 3:9 and so on. So for a basic parabola with an A value of one you would move right to the vertex once and then up/down once, then right 2 and up/down 4. You would also mirror this on the other side of the parabola.

Graphing From Vertex Form

Graphing from vertex form is simple. First you need to find the vertex. To find the location of the vertex you must look at your h value (which is the x coordinate of the vertex) and your k value (which is the y coordinate of the vertex). Using that information you plot the vertex. Now you would look at your a value to tell if the parabola opens up or down. Next you use the step pattern. Plot the points and connect them.
3.2 Graphing from Vertex Form

Factored Form y=(x-s)(x-r)

Axis of Symmetry

To find the axis of symmetry using factored form you must find the middle point between both of the zeroes. To do this you must take the numbers outside of the brackets and change their sign. For example in y=(x-4)(x-8) the -4 and -8 will become 4 and 8. Then, to find the axis of symmetry you add both numbers and divide them by two.

y=(x-4)(x-8)

4+8=12

12/2=6

The axis of symmetry for this equation would be 6.

Optimal Value

To find the optimal value in factored form you would have to sub in the axis of symmetry as x and solve for y.

y=(x-4)(x-8)

y=(6-4)(6-8)

y=(2)(-2)

y=-4

The optimal value would therefore equal -4 in this example.

Finding X intercepts/zeroes

In factored form all you have to do to find the zeroes is take the numbers outside of the bracket and change the signs in front of them. In y=(x-4)(x-8) the zeroes/x intercepts would be 4 and 8.

Graphing from Factored Form

Graphing in factored form is simple.


  • Find the axis of symmetry
  • Sub it into the equation to find the y value of the vertex
  • Plot the vertex & zeroes then connect them
3.5 Graphing from Factored Form

Expanding

Expanding is used to simplify brackets. The number outside of the bracket must be multiplied with every number inside of the bracket.

Example:

3(4+3x)

=(3*4)+(3*3x)

=12+9x

Multiplying Binomials

Multiplying binomials is like expanding, except you're using two brackets.


Example:

(x-4)(x+5)

=x^2+5x-4x-20

=x^2+x-20

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Standard Form y=ax^2+bx+c

Common Factoring

Common factoring can be used to factor out any number or variable that all of the terms can be divided by. Doing this simplifies your equation. For example in 10xy+40x+30x^3y all of the terms are divisible by five and x. That equation would become 10x(y+4+3x^2y).

Simple Factoring

To simple factor you need to take the equation and split it up into two brackets that equal the original equation when multiplied together. For this example we will use x^2+6x+8.

x^2+6x+8

(x+4)(x+2)

To properly factor you must find two numbers that add up to the B but when multiplied equal the C. 4 and 2 when added equal 6 and when multiplied equal 8.

Factoring Complex Trinomials

Example:

Factor the following trinomial.

2x^2 +10x+12

Solution:

Step 1:The first term is 2x^2 , which is the product of 2x and x. Therefore, the first term in each bracket must be x.

2x^2+10x+12


Step 2: First you want to simplify the equation by common factoring

2x+10x+12

=2(5x+6)


Step 3: Next you should look for factors of the two numbers in the brackets in this case those numbers are two and three.

2(5x+6)

=2(x+2)(x+3)

Therefore the factored form of 2x^2+10x+12 is 2(x+2)(x+3)

Perfect Squares

Perfect squares are trinomials that have one factor that adds up to equal B and multiplies to equal C. For example:

x^2+12x+36

(x+6)^2

This example is a perfect square because 6+6 equals 12(b) and 6*6 equals 36(C)

Another example:

x+6x+9

(x+3)^2

3+3 equals 6(B) and 3*3 equals 9(C) which makes this equation a perfect square.

Differences of Squares

Differences of squares are close to perfect squares but one factor has to be positive while the other is negative.


Example:

x^2-25 (There is no b because one factor is negative while the other is positive and they are the same number so they cancel each other out.)


Two factors must be the same and must still multiply to equal C.

x^2-25=(x+5)(x-5)


The simple way to find these factors is to square root your C and change the operations of that number.

Example:

x^2- 25

x^2 + 5^2

(x+5)(x-5)

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The quadratic formula is used to solve quadratic equations. It is very simple to use because all you have to do is take the numbers from the standard form equation then put them into the quadratic formula, and you're ready to solve!


Example:

3x^2+6x+2


If you take this equation and put it into standard form it would be:

-6+-√36-4(3)2

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2(3)

Solution:

-6+-√12


-6+3.46=-0.42

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6


-6-3.46=-1.57

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6


The zeroes of this equation would be 1.57 and 0.42

Discriminants

The discriminant is everything after the square root [√b^2-4(a)(c)] symbol. The discriminant is used to figure out how many solutions (zeroes) there are.


If D is greater than 0 it means there are two zeroes

If D is less than 0 it means there are no zeroes

If D is equal to 0 it means there is only one zero.

Links Between Topics

The topics that we have learned can be connected in many different ways. We can convert equations into other different equations. For example if we got a factored form equation and were told to graph it, we wouldn't be able to. Since we know that it is simple to graph in vertex form we would just change the question to vertex form and then graph it from there.

Word Problems

The hypotenuse of a right angle triangle is 25cm. The sum of the lengths on the other two sides is 21cm. Find the lengths of the sides.


ANSWER: 12 and 9



The equation shows the height (h) of a baseball in metres as a function of time (t) in seconds.

h=−5t2+40t+3



a) What was the initial height of the ball?


b) At what time does the ball hit the ground?


ANSWERS:

A)3M

B)8.07 Seconds

Reflection on Quadratics

Quadratics was a very interesting unit for me. At first I struggled and did not understand any of it until I got help from my friends and teacher. It was difficult for me to grasp most of the concepts but I now see how useful it really is to my life.