Understanding a graphing equation

[ILovePizza]

The height, h metres, of an infield fly ball “t” seconds after being hit is given by the equation h=30t – 5t^2.

a) Graph the equation on a graphing calculator.
b) Determine how high the ball is after 2.3 s.
c) Determine when the ball is at a height of 25 m.
d) Determine the maximum height of the ball.

I am new to graphing, and have only recently been introduced to a graphing calculator (TI-84 Plus Silver Edition).

The above question is part of our book, and has the solutions in the back; however, I have an old book, and someone has torn out my solutions, so I’m having difficulty understanding how to do these types of questions.

I’m not sure how to even start this one, and have played with the calculator, but again, I’m unsure where I should even enter the equation.

Thanks in advance for any assistance offered.

 

[mmm4444bot]

… someone has torn out my solutions, so I’m having difficulty understanding how to do these types of questions …

Did someone tear out all of the lessons, too? :wink:

I'm thinking that you don't have the user's guide for this calculator; I don't, either.

If memory serves, there should be a button below the bottom left corner of the display labeled, "Y=".

Press it, and enter 30*X-5*X^2.

Next to that button, there is a button labeled, "Window".

Press it, and set the following parameters.

xMin = 0
xMax = 6
xScl = 1
yMax = 50
yMin = 0
yScl = 5

Now, press the button that's labeled, "Graph".

'

… b) Determine how high the ball is after 2.3 s.
c) Determine when the ball is at a height of 25 m.
d) Determine the maximum height of the ball. …

For part (b), you simply replace the variable t with the given value of 2.3, and do the math to find the corresponding value of h.

For part (c), you simply replace the variable h with the given value of 25, and solve the equation for t.

For part (d), you can use the following formula to find the t-coordinate of the vertex point (i.e., the maximum value of h).

Given h = A*t^2 + B*t + C

The t-coordinate of the vertex is -B/(2A).

Calculate the t-coordinate of the vertex point, and then substitute it into the formula for h to find the corresponding height at the vertex.

Note: The independent variable in this formula is t, but you need to use X in your calculator when graphing because that model understands (X,Y) points (if memory serves).

If I wrote anything that you do not understand, then please ask specific questions.

Please post your work, if you want more help with this exercise, so that people might determine where to continue helping you.

 

[ILovePizza]

I appreciate your response; thank you!

I have to say I find this type of math very, very frustrating...almost to the point of complete demoralization. I have been working on this for some time now, and it just wears you down until you are completely frustrated. Sorry, just had to vent.....

Ok, I got the graph, and I understand that part now (I think...), but what does the graph provide, other than a pretty picture on a small screen (smile).

How to I, and where do I input 2.3 for t?

I think I’m just frustrated with this, and can’t think properly...

Sorry!

 

[mmm4444bot]

The information that I posted for parts (b) through (d) is for doing the calculations using paper and pencil.

I suppose that you could use the Trace button (with the left- and right-cursors keys) to move the "bug" along the graph and view the bug's coordinates at the bottom of the screen.

However, you won't be able to scroll to exactly X = 2.3; you would need to first repeatedly zoom in on that section of the graph, in order to get close to X = 2.3.

Same goes for trying to trace to the location where Y = 25.

I think that students should be able to answer these types of questions by hand, before they use a graphing calculator as a tool.

That model calculator does have a way for you to enter 2.3 as the value of X, while you are viewing the graph, and then the bug will jump to X = 2.3 exactly -- showing the corresponding value of Y. I do not remember which button you press to get to the CALC menu (short for "calculations"). If you can get the CALC menu to display, then I think you select the first choice (Value), followed by entering 2.3.

I'm not sure if that model will solve the equation 25 = 30*X - 5*X^2, automagically.

(If you go to ti.com, you can download a PDF copy of the user's manual.)

Do you understand the information that I posted for doing parts (b) through (d) by hand?

 

[ILovePizza]

I'm not sure what I'm doing any more....

I pressed the "Trace" button and received the following in regards to the graph: x=3 and y=45
Regardless, I tried to figure out the equation you noted: 25 = 30*X - 5*X^2

I came up with the following answer: x = 1, 5

Does any of this make sense?

Thanks for hanging in there!

 

[ILovePizza]

I have to call it a night; I've worked on this for so long, that my brain can't take it any-more. I'll check back before school in the morning.

Thanks for your help!

 

[Mrspi]

Does any of this make sense?

No, not really.

You SHOULD be able to draw the graph "by hand" as has been suggested.

But, you have a WONDERFUL calculator. Once you have had the calculator draw the graph for you, using the "trace" function should let you find (at least approximately) the value of the function for a given value of t (or x, as you entered the function in your calculator).

Are you looking for the value of the function when t = 2.3? Use the TRACE function....move the cursor until you get as close as you can to x = 2.3 (if you want to get closer, use the ZOOM feature of your calculator.)

You have a wonderful tool, one that those of us who studied math in the "old days" did NOT have available. The trade-off, I think, is that you young folks who rely on calculators don't know how to use them!

 

[mmm4444bot]

I pressed the "Trace" button and received the following in regards to the graph: x=3 and y=45

Yes, the point with coordinates (3, 45) is on the graph. Since x (this variable is actually t) represents the number of elapsed seconds since the baseball was hit, these coordinates tell us that the baseball is 45 feet above the ground three seconds after being hit.

That's probably the default location where the calculator put the "bug", after you pressed trace. Did you see the bug on the graph? (Maybe, it looks like an X, on that model.)

After pressing the trace button, you can trace along the graph by pressing the cursor-right and cursor-left buttons to move the bug.

Regardless, I tried to figure out the equation you noted: 25 = 30*X - 5*X^2

I came up with the following answer: x = 1, 5 << This is correct!

Do you understand what these two values are telling us?

(Of course, in terms of the given height function, these are values of t.)

t = 1 tells us that the baseball is 25 feet above the ground one second after it's hit (on its way up).

t = 5 tells us that the baseball is also 25 feet above the ground five seconds after it's hit (on its way down).

Ask one of your classmates how to get the bug to jump to x = 2.3 using the "Value" choice from the CALC menu.

To do part (b) by hand, we find the height of the baseball 2.3 seconds after being hit by substituting 2.3 for t in the given formula for h.

h = 30(2.3) - 5(2.3^2)

Ask one of your classmates how to get the calculator to show the maximum value of h. (Again, I think it's a selection from the CALC menu; possibly "Maximum".)

To do part (c) by hand, use the formula that I gave you to determine the value of t at the vertex point.

t = -b/(2a)

Since a = -5 and b = 30 in this quadratic polynomial, we get:

t = -30/[2(-5)]

Once you know this value of t, substitute it into the formula for h to find the maximum height.

After you become savvy with graphs of parabolas, and learn that the vertex lies on the axis of symmetry (halfway between the x-intercepts), you'll be able to see from the graph that the x-intercepts are 0 and 6, so the vertex must occur at the value of t halfway between 0 and 6 (which is 3).

Same result as calculating t = -b/(2a), just a different (graphical) method.

(The computer I'm using today cannot display .PDF files, otherwise I would have gone to ti.com to see how you get to the CALC menu. As Mrs. Pi noted, the graphing calculator is not much of a tool, unless you understand how to use it. Maybe, your school needs to do a better job of providing specific calculator instructions. Or, maybe, you were out sick that day. Talk with your classmates, and visit ti.com, if you have time.)

 

[ILovePizza]

Your help has been greatly appreciated! I feel much better after a night’s sleep; the lesson I should learn, is – not to spend so much time on one problem, as it can become overwhelming, and frustrating to where I am no longer productive. I need to walk away after 15, or so minutes, and return later to tackle it in a fresh light.

As for the calculator, it belongs to my brother, and is not one provided by the school; I will download the manual today.

Your guidance has shed some light of this issue, which I will capitalize upon today, and ask my friends questions, and bombard my teacher until she kicks me away from her desk (smile)...not really, I just need clarification on a couple of things.

Again, thank you for your patience, and guidance; I believe I’m starting to understand this....