PLEASE READ: This post has been made for a project at Pottsboro High School for Coach Johnson’s Algebra II Pre-AP Class. It was made to be an information resource hub where future students can come to find resources for their projects. However, feel free to use this information for anything else but your project!
The Vocab You Will Need:
- Axis of Symmetry — Used with Parabolas, these lines go through the lowest/highest part of the parabola, or u, and divide the parabola perfectly in half.
- Discriminant — This is the b^2 – 4ac part of the quadratic equation. It can be used to determine the amount of solutions in a problem.
- Rational — All Real Numbers: 3, 1/4, the square root of 9, etc. The discriminants of these answers are greater than zero. >0
- Irrational — Real Numbers that cannot be expressed regularly, such as π (Pi). The discriminants of these answers are equal to zero. =0
- Complex — Numbers that are not real, such as 5i. The discriminants of these answers are less than zero. <0
- Maximum/Minimum Point — The highest or lowest point on a parabola. The axis of symmetry is through this point and is also described as a vertex. See More Here.
- Quadratic Formula — This is one of the easiest, however longest, ways of how to find x in equations that have x squared. See the Equation here.
- Standard Form — This changes depending on what you’re working with/where you are in the world. For this project we are going to work with the Quadratic Standard Form. This is ax² + bx + c = 0. If you’d like more information on the standard form of other objects you can click here.
- Vertex — This dot shows the lowest or highest point on a parabola. Refer back to Maximum/Minimum Point for more information.
- Vertex Form — This form, as shown by: f (x) = a(x – h)2 + k, where h is the number of the vertex. If h is positive you’ll need to move the point left that many places, or if it is negative you will need to move it right. k is how many places the vertex goes up/down; this part follows a pattern normally where positive is up and negative is down. Click here for an picture of a parabola I made using the vertex form.
- Zeros/Roots/Intercepts — This is where the parabola touches the x axis, shown by any graphing calculator ever.
- Using Factoring
- To factor, you need to have a basic understanding of F.O.I.L
- To Foil an equation it must be in a format such as (x+2)(x-2).
- Interesting Factoid: Foil is derived from the distributive property: 7(x-3) which equals (7x-21)
- Below is an example of how foil works. Thanks Wikipedia!
- To factor, we do the opposite of foil
- We’ll use the answer to the first problem, (x+2)(x-2), which is x2 – 4
- To Factor this easy problem we must figure out what equals 4 in multiplication so, 1, 2, & 4.
- Since we know 2 squared is 4 and x squared is x2. We know this is a difference of squares problem, which means that the answer is:
- x2 – 4
- (x ) (x )
- (x – ) (x + )
- (x – 2) (x + 2)
- To factor, you need to have a basic understanding of F.O.I.L
- Using the Quadratic Formula
- Using a Table/Graph
- Using a table/graph is the easiest way to find zeros. Tables you just find where x equals zero and for graphs you find where the line hits the x-axis. See here for more info.
Graphing a Quadratic Function
- Direction of Opening
- To determine the direction of opening easily, just look at a in the standard form of the equation, ax2 + bx + c = 0. If A is positive, you can imagine it like an open bowl, if a is negative the bowl is upside-down. See More Info Here.
- Axis of Symmetry
- A very simple concept, to find the Axis of Symmetry line in a parabola, just find where the vertex in and drop a vertical line through the point. The easiest way to find this is using the Vertex Form. See Above.
- The Highest/Lowest Point on a parabola Click Here for more information about where the Vertex is.
- This point is where the parabola meets the y-axis. For an info-graphic showing the y-intecept, click Here.
- Point Symmetric to the Y-Intercept
- This is the point on the parabola that has the same y-value as the y-intercept. For an info-graphic explaining this, click here.
- Table of Values
- This is the table of x and y values for a set parabola. For an example see this one that I made.
- Maximum/Minimum Value
- This is the Maximum/Minimum value the parabola goes to. For an example, see this info-graphic.
- Double Root — These is a problem where x equals only one solution like in f(x) = x2 −10x + 25. In this equation, The answer is (x-5)(x-5) = (x − 5)^2, which equals x = 5. For a full dissection of this problem see this info-graphic of the information, or this website that this information was provided from.
- Two Real Solutions — This is a problem where the answers are exact numbers. For an example on this king of answer look at this info-graphic pulled from mathisfun.com
- Two Complex Solutions — This is a problem when both answers of a parabola contain imaginary numbers! EEEKK! (The Hardest One!!!) For this, look at this info-graphic describing complex solutions.
Writing a Quadratic Equation
- Given the Roots
- The easiest way to find the quadratic equation, ax2 + bx + c = 0
- For Example:
- Take your roots, for this problem we’ll say our root is a double root of 5
- x = 5
- (x – 5) (x – 5)
- x*x, x*-5, x*-5, -5*-5
- x2 -10x +25 = 0
- And Ta-Da!
- Given the Transformations
- This one is the hardest, being that it has the most steps. For this example, we’ll use the following directions: Left 5, down 4, flip over x
- So we’ll place these in the vertex form equation.
- f (x) = -1(x + 5)2 – 4
- f (x) = -1(x + 5)(x + 5) – 4
- f (x) = (–x – 5) (x+5) – 4
- f (x) = –x2 – 10x -29
- -x2 – 10x -29 = 0
- Here is the graph of the vertex form, and here is the graph of the standard form; you can see they are the exact same.
- Vertical Shift — This is the c value at the end of the quadratic equation. Also known as the y-intercept, the graph is moved up or down based on this number.
- ax2 + bx + c
- Horizontal Shift — This is how many points the graph is moved left or right. It is easily shown by the vertex form equation, f (x) = a(x – h)2 + k, where h is the horizontal shift, except in this case you do the opposite of that number.
- For more information on Horizontal & Vertical Shifts, see here.
- Vertical Stretch or Compression & Horizontal Stretch or Compression
- For this, the easiest way for me to explain it in a way the casual reader will understand is to direct them to this pdf.
- Reflections over the x-axis and y-axis.
- The easiest way to describe this with this example.
- f (x) = a(x – h)2 + k
- If a is negative in this equation, then your graph will be on the opposite side of the x axis.
- If a toy rocket is launched vertically upward from ground level with an initial velocity of 128 feet per second, then its height h after t seconds is given by the equation h(t) = −16t2 +128t (if air resistance is neglected).a. How long will it take for the rocket to return to the ground?
- 8 seconds for the rocket to return to the ground
- b. After how many seconds will the rocket be 112 feet above the ground?
- 1 second for the rocket to reach 112 feet above the ground.
For More Information on some of this information we’ve learned in Algebra II so far, look at mathisfun.com.
Again, this post has been made for a project at Pottsboro High School for Coach Johnson’s Algebra II Pre-AP Class. It was made to be an information resource hub where future students can come to find resources for their projects. However, feel free to use this information for anything else but your project!
Works Cited: All information has either been brought forth from previous knowledge, or great websites such as mathisfun.com. If you see property of yours on here, where you do not have credit, and would like it, or would like it to be removed please notify the author at: firstname.lastname@example.org