Vertex Form. Factored Form. Standard Form
Understanding quadratic equations allows one to apply their knowledge and solve real life problems. For example, when you throw a ball it goes up into the air, slows down and comes back down again. As a result of my understanding, I can figure out the max height of the ball.
Another place where quadratics is used Wonderland! That's right Wonderland. Engineers use their understanding of quadratics to decide the max and min height of roller coasters.
OUR LEARNING GOALS FOR THIS UNIT ARE
2. I am able to describe the transformations that the parabola has undergone using my understanding of the "a" value, "h" value and "k" value.
3. I am able to take my understanding of the equation: y=a(x-h)^2+k and use it to plot 3 points on a graph
Linear equations have a first difference that is the same:
Quadratic equations have a second difference that is the same:
1. Labeling Key Features of Quadratic Relations
**The graph of a quadratic equation is referred to as a parabola (curved or "u"shaped)**
Vertex: It is the place where the graph changes direction and can either be the highest or lowest point of the parabola.
Optimal Value: The optimal value is the max or min value of the parabola in other words- the highest or lowest point. It is represented by the k value. If the parabola opens down then the optimal value is the max value and if it opens up then it is the min value.
The Axis of Symmetry: This point divides the parabola into symmetrical halves. The AOS is represented by the h value in the vertex. The equation for the axis of symmetry is x=h.
X Intercepts: The point where the parabola interacts with the x intercept and where y is 0.
2. Describe Transformations
y= 0.5(x-h)^2 +K (compressed)
y= 5( x-h)^2 +k (stretched)
The h value determines if the parabola is horizontally translated to the left or the right.
For example: The x value was moved 5 units to the right
*If the h value is negative it is moved to the right ( so it's positive) and if it is positive it is moved to the left (so it is negative).
The K value determines where it is vertically translated. If it is positive then it is translated up and if it is negative it is translated down.
Describe the translation:
The parabola is vertical stretched by a factor of 5. It is horizontally translated 7 units to the right and it is vertically translated 5 units down
The parabola is reflected down on the x axis. The parabola is vertically stretched by a factor of 2. The parabola is horizontally translated 5 units to the left and is vertically translated 3 units up.
Now you try:
To solidify this concept watch this video of how to graph equations in vertex form using transformations: https://www.youtube.com/watch?v=PGdP9D47XSU
3. Plotting points on a graph
1. Determine the vertex
2. Plot the vertex
3. Use the step pattern (multiply the step pattern by the a factor if it's greater than 1) to make an accurate parabola.
Here is a video I created using "PowToon" to demonstrate this concept: https://www.powtoon.com/online-presentation/bZov0pd3Xcq/graphing-quadratic-equations/?mode=movie
Word Problems: Graphing Quadratic Functions Using Vertex Form
a) What was the maximum height of the football and how do you know?
The maximum height is 29 cm and I know this because it is the k value which is also the optimal value.
b) What was the height of the ball when it was kicked?
:. The height of the ball when it was kicked off the ground was 0.8 cm
c) How high was the ball after 2 seconds?
h=-4.9 (-0.4)^2+ 29
h= -4.9 (0.16)+29
The ball was 28.2 cm after 2 seconds
Our Learning Goals For this unit are
1. I am able to use my knowledge of GCF to factor standard form equations
2.I am able to plot points on a graph using the factored form equation
3.I am able to apply my knowledge of factored form equations by solving real world problems
To understand factoring you first must understand how to "expand binomials". Here is a quick video teaching you the concept: https://www.youtube.com/watch?v=JWJ-eUgP0uY
When factoring the first thing you should do is look for the GCF (flashback to grade 4 :) among the coefficients and variables.
What is the GCF between 2 and 6?
The answer is 2
What is the GCF the variables have in common?
The answer is h^2
This is referred to as "GREATEST COMMON FACTORING"
Aside from this rule, you can also use your previous knowledge of perfect squares when working with terms that are perfect squares.
What is the square root of -36?
6 and -6!
What is the square root of 144 and -169?
12 and 13!
NOW YOU TRY:
The square root of 100 is 10 and the square root of 9 is -3 and 3.
This is a special case and called "A DIFFERENCE OF SQUARES" because the second term is negative and a perfect square.
When there are 3 terms and perfect squares are involved in the equation it is referred to as a "PERFECT SQUARE TRINOMIAL"
What are the square roots of 9 and 16?
3 and 4 !
To check your answer multiple these values by 2 and make sure they equal the middle term.
Moreover, we will learn how to factor quadratic expressions of the form x^2+bx+c and this is referred to as "SIMPLE FACTORING" (when A=1).
**There is a coefficient infront of the x^2 and it is 1.
- find two integers that when multiplied together =2 and when added together =3
2 x 1
- what are two integers that when multiplied together =9 but when added=6
3 x 3=9
However, questions are not always this easy because "a" doesn't always equal 1
sometimes it looks like this :
where a is larger than 1. These questions are referred to as "COMPLEX FACTORING"( When A is greater than 1)
There are 2 ways to go about solving these questions:
1. Guess and check
-experiment with the factors of these integers until you get the right answer
Another method to factor "COMPLEX FACTORING" questions is by Decomposition which is demonstrated in this video: https://www.youtube.com/watch?v=1AlPfk8jx8U
Lastly the last scenario you will encounter is "FACTORING BY GROUPING". That's when there are no common variables or coefficients. This usually happens when there are 4 terms.
- Group the first the two terms and the other two terms
-Factor out the GCF
2.Creating a Graph
Suppose you're given this equation:
THINK: 2 integers that have a product of -8 are and equal 2
4 x -2=-8
X=-4 X= 2
-Add the r and s value and divide by 2. The value is the AOS (the AOS divides the parabola in half)
-To find the y value insert -1 into both x values
You know have all the parts to create and label a graph!
3. Solving word problems
A rectangle has area defined by 8X^2+2X-15
a) Factor to find algebraic expressions for the width and length of the rectangle
Method of choice: Guess and check
=8x^2+19x-15 --> wrong
b) If x represents 12 cm, determine the area and perimeter of the rectangle
A= LW P=2(L+W)
:. The area is 1161 cm^2 and the perimeter is 140 cm
Our learning goals for this unit are
2. I am able to turn the standard equation into vertex form by completing the square
3. I am able to plot points on a graph using the standard form equation
1. Identifying the X intercepts of the equation using the quadratic formula
To identify the x intercept of the equation you have to use the quadratic formula:
2. Turn Standard Form into Vertex Form by Completing the Square
Rewrite each relation in the form of y=a(x-h)^2+k
This how you do it:
1.Bracket the first values
2. Divide the b value by 2 and square it
3. Include the positive and negative integer of the b value you divided and squared in the bracket
4. Put the negative value outside the bracket.
5. When applicable use your knowledge of squares from the previous unit
**If x had a negative value or a value greater then 1 you would have to multiple the b value by it**
For further clarification on "completing a square" please refer to this video: https://www.khanacademy.org/math/algebra/quadratics/solving-quadratics-by-completing-the-square/v/solving-quadratic-equations-by-completing-the-square
Now you try
y=3x^2-12x-5~> The same rules apply but you will have to factor the a and b values by the GCF
1. If the answer is greater than 0 then there are 2 solutions
e.g, b^2-4ac 6^2-4ac
2. If the answer is less than 0 then there are no solutions
e.g, b^2-4ac 2^2-4(9)(7)
3. If the answer is equal to 0 then there is only one solution
e.g, b^2-4ac 8^2-4(2)(8)
3. Creating a graph Using the Standard Form Equation
2. Find the vertex
1.Find the X-intercept using the Quadratic Formula
y = x ^ 2 + 2x-3 = 0
2. Find the vertex
Get the X value and insert it into the equation to find the y value
AOS=r+s/2 and then sub into the equation
x = 1-3 / 2
x = -2/2
x = -1
y = -1 ^ 2 + 2 (-1) -3
y = 1-5
y = -4
:. The vertex is (-1,-4)
Plot these points on a graph
Let x represent the price reduction
**Find x using the quadratic formula**
Some connections between the strands were:
1. If an equation is in standard form and you want to quickly identify the vertex then you have to complete the square -> turn it into vertex form. Sometimes you need to factor out the GCF which is only possible by having a solid understanding of factoring. This ties together your understanding of factoring ,recognizing vertex format and plotting key features of a graph such as the vertex.
2.Factored form is connected to vertex form because they both allow you to easily identify the AOS. The AOS is represented by the h value in a vertex equation and the AOS is calculated by isolating the x's and adding them together and dividing by 2 in a factored equation. This is important because the AOS divides the parabola into symmetrical halves.
3.Vertex Form is connected to graphing because you can recognize the optimal value,the vertex, the AOS, the direction, shape and transformations the parabola has undergone. This is helpful because all the key parts of a graph are identified just by looking at the equation.
4.The connection between standard form and factored form is they both help you to identify the roots.You can factor a standard equation and isolate for the x's and divide by 2 or you can solve for the x's by substituting the values from the standard equation into the quadratic formula.
5.Standard form connects to graphing because the C value represents the y intercept and "a" gives you the shape and direction .You then use the quadratic formula to find the x value and from there substitute for the y value so you know the vertex. You now have the key parts that make up a graph.
6. Vertex form connects to graphing because "a" gives you the shape and direction of the opening, the value of r and s gives you the x intercepts/AOS, you sub x into the equation to get the optimal value and set x=0 to get y. These are all the key points of a graph