Interactive Manual › Chapter 4

Chapter 4: Parametric Graphing

Graph curves in which X and Y are both defined by a third variable T: Par mode, the X1T/Y1T… editor, Tmin/Tmax/Tstep, and exploring parametric graphs with TRACE, ZOOM and the CALC menu.

Getting Started: Path of a Ballp. 91

A ball is hit from ground level at 30 meters per second, at an angle of 25 degrees above the horizontal. Ignoring everything except gravity, how far does it travel, when does it land, and how high does it climb? Parametric graphing is made for questions like this, because position depends on time.

With starting speed v₀ and launch angle θ, the position at time T has a horizontal and a vertical part. The vertical one loses height to gravity, g = 9.8 m/s².

The three parametric equations used in the example
Quantity	Formula	Entered as
Horizontal position	X(t) = t·v₀·cos(θ)	`X1T = 30T cos(25°)`
Vertical position	Y(t) = t·v₀·sin(θ) − ½·g·t²	`Y1T = 30T sin(25°) − (9.8/2)T²`
Vertical component vector	drops straight down to the ground	`X2T = 0`, `Y2T = Y1T`
Horizontal component vector	slides along the ground	`X3T = X1T`, `Y3T = 0`

The tutorial below graphs the ball together with its two component vectors, so you see the whole motion at once. Do the steps in order; the calculator highlights each key.

Tutorial

Graph the path of a ball and read distance, height and time

You will switch to Par and Simul modes, turn the axes off, define three parametric equations, set the window, graph, and then trace.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the MODE screen.
MODE
Move down to the Func / Par / Pol / Seq row (it is the fourth row) and select Par, so that the Y= editor and graph work with parametric equations.
▼︎▼︎▼︎▶︎ENTER
Move down to the last row (Sequential / Simul) and select Simul. All three equations will now be drawn at the same moment instead of one after the other.
▼︎▼︎▶︎ENTER
Open the FORMAT screen, where the graph format is set.
2ndFORMAT
Move down to the axes row and select AxesOff. The graph will be drawn without axes.
▼︎▼︎▼︎▶︎ENTER
Leave the format screen and open the parametric Y= editor. It has an X line and a Y line for each equation.
2ndQUITY=
Define X1T = 30T cos(25°). The X,T,θ,n key types T in Par mode. The degree sign is item 1 of the ANGLE menu, 2ndANGLE; close the parenthesis of cos( and press ENTER.
30X,T,θ,nCOS252ndANGLE1)ENTER
Define Y1T = 30T sin(25°) − (9.8/2)T². Type the first part, press −, then open the fraction template with ALPHAF1 and choose 1:n/d. Type 9.8, move right with ▶︎ to the denominator, type 2, move right again to leave the fraction, and finish with T² (X,T,θ,n then x²).
30X,T,θ,nSIN252ndANGLE1)−ALPHAF119.8▶︎2▶︎X,T,θ,nx²ENTER
The vertical component vector is made of X2T and Y2T. Define X2T = 0.
0ENTER
Define Y2T = Y1T by pasting it from the Y-VARS menu. Press VARS, move to the Y-VARS tab with ▶︎, choose 2:Parametric, then 2:Y₁T, and press ENTER to store the line. (You can also reach these variables from the quick menu ALPHAF4; pick its PAR tab.)
VARS▶︎22ENTER
The horizontal component vector is made of X3T and Y3T. Paste X3T = X1T the same way (2:Parametric, 1:X₁T).
VARS▶︎21ENTER
Define Y3T = 0.
0ENTER
Now choose graph styles. Press ◀︎ ◀︎ to reach the style icon at the left of the line you are on, then ▲︎ to the icon of Y3T, and press ENTER once to change it from line to thick. Then ▲︎ to X3T and press ENTER once for thick as well. These are the two lines of the horizontal vector.
◀︎◀︎▲︎ENTER▲︎ENTER
For the vertical vector, go up to Y2T and press ENTER twice (line, thick, path). Go up to X2T and do the same. Then repeat for Y1T and X1T so the ball itself is drawn with the path style too. (The path icon draws a small circle that leads the curve.)
▲︎ENTERENTER▲︎ENTERENTER▲︎ENTERENTER▲︎ENTERENTER
Open the window editor and enter the values. Each ENTER stores a value and moves down to the next variable: Tmin=0, Tmax=5, Tstep=.1, Xmin=⁻10, Xmax=100, Xscl=50, Ymin=⁻5, Ymax=15, Yscl=10. Use the (−) key for the minus signs.
WINDOW0ENTER5ENTER.1ENTER(−)10ENTER100ENTER50ENTER(−)5ENTER15ENTER10ENTER
Press GRAPH. You watch the ball fly while the two component vectors are drawn alongside it. X, Y and T are updated as the plot runs. (To make the ball look like it is flying through the air, set the style of X1T and Y1T to animate.)
GRAPH
Now answer the questions with TRACE. Tracing begins at Tmin on the first equation (X1T, Y1T) and moves one Tstep at a time. X is the distance, Y the height and T the time, shown at the bottom of the screen.
TRACE
Type a T value to jump straight to it. At T=1.3 the ball is near the top of its flight: it is about 8.2 meters high, and about 35.3 meters downrange. (The exact peak, at T≈1.295, is 8.20 meters.)
1.3ENTER
Jump to T=2.6. The height has just dropped below zero, so the ball has landed a moment earlier (at T = 2·30·sin 25°/9.8 ≈ 2.59 seconds). The distance covered is about 70.4 meters.
2.6ENTER

Note You can see every window variable, including ΔX and ΔY, on the VARS Window menu (VARS then 1:Window). They cannot be typed in the WINDOW editor.

Defining and Displaying Parametric Graphsp. 93

In a parametric graph, X and Y are each a function of a third variable, T. As T grows from Tmin to Tmax, the calculator plots the point (X(T), Y(T)) and joins the points. This is how you graph things that are not functions of X (circles, loops, spirals) and anything that unfolds over time.

TI-84 Plus Graphing Mode Similaritiesp. 93

Defining a parametric graph works almost exactly like defining a function graph, so this chapter only covers what is different. If you have not yet used the Y= editor, window settings, TRACE and ZOOM in Func mode, read Chapter 3 first.

Setting Parametric Graphing Modep. 93

Press MODE and choose Par on the fourth row. Do this before you enter window variables or equations: the Y= editor, the WINDOW editor and the stored equations all depend on the current graphing mode.

The graphing-mode row of the MODE screen
Graphing mode	What you enter in the Y= editor
Func	Y1 … Y0, each a function of X
Par	X1T/Y1T … X6T/Y6T, each defined by T
Pol	r1 … r6, each a function of θ
Seq	u, v, w, each a sequence in n

Tutorial

Switch to parametric graphing mode

On a freshly reset calculator the mode is Func. You will change it to Par and prove it by opening the Y= editor.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the MODE screen.
MODE
Move down to the Func / Par / Pol / Seq row and select Par.
▼︎▼︎▼︎▶︎ENTER
Return to the home screen.
2ndQUIT
Open the Y= editor. It now lists X₁T= and Y₁T= instead of Y₁=.
Y=

Displaying the Parametric Y= Editorp. 93

After you select Par mode, Y= shows the parametric editor. You can define up to six parametric equations, X1T/Y1T through X6T/Y6T. Each equation has two lines, an X component and a Y component, and both are written in terms of T. A very common use is to graph a quantity against time.

X1T … X6T

X1T=expression in TX2T=expression in T…X6T=expression in T

The horizontal (X) component of each of the six parametric equations. Both X and Y must be defined for the equation to graph. From the home screen or a program you can paste a name from the Y-VARS Parametric menu.

Where: VARS Y-VARS tab, 2:Parametric, odd-numbered items (1:X₁T, 3:X₂T, …)

Keys: VARS▶︎21

X1T=30T cos(25°) defines the horizontal position of a launched ball.

Y1T … Y6T

Y1T=expression in TY2T=expression in T…Y6T=expression in T

The vertical (Y) component of each of the six parametric equations. Every XnT line needs a matching YnT line.

Where: VARS Y-VARS tab, 2:Parametric, even-numbered items (2:Y₁T, 4:Y₂T, …)

Keys: VARS▶︎22

Y1T=30T sin(25°)−(9.8/2)T² defines the vertical position of the same ball.

Selecting a Graph Stylep. 93

The little icon at the left end of each line is its graph style. In Par mode the default is line, which connects the plotted points. Five styles are available for parametric equations: line, thick, path, animate and dot. (The two shading styles, above and below, are only for Func mode.)

Graph styles in Par mode, in the order ENTER cycles through them
Style	What it draws
Line	Connects the plotted points with a thin line (the default)
Thick	The same line, drawn thicker
Path	A circle moves along the curve and leaves a trail, so you see the order of plotting
Animate	A circle moves along the curve and leaves no trail
Dot	Only the individual plotted points

Tutorial

Change the graph style of a parametric equation

Each press of ENTER on the style icon moves to the next style: line, thick, path, animate, dot, then back to line.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the Y= editor. The cursor is on the first line, X1T.
Y=
Press ◀︎ until the cursor is on the style icon at the left edge of the line. Then press ENTER once to make X1T thick.
◀︎◀︎ENTER
Press ENTER three more times to pass through path and animate and arrive at dot. One more press would return to line.
ENTERENTERENTER
Press ENTER once more to go back to the line style, then GRAPH to see the circle.
ENTERGRAPH

Defining and Editing Parametric Equationsp. 94

Define or edit a parametric equation exactly as you would a function in Chapter 3. The independent variable is T, and in Par mode there are two ways to type it:

Press X,T,θ,n, which becomes T in Par mode.
Press ALPHAT, the green T above the 4 key.

A parametric equation always has two parts, X and Y, and you must define both of them. A graph needs both lines to have content.

Tutorial

Define and graph a circle

X = 5 cos(T) and Y = 5 sin(T) trace a circle of radius 5 as T runs from 0 to 2π.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the Y= editor. In Par mode it shows X₁T and Y₁T.
Y=
Define X1T = 5 cos(T): type 5, press COS, then X,T,θ,n for T, close the parenthesis and press ENTER.
5COSX,T,θ,n)ENTER
Define Y1T = 5 sin(T) on the next line the same way.
5SINX,T,θ,n)ENTER
Press ZOOM 6:ZStandard. It sets Xmin, Xmax, Ymin and Ymax to ⁻10 and 10 and resets the T window as well (Tmin=0, Tmax=2π, Tstep=π/24), then graphs.
ZOOM6
The circle looks slightly squashed because the pixels are not square. Press ZOOM 5:ZSquare to correct the proportions.
ZOOM5

Tutorial

Type the variable T both ways

XTTN and ALPHA 4 give the same T in Par mode, so these two equations are identical.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the Y= editor and type X1T = 2T with the X,T,θ,n key.
Y=2X,T,θ,nENTER
Now type Y1T = 2T using ALPHAT for T (the shorthand T in the key list means ALPHA 4).
2ALPHATENTER
Both lines now read 2T. Press GRAPH to see the straight line they draw.
GRAPH

Selecting and Deselecting Parametric Equationsp. 94

Only selected equations are graphed. In the Y= editor, an equation is selected when the = signs of both its X and Y lines are highlighted. Any number of the six equations can be selected.

To switch an equation on or off, put the cursor on the = sign of either its X line or its Y line and press ENTER. Selecting one line changes the other too, so the pair always stays together.

Tutorial

Deselect and reselect an equation

Two equations are defined: a circle and a line. You will switch the circle off so that only the line is graphed.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the Y= editor. The cursor starts on the first line, X1T.
Y=
Press ◀︎ once to move from the start of the expression onto the = sign.
◀︎
Press ENTER. The = signs of both X1T and Y1T lose their highlight, which means the circle is now deselected.
ENTER
Press ENTER again to select the circle once more.
ENTER

Setting Window Variablesp. 94

Press WINDOW to see and change the window variables, which define the viewing window. These are the defaults in Par mode with Radian angle mode:

Default parametric window
Variable	Default	Meaning
Tmin	0	Smallest T value to evaluate
Tmax	6.2831853…	Largest T value to evaluate (2π)
Tstep	.1308996…	Increment of T between points (π÷24)
Xmin	⁻10	Smallest X value shown
Xmax	10	Largest X value shown
Xscl	1	Spacing between X tick marks
Ymin	⁻10	Smallest Y value shown
Ymax	10	Largest Y value shown
Yscl	1	Spacing between Y tick marks

Note The default Tstep is small enough for a smooth circle, but you may need to change the T variables so that enough points are plotted. A Tstep that is too large gives a jagged curve; one that is too small makes the graph slow.

Tmin

Tmin

The smallest value of T that is evaluated. Plotting and tracing start here. The default is 0.

Where: VARS 1:Window, T/θ tab, 1:Tmin

Keys: VARS1▶︎1

0→Tmin

Tmax

Tmax

The largest value of T that is evaluated. The default is 2π (about 6.2831853). Tmax should be greater than Tmin.

Where: VARS 1:Window, T/θ tab, 2:Tmax

Keys: VARS1▶︎2

5→Tmax

Tstep

Tstep

The amount T increases from one plotted point to the next, and the distance of one TRACE move. The default is π÷24. It must be greater than 0.

Where: VARS 1:Window, T/θ tab, 3:Tstep

Keys: VARS1▶︎3

.1→Tstep

Tutorial

Change the T window variables

You will set Tmin, Tmax and Tstep in the WINDOW editor and see the circle become a half-circle.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the WINDOW editor. The first three lines are the T variables.
WINDOW
Press ENTER to keep Tmin = 0. Then change Tmax to π: press 2ndπ for π and ENTER.
ENTER2ndπENTER
Change Tstep to π÷24: type 2ndπ, ÷, 2 4 and ENTER. (Because you typed an expression, the calculator stores its value.)
2ndπ÷24ENTER
Press GRAPH. T now only runs from 0 to π, so you see just the upper half of the circle.
GRAPH

Tutorial

See what Tstep does

A large Tstep plots few points, so the curve is only a rough polygon.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the WINDOW editor and move down to Tstep.
WINDOW▼︎▼︎
Type a coarse step of 1 and press ENTER. Fewer than 7 points will be plotted for T from 0 to 2π.
1ENTER
Graph it. The circle is drawn as a rough polygon.
GRAPH
Go back and restore a smooth value: open WINDOW, move to Tstep, and enter π÷24 again.
WINDOW▼︎▼︎2ndπ÷24ENTERGRAPH

Setting the Graph Formatp. 94

Press 2ndFORMAT (FORMAT) to see the current graph format. Chapter 3 describes every setting in detail. The format settings are shared by all graphing modes, so what you choose here also applies to Func, Pol and Seq graphs. (Seq mode has one extra axes setting.)

The FORMAT screen
Row	Choices	Effect
Coordinates	RectGC, PolarGC	Shows cursor position as X and Y, or as R and θ as well
Coordinate display	CoordOn, CoordOff	Show or hide the cursor coordinates at the bottom
Grid	GridOff, GridOn	Dots at the tick marks
Axes	AxesOn, AxesOff	Draw or hide the axes (the example turns them off)
Labels	LabelOff, LabelOn	Label the axes X and Y
Expression	ExprOn, ExprOff	Show the equation being traced

Tutorial

Switch to PolarGC format

PolarGC makes TRACE and the free-moving cursor report polar coordinates too.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the FORMAT screen.
2ndFORMAT
The cursor starts on RectGC. Move right to PolarGC and press ENTER to select it.
▶︎ENTER
Go back to the home screen. From now on cursor coordinates in parametric graphs include R and θ.
2ndQUIT

Displaying a Graphp. 95

Press GRAPH and the calculator plots the selected equations. For each value of T from Tmin up to Tmax in steps of Tstep, it computes X and Y and plots the point. The window variables decide which part of the plane you see. While it draws, the variables X, Y and T hold the values of the point being plotted.

Smart Graph works with parametric graphs: the calculator redraws only when something that affects the graph has changed since the last time it was shown.

Window Variables and Y.VARS Menusp. 95

From the home screen, or from inside a program, you can:

refer to the X or Y part of an equation by its name (for example X1T),
store a new expression in an equation (as a string, e.g. "5cos(T)"→X1T),
select or deselect parametric equations, and
store values directly to window variables (for example 3→Tmax).

FnOn

FnOnFnOn function#[,function#,…]

Selects equations. With no argument, selects all equations of the current graphing mode. With numbers, selects just those equations: in Par mode, FnOn 1 selects both X1T and Y1T.

Where: VARS Y-VARS tab, 4:On/Off, 1:FnOn

Keys: VARS▶︎41

FnOn 1,3 selects equations 1 and 3

FnOff

FnOffFnOff function#[,function#,…]

Deselects equations. With no argument, deselects all of them; with numbers, only those equations (in Par mode, both the X and the Y component of each).

Where: VARS Y-VARS tab, 4:On/Off, 2:FnOff

Keys: VARS▶︎42

FnOff 2 deselects equation 2 (X2T and Y2T)

Tutorial

Store window values and equations from the home screen

Anything you can do in the WINDOW and Y= editors you can also do with commands on the home screen.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Store 3 into Tmax. Type 3, press STO►, then choose Tmax from the VARS Window menu: VARS 1:Window, move to the T/θ tab with ▶︎, 2:Tmax. Press ENTER.
3STO►VARS1▶︎2ENTER
Store an equation. Type a string "2T" (the quote mark is ALPHA"), press STO►, then paste X1T from the Y-VARS menu (VARS, ▶︎, 2:Parametric, 1:X₁T) and press ENTER.
ALPHA"2X,T,θ,nALPHA"STO►VARS▶︎21ENTER
Turn equation 1 off with FnOff 1. Paste FnOff from the Y-VARS menu (VARS, ▶︎, 4:On/Off, 2:FnOff), type 1 and press ENTER.
VARS▶︎421ENTER
Turn it back on with FnOn 1.
VARS▶︎411ENTER

Exploring Parametric Graphsp. 95

Once a parametric graph is on screen you can explore it with the free-moving cursor, TRACE, the ZOOM menu and the CALC menu, which all work the way they do for function graphs, with the differences below.

Free-Moving Cursorp. 95

The free-moving cursor works as it does in Func mode: after you press GRAPH, the arrow keys move a cross anywhere on the screen.

In RectGC format, moving the cursor updates X and Y. They are shown at the bottom of the screen when CoordOn is selected.
In PolarGC format, X, Y, R and θ are all updated. With CoordOn selected, the screen shows R and θ.

Tutorial

Move the free cursor on a parametric graph

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Graph the circle with the standard window.
ZOOM6
Move the cursor around with the arrow keys. The coordinates at the bottom of the screen follow it.
▶︎▶︎▲︎▲︎

TRACEp. 96

Press TRACE to trace. The trace cursor sits on the first selected equation at Tmin, and the ◀︎ and ▶︎ keys move it one Tstep along the curve. If ExprOn is selected, the equation is shown on the screen.

In RectGC format, the screen shows X, Y and T (when CoordOn is on).
In PolarGC format, X, Y, R, θ and T are updated, and with CoordOn the screen shows R, θ and T.
X and Y (or R and θ) are always computed from T.

To move five plotted points at a time, press 2nd◀︎ or 2nd▶︎. If the curve goes beyond the top or bottom of the screen, the coordinates at the bottom keep updating correctly even though the cursor is off screen.

Note Quick Zoom (pressing ENTER while tracing to zoom in on the cursor) is available in parametric graphing. Panning is not: the cursor does not push the window sideways when it reaches the edge.

Tutorial

Trace a curve one Tstep and five points at a time

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Graph the circle with the standard window and its default Tstep.
ZOOM6
Start tracing. The cursor is at Tmin = 0, the rightmost point of the circle (X=5, Y=0).
TRACE
Press ▶︎ once. T increases by Tstep (π÷24 ≈ 0.1309) and the cursor moves counterclockwise.
▶︎
Press 2nd▶︎ to jump five points at once.
2nd▶︎
Press 2nd◀︎ to jump five points back.
2nd◀︎

Moving the trace cursor to any valid T value. While tracing, just type a number. When you press the first digit, a T= prompt appears in the bottom-left corner with your digits. You may even enter an expression there. The T value must be valid for the current window. Press ENTER and the cursor moves to that T.

Tutorial

Jump the trace cursor to a chosen T value

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Start tracing.
TRACE
Type 1. A T=1 prompt appears in the bottom-left corner. Press ENTER to move there. The point at T=1 is (5 cos 1, 5 sin 1).
1ENTER
The prompt accepts expressions too: type 3÷2 (3 ÷ 2) and press ENTER. The cursor moves to T=1.5, almost at the top of the circle.
3÷2ENTER

ZOOMp. 96

ZOOM operations work as they do in Func mode, and they change only the X and Y window variables (Xmin, Xmax, Xscl, Ymin, Ymax, Yscl). The T variables (Tmin, Tmax, Tstep) are changed only by ZStandard.

The zoom-memory variables for parametric graphing are on the VARS Zoom menu, ZT/Zθ tab. ZoomSto saves the current T window in them and ZoomRcl restores it.

ZTmin

ZTmin

Zoom-memory copy of Tmin. ZoomSto saves Tmin here, and ZoomRcl loads it back.

Where: VARS 2:Zoom, ZT/Zθ tab, 1:ZTmin

Keys: VARS2▶︎1

ZTmax

ZTmax

Zoom-memory copy of Tmax. ZoomSto saves Tmax here, and ZoomRcl loads it back.

Where: VARS 2:Zoom, ZT/Zθ tab, 2:ZTmax

Keys: VARS2▶︎2

ZTstep

ZTstep

Zoom-memory copy of Tstep. ZoomSto saves Tstep here, and ZoomRcl loads it back.

Where: VARS 2:Zoom, ZT/Zθ tab, 3:ZTstep

Keys: VARS2▶︎3

Tutorial

Which ZOOM commands change the T window?

You will set Tmax to 3, show that ZDecimal leaves it alone, and that ZStandard resets it.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Set Tmax = 3 from the home screen: type 3, press STO►, choose Tmax on the T/θ tab of the VARS Window menu, and press ENTER.
3STO►VARS1▶︎2ENTER
Press ZOOM 4:ZDecimal. It changes the X and Y window but Tmax stays 3.
ZOOM4
Press ZOOM 6:ZStandard. This one resets the T window too, so Tmax returns to 2π.
ZOOM6

Tutorial

Save and recall the T window with ZoomSto

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the ZOOM menu and go to the MEMORY tab with ▶︎. Choose 2:ZoomSto. The current window, T variables included, is saved in the zoom variables.
ZOOM▶︎2
Press ZOOM, ▶︎, 3:ZoomRcl to bring the saved window back at any time.
ZOOM▶︎3

CALCp. 96

CALC works as in Func mode, but the menu is shorter. Press 2ndCALC (CALC) on a parametric graph to see the four items available.

value

value

Asks for a value of T (T=) and then shows X and Y at that point on the current curve. The value must be within the window.

Where: 2ndCALC 1:value

Keys: 2ndCALC1

dy/dx

dy/dx

Shows the slope dy/dx of the curve at the T value you enter. This is the derivative of Y with respect to X, which equals (dy/dt)÷(dx/dt).

Where: 2ndCALC 2:dy/dx

Keys: 2ndCALC2

dy/dt

dy/dt

Shows dy/dt, how fast Y changes with T (for the ball, its vertical velocity), at the T value you enter.

Where: 2ndCALC 3:dy/dt

Keys: 2ndCALC3

dx/dt

dx/dt

Shows dx/dt, how fast X changes with T (for the ball, its horizontal velocity), at the T value you enter.

Where: 2ndCALC 4:dx/dt

Keys: 2ndCALC4

Tutorial

Find the velocity of the ball at T=1

The ball example is already defined. You will use CALC to read its vertical and horizontal speed after one second.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Graph the ball so that CALC has a curve to work on.
GRAPH
Open the CALCULATE menu.
2ndCALC
Choose 3:dy/dt. A T= prompt appears at the bottom of the screen.
3
Type 1 and press ENTER. The screen shows dy/dt ≈ 2.8785 (m/s): the ball is still rising, at a slower speed than the 12.68 m/s it started with, because 9.8 m/s has been lost to gravity.
1ENTER
Now ask for dx/dt: open CALC, choose 4:dx/dt, type 1 and press ENTER. It is ≈ 27.189, the constant horizontal speed 30 cos 25°.
2ndCALC41ENTER
The slope of the path at that moment is dy/dx = (dy/dt)÷(dx/dt). Choose 2:dy/dx, type 1 and press ENTER. The result is about 0.10587, which is 2.8785÷27.189.
2ndCALC21ENTER
Last, 1:value gives the position itself. Choose it, type 1 and press ENTER: the cursor jumps to T=1, where X ≈ 27.19 m and Y ≈ 7.78 m.
2ndCALC11ENTER

1:X₁T	The X component of equation 1
2:Y₁T	The Y component of equation 1
3:X₂T	The X component of equation 2
4:Y₂T	The Y component of equation 2
5:X₃T	The X component of equation 3
6:Y₃T	The Y component of equation 3
7:X₄T	The X component of equation 4
8:Y₄T	The Y component of equation 4
9:X₅T	The X component of equation 5
0:Y₅T	The Y component of equation 5
A:X₆T	The X component of equation 6 (scroll down)
B:Y₆T	The Y component of equation 6 (scroll down)

1:Tmin	Smallest T value
2:Tmax	Largest T value
3:Tstep	T increment
4:θmin	Smallest θ value (polar)
5:θmax	Largest θ value (polar)
6:θstep	θ increment (polar)

1:ZTmin	Stored Tmin for ZoomRcl
2:ZTmax	Stored Tmax for ZoomRcl
3:ZTstep	Stored Tstep for ZoomRcl
4:Zθmin	Stored θmin (polar)
5:Zθmax	Stored θmax (polar)
6:Zθstep	Stored θstep (polar)