sdmurray

07-02-2011, 10:46 AM

In the community you live, you have been asked to serve on the planning committee for

the Fourth of July festival. Your committee has decided to begin meeting early to avoid

some of the problems experienced in previous years. Unfortunately, those problems

stemmed from poor budgeting and record keeping. To ensure that the festival at least

breaks even and does not cost more than it brings into the town, you are reviewing

information kept from the previous year. You are particularly interested in the big name

band brought in, and determining how to make the most profit from ticket sales this

coming year. You see this information:

Revenue from ticket sales:

Cost of band:

Profit: 160 (given in thousands, use 160)

1. The equation to determine profit is Revenue – Cost = Profit. Write the equation

that explains the profit made last year. Be certain to combine like terms.

NOTE! Please post a note to me in your IF when you arrive at this equation. I

want to check your answer; if the answer is not correct, the rest of the appendix

will not be correct. Make the subject line: Check #1 on Appendix

2. If you graphed this equation, would the graph open up or down? How could you

tell without graphing it?

3. Given the shape of the graph, what happened to ticket sales as time passed?

4. Given the equation you arrived at in question #1, what two days were the break-

even points for ticket sales? (The break-even points are the x-intercepts, where

the graphed parabola crosses the x axis).

AAGN0TY9T1Hoey—MAT117

5. At what day did the peak of profit from ticket sales occur? (hint, this is the line of

symmetry)

6. At that day, what was the profit from ticket sales? (hint, this is the maximum)

7. What was the vertex of the graphed line?

8. Logical conclusion: As you look at the graph, what day do you believe the

committee should have stopped attempting to sell tickets? Explain.

How to find the line of symmetry and minimum or maximum of a parabola:

Warning. I think through this process differently than the book explains, though we arrive at the same solutions. That is common in math. I’m going to show you the way I figure it out. If you understand the book’s explanation, then by all means, use that one. You will arrive at the same answer, either way.

The vertex of a parabola is the highest or lowest point of the curve. The line of symmetry is the x value of the vertex. The minimum or maximum is the y value of the vertex.

MINIMUM (opens up) MAXIMUM (opens down)

We have a minimum if the U opens up, because the curve is down on the bottom.

We have a maximum if the U opens downward, because the curve is at the top.

To me, the simplest way to find the minimum or maximum is by first finding the axis of symmetry. The axis of symmetry is the line that cuts through the perfect center of the parabola. The axis of symmetry will tell us the x value. The axis of symmetry in this graph is at -4. Therefore, x = -4. The minimum or maximum is going to be the y value. We can visually see the y is at 0, so the minimum would be 0 for this graph. The vertex would be (-4, 0).

How do you figure that out if you don’t see the graph? You use math.

The formula for finding the line of symmetry is:

x=-b/2a

Let’s say we have the polynomial of: -2(x+3)^2

If we multiply that out, we would get:

-2(x+3)(x+3)

-2(x^2+6x+9)

-2x^2-12x-18=0

Now, we can see the “a” term is -2. The “b” term is -12. The “c” term is -18. To find the line of symmetry, which will be the x value, we’ll use the equation:

x=-b/2a=-(-12)/2(-2) =-(-12)/(-4)=-3

Catch all of those negatives. The -12/-4 would be positive 3, but there is a negative in front of the fraction, which makes it negative again.

So, if we graphed the equation -2(x+3)^2 we would find the perfect middle of it would be at x = -3. Now, to find the y value, or the minimum or maximum, we’ll substitute -3 for x in our polynomial:

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

y=-2(0)^2

y=0

So, the vertex of our polynomial, when graphed, is at (-3, 0). We know that we will have a maximum of 0, because zero is the y of our ordered pair. How do we know this will be a maximum? Here is the trick. Look at our “a” term. It was -2. Notice it was a negative? If the a is negative, we have a maximum. If the a is positive, we have a minimum.

a < 0 (negative) = maximum because parabola opens down

a > 0 (positive) = minimum because parabola opens up

Here is the graph of -2(x+3)^2

Class, as you wrap up your test this weekend, I wanted to remind you of this really important difference.

When a question asks you for the solutions for x, you are to write it with a comma but NO parenthesis. For example, if you have:

(x + 2)(x - 4)

the two solutions for x are:

x = -2, 4

However, if you are asked to write the x INTERCEPTS, remember that an intercept is the point on a graph where the line crosses the x axis. That's referring to a coordinate, or ordered pair of (x, y). The y is always going to be 0, because it's right on the x axis. For the example above of (x + 2)(x - 4), the x-intercepts would be written as:

(-2, 0), (4, 0)

Notice I gave two ordered pairs with parenthesis around them, and there is a comma between them. If you don't format it right, it's going to be counted wrong, and for good reason. If you just wrote (-2, 0)(4, 0) that would be insinuating you would somehow try to multiply those two together.

Last, if you are asked to write the vertex, the vertex is an ordered pair that contains an x and a y coordinate. That would be written as (-2, 4).

the Fourth of July festival. Your committee has decided to begin meeting early to avoid

some of the problems experienced in previous years. Unfortunately, those problems

stemmed from poor budgeting and record keeping. To ensure that the festival at least

breaks even and does not cost more than it brings into the town, you are reviewing

information kept from the previous year. You are particularly interested in the big name

band brought in, and determining how to make the most profit from ticket sales this

coming year. You see this information:

Revenue from ticket sales:

Cost of band:

Profit: 160 (given in thousands, use 160)

1. The equation to determine profit is Revenue – Cost = Profit. Write the equation

that explains the profit made last year. Be certain to combine like terms.

NOTE! Please post a note to me in your IF when you arrive at this equation. I

want to check your answer; if the answer is not correct, the rest of the appendix

will not be correct. Make the subject line: Check #1 on Appendix

2. If you graphed this equation, would the graph open up or down? How could you

tell without graphing it?

3. Given the shape of the graph, what happened to ticket sales as time passed?

4. Given the equation you arrived at in question #1, what two days were the break-

even points for ticket sales? (The break-even points are the x-intercepts, where

the graphed parabola crosses the x axis).

AAGN0TY9T1Hoey—MAT117

5. At what day did the peak of profit from ticket sales occur? (hint, this is the line of

symmetry)

6. At that day, what was the profit from ticket sales? (hint, this is the maximum)

7. What was the vertex of the graphed line?

8. Logical conclusion: As you look at the graph, what day do you believe the

committee should have stopped attempting to sell tickets? Explain.

How to find the line of symmetry and minimum or maximum of a parabola:

Warning. I think through this process differently than the book explains, though we arrive at the same solutions. That is common in math. I’m going to show you the way I figure it out. If you understand the book’s explanation, then by all means, use that one. You will arrive at the same answer, either way.

The vertex of a parabola is the highest or lowest point of the curve. The line of symmetry is the x value of the vertex. The minimum or maximum is the y value of the vertex.

MINIMUM (opens up) MAXIMUM (opens down)

We have a minimum if the U opens up, because the curve is down on the bottom.

We have a maximum if the U opens downward, because the curve is at the top.

To me, the simplest way to find the minimum or maximum is by first finding the axis of symmetry. The axis of symmetry is the line that cuts through the perfect center of the parabola. The axis of symmetry will tell us the x value. The axis of symmetry in this graph is at -4. Therefore, x = -4. The minimum or maximum is going to be the y value. We can visually see the y is at 0, so the minimum would be 0 for this graph. The vertex would be (-4, 0).

How do you figure that out if you don’t see the graph? You use math.

The formula for finding the line of symmetry is:

x=-b/2a

Let’s say we have the polynomial of: -2(x+3)^2

If we multiply that out, we would get:

-2(x+3)(x+3)

-2(x^2+6x+9)

-2x^2-12x-18=0

Now, we can see the “a” term is -2. The “b” term is -12. The “c” term is -18. To find the line of symmetry, which will be the x value, we’ll use the equation:

x=-b/2a=-(-12)/2(-2) =-(-12)/(-4)=-3

Catch all of those negatives. The -12/-4 would be positive 3, but there is a negative in front of the fraction, which makes it negative again.

So, if we graphed the equation -2(x+3)^2 we would find the perfect middle of it would be at x = -3. Now, to find the y value, or the minimum or maximum, we’ll substitute -3 for x in our polynomial:

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

y=-2(0)^2

y=0

So, the vertex of our polynomial, when graphed, is at (-3, 0). We know that we will have a maximum of 0, because zero is the y of our ordered pair. How do we know this will be a maximum? Here is the trick. Look at our “a” term. It was -2. Notice it was a negative? If the a is negative, we have a maximum. If the a is positive, we have a minimum.

a < 0 (negative) = maximum because parabola opens down

a > 0 (positive) = minimum because parabola opens up

Here is the graph of -2(x+3)^2

Class, as you wrap up your test this weekend, I wanted to remind you of this really important difference.

When a question asks you for the solutions for x, you are to write it with a comma but NO parenthesis. For example, if you have:

(x + 2)(x - 4)

the two solutions for x are:

x = -2, 4

However, if you are asked to write the x INTERCEPTS, remember that an intercept is the point on a graph where the line crosses the x axis. That's referring to a coordinate, or ordered pair of (x, y). The y is always going to be 0, because it's right on the x axis. For the example above of (x + 2)(x - 4), the x-intercepts would be written as:

(-2, 0), (4, 0)

Notice I gave two ordered pairs with parenthesis around them, and there is a comma between them. If you don't format it right, it's going to be counted wrong, and for good reason. If you just wrote (-2, 0)(4, 0) that would be insinuating you would somehow try to multiply those two together.

Last, if you are asked to write the vertex, the vertex is an ordered pair that contains an x and a y coordinate. That would be written as (-2, 4).