Interactive Manual › Chapter 5
Chapter 5: Polar Graphing
Graph polar equations r=f(θ): Pol mode, the polar Y= editor (r1 to r6), the θmin, θmax and θstep window variables, graph formats, TRACE, ZOOM and CALC in polar graphs.
Getting Started: Polar Rosep. 97
The polar equation R=A sin(Bθ) draws a flower-like curve called a rose. In this walkthrough you graph the rose for A=8 and B=2.5, fix the window so the whole rose shows, and then try other values of A and B.
Graph the rose r1=8 sin(2.5θ)
You will switch to Pol mode, define r1, graph it in the standard window, extend θmax to 4π and square the window.
▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go
- Open the MODE screen. Move down to the graphing-mode row (the fourth row, where Func is highlighted), move right twice to Pol, and press ENTER to select it. Leave the other settings on their defaults (the left-hand options).MODE▼︎▼︎▼︎▶︎▶︎ENTER
- Open the polar Y= editor. It lists r1 to r6 instead of Y1 to Y6.Y=
- Define r1 as 8 sin(2.5θ). The X,T,θ,n key types θ in Pol mode. Press ENTER to store the equation.8SIN2.5X,T,θ,n)ENTER
- Press GRAPH to plot r1 once in the current window.GRAPH
- Choose 6:ZStandard from the ZOOM menu (ZOOM6) to graph in the standard window. The graph shows only five petals and looks lopsided, because ZStandard sets θmax to 2π and makes the window (not the pixels) square.ZOOM6
- Open the window variables and move down one row to θmax. Enter 4π (type 4, then 2ndπ for π). This lets the curve make two full turns.WINDOW▼︎42ndπ
- Press GRAPH to plot with the new θmax. The rose is now complete but still looks squashed.GRAPH
- Choose 5:ZSquare (ZOOM5) to graph again with equal scaling on both axes. Now the full, symmetrical rose appears. ZSquare changes only the X and Y window, so θmax stays at 4π.ZOOM5
- To explore, repeat these steps with other numbers for A and B in
r1=A sin(Bθ)and watch how the petals change. A sets how far the petals reach, and B sets how many there are.
Defining and Displaying Polar Graphsp. 98
Polar graphing works almost exactly like function graphing. This chapter only covers what is different, so it helps to know how the two modes compare and to have read the function graphing chapter.
TI-84 Plus Graphing Mode Similaritiesp. 98
The steps for defining a polar graph follow the steps for defining a function graph: set the mode, enter equations in the Y= editor, set the window and format, then press GRAPH. The sections below describe the points where polar graphing differs.
Setting Polar Graphing Modep. 98
Press MODE to open the MODE screen. To graph polar equations you must select Pol in the graphing-mode row. Do this before you enter window variables and before you enter polar equations, because the Y= editor and the window variables change to match the mode.
Select Pol graphing mode
▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go
- Open the MODE screen.MODE
- Move down three rows to the line that reads Func Par Pol Seq.▼︎▼︎▼︎
- Move right twice so the cursor sits on Pol, then press ENTER to select it.▶︎▶︎ENTER
- Leave the MODE screen and return to the home screen.2ndQUIT
- Check the mode: opening the Y= editor now shows the polar equations r1 to r6.Y=
Displaying the Polar Y= Editorp. 98
After selecting Pol mode, press Y= to open the polar Y= editor. It holds up to six polar equations, named r1 through r6. Each one is written in terms of the independent variable θ.
r₁(θ)r₂(θ)r₃(θ)r₄(θ)r₅(θ)r₆(θ)The six polar equation variables. Use them like functions on the home screen, store text to them, or paste them from the YVARS shortcut menu (POL tab) or the VARS Y-VARS Polar menu.
"8sin(2.5θ)"→r₁stores the rose equation.r₁(0)evaluates r1 at θ=0.
Selecting Graph Stylesp. 98
The icon to the left of each of r1 to r6 shows that equation's graph style. In Pol mode the default is line, which joins the plotted points with straight segments.
| Style | Effect |
|---|---|
| Line (default) | Connects the plotted points. |
| Thick | Connects the points with a heavier line. |
| Path | Draws a small circle that travels along the curve as it is plotted. |
| Animate | Moves a small circle along the curve without leaving a trail. |
| Dot | Plots each point as a single dot without connecting them. |
To change a style, open the Y= editor, move the cursor to the icon column (◀︎ from the equation) and press ENTER repeatedly to cycle through the styles.
Change the graph style of r1
▶ 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 r1.Y=
- Press ◀︎ twice: the first press highlights the
=sign, the second moves onto the style icon at the far left.◀︎◀︎ - Each press of ENTER changes the icon to the next style: line, thick, path, animate, dot, and then back to line. Press it once for thick.ENTER
Defining and Editing Polar Equationsp. 98
Define or edit a polar equation the same way you would a function: put the cursor on an equation line, type the expression and press ENTER. The independent variable is θ, and there are two ways to type it in Pol mode.
- Press X,T,θ,n.
- Press ALPHAθ (the θ above the 3 key, shown as
ALPHA [θ]).
Define a polar equation
This enters r1=2 cos(θ) using the θ key, and r2=4 sin(θ) using ALPHA θ.
▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go
- Open the polar Y= editor.Y=
- Enter
2 cos(θ)for r1, typing θ with X,T,θ,n.2COSX,T,θ,n)ENTER - Now enter
4 sin(θ)for r2, typing θ with ALPHAθ instead.4SINALPHAθ)ENTER
Selecting and Deselecting Polar Equationsp. 98
Only selected equations are graphed. An equation is selected when the = sign next to it is highlighted. You may select any or all of r1 to r6.
To change an equation's status, move the cursor onto its = sign and press ENTER.
Deselect and reselect a polar equation
▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go
- Open the Y= editor. r1 is selected (its
=is highlighted).Y= - Press ◀︎ once. The cursor leaves the equation and highlights the
=sign of r1, which is where selection is switched.◀︎ - Press ENTER to deselect r1. The
=is no longer highlighted, so GRAPH would skip this equation.ENTER - Press ENTER again to select r1 once more.ENTER
Setting Window Variablesp. 99
Press WINDOW to see the window variables. They set the viewing window and the range of θ values to evaluate. The defaults for Pol mode with Radian angles are listed below.
| Variable | Default | Meaning |
|---|---|---|
| θmin | 0 | Smallest θ value to evaluate |
| θmax | 2π (6.2831853…) | Largest θ value to evaluate |
| θstep | π÷24 (.1308996…) | Increment between θ values |
| 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 |
Check and change the θ window variables
▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go
- Open the window editor. The cursor starts on θmin.WINDOW
- Move down to θmax and set it to 4π. Type 4, then 2ndπ for π, then ENTER.▼︎42ndπENTER
- Now θstep is highlighted. Change it to π÷48 for a smoother curve: press 2ndπ, ÷, 48 and ENTER.2ndπ÷48ENTER
Setting the Graph Formatp. 99
Press 2ndFORMAT to see the current graph format settings. The function graphing chapter describes each one in detail. The format settings are shared with the other graphing modes.
Switch to PolarGC format
RectGC shows rectangular coordinates (X, Y) when you move the cursor. PolarGC shows polar coordinates (R, θ).
▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go
- Open the FORMAT screen.2ndFORMAT
- Move down to the second row, move right to PolarGC and press ENTER.▼︎▶︎ENTER
- Leave the FORMAT screen. Cursor and trace coordinates will now show R and θ.2ndQUIT
Displaying a Graphp. 99
When you press GRAPH, the calculator plots every selected polar equation. For each θ from θmin to θmax in steps of θstep it works out R, converts the point to X and Y, and plots it. The window variables decide which part of the plane you see.
While a graph is being drawn, the variables X, Y, R and θ are updated with the current point.
Smart Graph works for polar graphs too: if nothing has changed since the last time the graph was drawn, pressing GRAPH just redisplays it.
Graph the cardioid r1=3+3 cos(θ)
A cardioid is a heart-shaped curve. This one is r1=3+3cos(θ), stored in r1 by a silent setup step.
▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go
- Press GRAPH to plot the selected equation across θ from 0 to 2π.GRAPH
Window Variables and Y.VARS Menusp. 99
From the home screen or inside a program you can work with polar equations and window variables directly.
- Use an equation's name (r1 to r6) as a variable. The names are on the YVARS shortcut menu (ALPHAF1).
- Store polar equations.
- Select or deselect polar equations.
- Store values directly to window variables such as θmin, θmax and θstep.
Store an equation in r1 and a value in θmax from the home screen
▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go
- Type the equation as text in quotation marks: ALPHA" gives the quote mark, then
3cos(θ), then the closing quote.ALPHA"3COSX,T,θ,n)ALPHA" - Press STO►, then open the YVARS shortcut menu with ALPHAF4. It has three tabs (Y, PAR, POL) and opens on POL because the calculator is in Pol mode.STO►ALPHAF4
- Choose 1:r₁ and press ENTER. The text is stored as the definition of r1.1ENTER
- Window variables can be stored to as well. Type 4π, press STO►, then open the window-variable list with VARS and choose 1:Window….42ndπSTO►VARS1
- Move right to the T/θ tab and choose 5:θmax, then press ENTER.▶︎5ENTER
Exploring Polar Graphsp. 100
After a polar graph is displayed, you can use the free-moving cursor, TRACE, the ZOOM menu and the CALC menu to explore it.
Free-Moving Cursorp. 100
The free-moving cursor works as it does in Func mode. Press the arrow keys on the graph to move it anywhere.
| Format | Variables updated as the cursor moves | Shown when CoordOn is selected |
|---|---|---|
| RectGC | X, Y | X, Y |
| PolarGC | X, Y, R, θ | R, θ |
TRACEp. 100
Press TRACE to start tracing. The trace cursor moves along the curve one θstep at a time with ◀︎ and ▶︎. A trace starts on the first selected equation at θmin. If ExprOn is chosen in the format, the equation is shown on screen.
| Format | Variables updated by TRACE | Shown when CoordOn is selected |
|---|---|---|
| RectGC | X, Y, θ | X, Y, θ |
| PolarGC | X, Y, R, θ | R, θ |
To jump five plotted points at a time, press 2nd◀︎ or 2nd▶︎. If the cursor goes off the top or bottom of the screen, the coordinates at the bottom keep changing correctly.
Trace a polar curve
▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go
- Press TRACE. The cursor appears on r1 at θmin=0, where r=6, so X is 6 and Y is 0.TRACE
- Press ▶︎ to move one θstep (π÷24) along the curve.▶︎
- Press 2nd▶︎ to jump five plotted points at once.2nd▶︎
Moving the Trace Cursor to Any Valid Theta Valuep. 100
While tracing, type a number to jump straight to that θ value on the current equation. As soon as you type the first digit, a θ= prompt with your number appears at the bottom-left of the screen. You may type an expression there too. The value must lie inside the current viewing window. Press ENTER to move the cursor.
Trace to a specific θ value
▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go
- Start tracing. The cursor begins at θmin on r1.TRACE
- Type 2 to open the
θ=prompt, then ENTER to jump the cursor to θ=2.2ENTER
ZOOMp. 101
ZOOM operations work as they do in Func mode. They change only the X window variables (Xmin, Xmax, Xscl) and the Y window variables (Ymin, Ymax, Yscl).
The θ window variables (θmin, θmax, θstep) are left alone, with one exception: 6:ZStandard resets them too. In the VARS ZOOM secondary menu (ZT/Zθ), items 4:Zθmin, 5:Zθmax and 6:Zθstep are the zoom-memory variables for Pol mode.
See which window variables ZOOM changes
▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go
- Open the ZOOM menu and choose 5:ZSquare. It adjusts X and Y but leaves θmax at 4π.ZOOM5
- Now choose 6:ZStandard. This one also resets θmax to 2π.ZOOM6
CALCp. 101
CALC operations work as they do in Func mode. Press 2ndCALC to open the CALCULATE menu. In Pol mode it has these three items.
value (enter θ at the prompt)Evaluates the selected polar equation at a θ value you type, moves the cursor to that point and shows its X and Y (and θ). Press ENTER to accept the value.
dy/dx (enter θ at the prompt)Finds the slope dy/dx of the selected polar curve at the θ value you enter.
dr/dθ (enter θ at the prompt)Finds dr/dθ, the rate at which r changes with θ, for the selected polar equation at the θ value you enter.
Evaluate a polar curve with CALC value
▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go
- Press 2ndCALC to open the CALCULATE menu from the graph. In Pol mode it has three items.2ndCALC
- Choose 1:value. The graph asks for a θ value.1
- Type 2 and press ENTER. The cursor moves to θ=2 on the curve and X and Y are filled in (r is 3+3cos(2), about 1.75).2ENTER
Find the slope of a polar curve with dy/dx
▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go
- Open the CALCULATE menu and choose 2:dy/dx.2ndCALC2
- Type 1 as the θ value and press ENTER. The cursor moves to θ=1 and the slope dy/dx is shown at the bottom of the screen.1ENTER