Angry Birds Project, Day 1 and 2

For those of you who have been in and out of class during this 3-day week, we did a 2-day project on how parabolas and quadratic functions related to the game Angry Birds. Each bird launched makes a parabola - upside-down, of course, because the path of the bird travels in an n-shape rather than a u-shape.

For the first day, we analyzed a graph of a traveling angry bird. We compared the path that the bird travels with the vertex, horizontal (x-axis) distance traveled, and the vertical (y-axis) height of the parabola.

In the second day, we designed our own Levels for the angry birds. Each level had to contain at least three targets, and you had to give me a quadratic equation (in vertex form) that would get the bird to hit each target in your Level. Shout out to Mia Goodwin, Hunter Ezell, and Tayda Galloway for some amazing-looking Levels! Shout out to Bruce Johnson for making his targets angry-squids! And shout out to Kindall because all of my 7th period students really liked your Level - great job!

Graphing a Parabola in Vertex Form

Vertex form is different from standard form, it looks like: y = 2(x - 3)^2 + 4. The 3 inside the parentheses tells us where to move on the x-axis, but the OPPOSITE because it's inside parentheses. The 4 at the end tells us where to move on the y-axis (not opposite because it's not inside parentheses). Together, (3, 4) makes the vertex of the parabola.

We can treat the 2 on the outside of the parabola like a slope. From your vertex, go up 2, over one on both sides of the function. Connect them with a curve at the bottom to make your parabola.

There are a few special cases that we should note:

y = 1/2(x + 3)^2 + 3 - the 1/2 in front of the parentheses tells us the parabola will get wider, not thinner.
y = (x + 3)^2 - this means the vertex will remain on the x - axis
y = 3x^2 + 1 - this means the vertex will remain on the y-axis
y = x^2 - when there are no numbers, the parabola remains on the origin of the coordinate plane.

Graphing a Parabola in Standard Form

Standard form of a parabola is when we see f(x) = x^2 + 5x + 4: a quadratic term with x-squared, a linear term with x, and a constant term which can be any number. When we do this, our goal is to first factor our function by grouping with parentheses. Ours would factor into (x + 1)(x + 4). Then, set them equal to zero and solve: this gives you the x-intercepts. x + 1 =0 and x + 4 = 0. When you solve to get x by itself, you get x = -1 and x = -4. Graph these and find their midpoint on the x-axis (x = -2.5). This is the x-coordinate of the vertex of the parabola. Then, use the formula y = -b/2a to find the y-coordinate of the vertex. Ours would be y = -5(2)(1) = -2.5. Thus, our vertex is (-2.5, -2.5) and we can plot that point on the graph. These three points form the foundation of the parabola: connect them and graph.

Note that this vertex would be at the bottom, so it would be a minimum of the function: it is the lowest point in the function. If the parabola was upside-down, the vertex would be a maximum because it would be at the top of the function.

Parabola Vocabulary

Parabolas are functions that contain x-squared within them in some form. They can have up to three terms: a quadratic (x-squared) term, a linear term (x), and a constant term (a number), so that the parabola takes the form f(x) = ax^2 + bx + c. For example, f(x) = 2x^2 + 5x + 9. However, this does not have to be the case. As long as the function contains x-squared within it, it can be a parabolic function.

A parabola is shaped like a "U" if the number in front of x-squared is positive, and an upside-down U (n) if the number in front of the x-squared is negative. The middle of a parabola is the axis of symmetry, and the top/bottom of the axis of symmetry is called the vertex. The larger the number in front of the x-squared term, the thinner the function will be. The smaller the number in front of the x-squared term, the wider the function will be.

Systems of Inequalities

This is the last topic in systems of equations! Systems of inequalities are very similar, except instead of finding x and y, we are graphing both inequalities on a coordinate plane and shading the solution area for both.

For example:
y > 2x + 1
y < -3x - 3

To solve this question, graph both systems on the same coordinate plane and shade the side of the graph that tests true when you plug in (0, 0) to both inequalities. The area of shading that overlaps for both inequalities will be your solution region. This means that ANY point that you plug in within that region will work for both x and y.

Review for the Systems of Equations Quiz

Tomorrow is the last quiz of the nine weeks! Be sure to bring your 1/2 page cheat sheet and study your examples from today so that you can finish the nine weeks strong! This quiz is not meant to be tricky or confusing. I hope that you all do well and get A's - it is exactly like the review we did in class today.

In case you missed class, we went over the following topics for the quiz tomorrow:
-Solving with substitution
-Solving with elimination
-Solving systems with 3-variables using Cramer's rule (determinants)
-Plotting a point in 3-D
-Plotting a line in 3-D (looks like a triangle shaded in on a coordinate plane).

Know all of this? Then you should be all set to ace the quiz tomorrow! If you put a BOX around your name on the top of your page I will add TWENTY BONUS POINTS to your quiz score!

Plotting Points and Graphing Lines in 3-Dimensions

A 3-D coordinate plane looks like this:

where the x-axis comes out towards you, the y-axis is now the horizontal axis, and the z-axis is now the vertical axis. To plot a point in 3-D, you must first go over by your x, then over by your y, then up/down by your z. Your answer might not look like it makes sense - AND THAT'S OKAY! We're trying to graph something in 3-D on a 2-D piece of paper, so naturally, it's not going to look 1005 perfect. Just make sure you get the point in the general correct area of the graph.

To graph a line in 3-D, such as 3x + 4y + 3z = 12, we substitute zeroes in for every variable that we're not trying to find. For example, when trying to find x, I would keep x and plug in 0's for y and z, like this: 3x + 4(0) + 3(0) = 12. The zeroes would cancel out, and when solving for x, divide by 3 on both sides to get x = 4. Plot this on the 3-D plane. Repeat this process for the other two variables and plot them as well. Your final answer should be three points: connect them together and make a triangle, then shade in the triangle to represent the plane of answers.

Solving 3-Variable Systems of Equations

There are many ways to solve 3-variable systems of equations. The three most common are substitution, elimination, and Cramer's Rule. I chose to teach Cramer's rule because it builds off of previous concepts, and because it is the most straightforward (in my opinion) of the three methods.

To begin, you'll have a problem that looks something like this:
2x + y - z = 5
3x + 5y - 4z = 10
x - 2y + 2z = 12

where you will need to solve for x, y, and z, and your answer will be a coordinate triplet, meaning that it can be graphed on a 3-D coordinate plane. First, make a 3 x 3 matrix of all the coefficients to the left side of the equals sign. Find that determinant, and label it "D."

Next, make another 3 x 3 matrix but leave out the column of x's (so your first column should be blank). Where the x coefficients used to be, insert the numbers to the right of the equals sign. Find that determinant, label it "Dx." Repeat this process for both y and z, leaving out those variables' numbers and replacing them with the numbers to the right of the equals sign for each matrix. Find those determinants and label them "Dy" and "Dz."

Almost done! To get your final values of x, y, and z, simply divide the following to get your answer:
x = Dx/D,      y = Dy/D,      z = Dz/D

Solving 2-Variable Equations using Elimination

We solve 2-variable systems of equations using elimination whenever both x and y are on the same side of the equals sign for both equations. The idea of "elimination" is that we eliminate, or cross out one of the variables so that we can solve for one of them. Then, once you have one variable, plug it back in and solve for the other one. Express your answer as a coordinate pair.

Example: Solve the system of equations.

3x + 4y = 2
-3x - y = 7        -Cross out your x's first because 3x cancels out with -3x. Add the remaining terms.
       3y = 9       -Solve for y
         y = 3

Now, plug y = 3 into either of the equations and solve for x.
3x + 4(3) = 2
3x + 12 = 2    -Multiply
3x = -10         -Subtract 12 on both sides
x = -3.33        - Divide by 3 on both sides

Answer: (-3.33, 3) Express it as a coordinate pair.

Guest Speaker and Random Meeting

We had a guest speaker yesterday from Arkansas State University! Shy the master's student told us about the Oral Communications class that she teaches to college freshman, and what it's like to be a freshman on the A-State campus. Expectations are a lot different in college - especially when your class only meets one day a week but you have it for 3-hours at a time!

I was pulled from my second period class today (just as we were about to graph y = sinx - boo!) to attend a training session at the Junior High. My apologies for this one - I didn't know about it in advance. A BIG THANK YOU to everyone who practiced their math ACT while I was gone and a HUGE Shout Out to my aid Bre Womack for setting everything up while the sub was there!

Solving 2-Variable Systems of Equations by Substitution

We're done with matrices and are moving onto systems of equations. These are linear equations (so we expect at least two variables) that intersect each other at some point. The point is the answer to our system of equations. With two variables, that means it has to have at least an x answer and a y answer. We display our answer as a coordinate pair (x,y).

To solve using substitution, we have two equations, and expect one variable to be by itself on one side of the equals sign. We substitute that variable, solve for the remaining variable, and find the solution to one variable. Then, once we have that solution, we plug it back into the first equation to get the answer to our other variable.

FOR EXAMPLE:

Solve the system of equations:
y = 2x
6x - 2y = 12

First, we know that y = 2x. That means whenever we see a "y" in the second equation, we can put "2x"

Second equation: 4x - 2(2x) = 12    - substitute
                           6x - 4x = 12         - multiply
                                   2x = 12        - combine like terms
                                     x = 6          - divide by 2 on both sides to get x = 6

Then go back to the 1st equation y = 2x. Since we know what x is (we just found it, x = 6), we can plug the 6 in for the x and solve for the y:

y = 2x
y = 2(6)  - plug in x = 6
y = 12    - multiply to solve

Note that it doesn't matter which variable we find first - the x or the y - because we're going to need both of them to represent the answer. Write your answer as a coordinate pair: (6, 12) since it represents a point on the coordinate plane where your lines intersect.

Matrix Test and Finding the Intersection of Two Lines

Good job on those matrix tests! People had the most trouble with the "solving matrix equations" questions, but other than that, I'd say that the average person did fairly well on everything else and at least half of you managed to answer one of the bonus questions correctly - great job!

Our next unit of study will be systems of equations in two and three dimensions. To prepare for this, I left some subwork today (I was on the field trip to Little Rock) where you had to plot two lines on the same coordinate plane and then tell where those two lines intersected, or crossed, one another. More information on that tomorrow...

If I didn't get your review sheet then tomorrow is the last late day that you can turn it in for partial credit, otherwise, just keep it for no credit.

Sub Day (extra day to study matrices!!)

I was at a 1/2 day meeting today with the math department, so, you lucky ducks, your matrix test is pushed back until tomorrow. The study guide given to you by the sub was due back by the end of the class period and should have helped refresh your memory for some of the key concepts you'll need for tomorrow.

Review sheets are also due tomorrow - bring them back!