Chapter 3: Function Graphing

TI-84 Plus Graphing Mode Similaritiesp. 66

This chapter describes function graphing, but the steps are nearly the same in all four graphing modes. The things that differ in parametric, polar and sequence graphing are covered in the chapters for those modes.

Defining a Graphp. 66

To define a graph in any graphing mode, work through these steps. You can skip the ones you do not need.

1. Press MODE and choose the graphing mode (see Setting the Graph Modes).
2. Press Y= and enter, edit or select one or more functions (Defining Functions).
3. Turn off any stat plots you do not want drawn (Turning On or Turning Off a Stat Plot).
4. Set the graph style for each function (Setting Graph Styles).
5. Press WINDOW and set the viewing window variables (Setting the Viewing Window Variables).
6. Press 2ndFORMAT (FORMAT) and choose the graph format settings (Setting the Graph Format).

Displaying and Exploring a Graphp. 66

Once the graph is defined, press GRAPH to draw it. Then use the tools in this chapter to study it: the free-moving cursor, TRACE, the ZOOM instructions and the CALC menu.

Saving a Graph for Later Usep. 66

You can store everything that defines the current graph in one of ten graph database variables, GDB1 to GDB9 and GDB0. Recalling the database later restores the graph. A graph database holds:

• the Y= functions
• the graph style of each function
• the window settings
• the format settings

You can also store a picture of the current graph screen in one of ten picture variables, Pic1 to Pic9 and Pic0, and superimpose stored pictures on later graphs. Both features belong to the drawing and variables chapters of the guidebook (GDB and Pic commands).

Checking and Changing the Graphing Modep. 67

Press MODE to see the mode settings. To graph functions Y=f(X), the graphing mode must be Func, and it must be chosen before you enter window values or functions, because each mode has its own set of both.

• Func: function graphing, Y=f(X).
• Par: parametric graphing.
• Pol: polar graphing.
• Seq: sequence graphing.

Other mode settings also change how graphs come out:

Setting Modes from a Programp. 67

A program can change the graphing mode and any other mode. In the program editor, on an empty line, press MODE, move the cursor to the mode you want and press ENTER. The name of the mode is pasted at the cursor, and the mode changes when the program runs.

Displaying Functions in the Y= Editorp. 68

Press Y= to open the Y= editor. It holds ten functions, stored in the variables Y1 to Y9 and Y0. You can graph any number of them at the same time. A function takes part in the graph only when its = sign is highlighted (see Selecting and Deselecting a Function).

Plot1Plot2Plot3
Y1=X²
Y2=X+2
Y3=
Y4=
Y5=
Y6=
Y7=

Defining or Editing a Functionp. 68

1. Press Y=.
2. Press ▼︎ to move to the function you want to define or edit. To erase a function, press CLEAR while the cursor is on it.
3. Enter or edit the expression (details below).
4. Press ENTER or ▼︎ to move on to the next function.

In the expression you may use functions, variables, lists and matrices. If the expression gives a non-real result at some X (for example the square root of a negative number), that point is simply not plotted and no error appears.

The independent variable is X. In Func mode the X,T,θ,n key types X for you; you can also press ALPHAX. The shortcut menus FRAC, FUNC, MTRX and YVAR are on ALPHAF1, ALPHAF2, ALPHAF3 and ALPHAF4.

As soon as you type the first character the = sign is highlighted: the function is selected, and the expression is stored in Yn as you type.

Defining a Function from the Home Screen or a Programp. 68

From a blank line on the home screen or in a program, put the expression in quotation marks, store it with STO►, and finish with a Y variable from the YVAR menu. The result is a statement of the form "expression"→Yn. When it runs, the expression is stored in Yn, the function is selected, and the calculator shows Done.

1. Press ALPHA" (the " mark), enter the expression, then press ALPHA" again.
2. Press STO►.
3. Press ALPHAF4 to open the YVAR menu, pick the function name, and press ENTER.

Evaluating Y= Functions in Expressionsp. 69

Selecting and Deselecting a Functionp. 69

The calculator graphs only the functions that are selected; a function is selected when the = in its Y= line is highlighted. Any or all of Y1 to Y9 and Y0 may be selected.

1. Press Y=.
2. Move the cursor to the function.
3. Press ◀︎ to put the cursor on the = sign.
4. Press ENTER to switch between selected and deselected.

Turning On or Turning Off a Stat Plot in the Y= Editorp. 70

The top line of the Y= editor shows Plot1, Plot2 and Plot3. A plot that is on has its name highlighted. To switch one, press ▲︎ (and ◀︎ / ▶︎) to put the cursor on its name and press ENTER. Stat plots are drawn together with the selected functions, so turn them off when you only want the functions.

Selecting and Deselecting Functions from the Home Screen or a Programp. 70

On a blank line of the home screen or of a program, press VARS ▶︎ to open the Y-VARS menu, choose 4:On/Off, then 1:FnOn or 2:FnOff. Type the numbers (1 to 9, or 0; not the name Yn) of the functions to switch, separated by commas. With no numbers, FnOn or FnOff affects every function in the current mode. Press ENTER to run it; the calculator shows Done.

Graph Style Icons in the Y= Editorp. 71

Graph styles let you tell functions apart when you graph several together: for example Y₁ as a solid line, Y₂ as a dotted line and Y₃ as a thick line. The style icon is the small symbol at the far left of each Y= line. Not every style exists in every graphing mode; the chapters for the other modes list theirs.

Setting the Graph Stylep. 71

1. Press Y=.
2. Press ▼︎ or ▲︎ to move the cursor onto the function.
3. Press ◀︎ twice: once to pass the = sign and once more to reach the icon in the first column. A different (insert) cursor appears. Steps 2 and 3 can be done in either order.
4. Press ENTER repeatedly to rotate through the seven styles, in the order of the table above.
5. Press ▶︎, ▲︎ or ▼︎ when the style you want is showing.

Shading Above and Belowp. 72

When two or more functions use Above or Below, the calculator gives each a different shading pattern so overlaps stay readable. The patterns rotate in this order:

1. Vertical lines for the first shaded function.
2. Horizontal lines for the second.
3. Diagonal lines sloping down to the right for the third.
4. Diagonal lines sloping up to the right for the fourth.
5. The fifth shaded function starts over with vertical lines, and so on.

Where shaded areas overlap, the patterns are drawn on top of each other.

Setting a Graph Style from a Programp. 72

In the program editor press PRGM to open the control menu and choose H:GraphStyle(. The first argument is the number of the Y= function in the current graphing mode, and the second is a style number from 1 to 7 (see the table above).

The TI-84 Plus Viewing Windowp. 73

The viewing window is the part of the coordinate plane the graph screen shows: from Xmin to Xmax horizontally and from Ymin to Ymax vertically. Xscl is the distance between tick marks on the x-axis and Yscl the distance on the y-axis. Set both scales to 0 to remove the tick marks.

Displaying the Window Variablesp. 73

Press WINDOW to see the current values. The variables listed depend on the graphing mode. These are the Func-mode defaults:

WINDOW
Xmin=⁻10
Xmax=10
Xscl=1
Ymin=⁻10
Ymax=10
Yscl=1
Xres=1

Xres sets the pixel resolution of function graphs, from 1 to 8 (default 1). At Xres=1 a function is evaluated and plotted at every pixel column; at Xres=8 only at every eighth column. Smaller values give smoother graphs but take longer to draw.

Changing a Window Variable Valuep. 73

1. Press ▼︎ or ▲︎ to move to the variable.
2. Edit the value. It may be an expression. Typing a new number replaces the old value; you can also move to a digit and edit it in place.
3. Press ENTER, ▼︎ or ▲︎. An expression is evaluated and the result is stored.

Storing to a Window Variable from the Home Screen or a Programp. 73

1. On a blank line, enter the value (an expression is fine).
2. Press STO►.
3. Press VARS and choose 1:Window. You see the X/Y menu of Func window variables. Press ▶︎ for the T/θ menu (Par and Pol variables), and ▶︎ twice for the U/V/W menu (Seq variables).
4. Choose the variable. Its name is pasted at the cursor.
5. Press ENTER. The value is stored and displayed.

ΔX and ΔYp. 74

ΔX and ΔY are the distances from the center of one pixel to the center of the next, horizontally and vertically. They measure graphing accuracy. The calculator works them out from the window every time it draws a graph:

• ΔX = (Xmax − Xmin) / 94
• ΔY = (Ymax − Ymin) / 62

They are items 8 and 9 on the VARS 1:Window X/Y menu; ΔX is also on the WINDOW screen. You can store a value into ΔX or ΔY. If you do, Xmax or Ymax is recalculated from the new ΔX or ΔY together with Xmin or Ymin.

Displaying the Format Settingsp. 74

Press 2ndFORMAT (FORMAT) to see the format settings. Each row has two choices, and exactly one of them is highlighted.

RectGC PolarGC
CoordOn CoordOff
GridOff GridOn
AxesOn AxesOff
LabelOff LabelOn
ExprOn ExprOff

You can also reach this screen from MODE: set the GOTO FORMAT GRAPH line to YES and press ENTER. After making your changes, press MODE to return to the MODE screen. Format settings apply to every graphing mode. Seq mode has one extra format setting (explained in the sequence graphing chapter).

Changing a Format Settingp. 75

1. Use ▼︎, ▶︎, ▲︎ and ◀︎ to move the cursor to the setting you want.
2. Press ENTER to select the highlighted setting.

RectGC, PolarGCp. 75

RectGC shows the cursor position as rectangular coordinates X and Y. PolarGC shows it as polar coordinates R and θ. This setting decides which variables are updated when a graph is drawn, when you move the free-moving cursor and when you trace:

• RectGC updates X and Y. With CoordOn, X and Y are displayed.
• PolarGC updates X, Y, R and θ. With CoordOn, R and θ are displayed.

CoordOn, CoordOffp. 75

CoordOn shows the cursor coordinates at the bottom of the graph. If ExprOff is also selected, the number of the function you are tracing appears in the top-right corner. CoordOff shows neither the coordinates nor the function number.

GridOff, GridOnp. 75

Grid points are laid out in rows that line up with the tick marks on each axis. GridOff shows no grid; GridOn shows it.

AxesOn, AxesOffp. 75

AxesOn draws the axes; AxesOff hides them. With AxesOff, the LabelOff / LabelOn setting is overridden because there are no axes to label.

LabelOff, LabelOnp. 76

LabelOn puts the letters X and Y beside the axes, and LabelOff leaves them off. Labels show only when AxesOn is also selected.

ExprOn, ExprOffp. 76

These control whether the Y= expression is shown while a trace cursor is active. The setting applies to stat plots too. With ExprOn, the expression appears in the top-left corner of the graph screen. With ExprOff and CoordOn, a number in the top-right corner tells you which function you are tracing.

Displaying a New Graphp. 76

Press GRAPH to draw the selected functions. TRACE, the ZOOM instructions and the CALC operations also draw the graph for you. While it is being plotted the busy indicator shows in the corner, and the variables X and Y are updated as the plot moves along.

Pausing or Stopping a Graphp. 76

• Press ENTER to pause the plot, and ENTER again to resume it.
• Press ON to stop the plot. Press GRAPH to start it over.

Smart Graphp. 76

Smart Graph redraws the previous graph instantly when you press GRAPH, but only if nothing that affects the picture has changed since it was last displayed. If you have done any of the following, the graph is plotted again from the new values:

• changed a mode setting that affects graphs
• changed a function in the current picture
• selected or deselected a function or stat plot
• changed the value of a variable used in a selected function
• changed a window variable or a graph format setting
• cleared drawings with ClrDraw
• changed a stat plot definition

Overlaying Functions on a Graphp. 77

You can add a new function to an existing graph without redrawing the old ones. Store sin(X) in Y₁ and press GRAPH. Then store cos(X) in Y₂ and press GRAPH again: Y₂ is drawn over the original Y₁ graph.

Graphing a Family of Curvesp. 77

If a list appears in a Y= expression, the calculator plots the function once for each list element. In Simul graphing-order mode all functions are plotted for the first element of each list, then for the second, and so on.

• {2,4,6}sin(X) graphs three functions: 2 sin(X), 4 sin(X) and 6 sin(X).
• {2,4,6}sin({1,2,3}X) graphs 2 sin(X), 4 sin(2X) and 6 sin(3X).

Graphing Accuracyp. 78

The free-moving cursor jumps from pixel to pixel. A pixel that looks as if it lies on a curve may be slightly off it, so the coordinates shown can differ from a true point of the function. To follow a function exactly, use TRACE instead.

The displayed coordinates are only accurate to within the width and height of one pixel. As Xmin and Xmax (and Ymin and Ymax) get closer together, for example after a Zoom In, each pixel covers less distance, so the coordinates get more precise.

Beginning a Tracep. 78

Press TRACE to move a cursor along a function, from one plotted point to the next. If the graph is not showing, TRACE draws it first. The cursor starts on the first selected function in the Y= editor, at the middle X value of the window. With CoordOn, its coordinates appear at the bottom, and with ExprOn the function appears in the top-left corner.

Moving the Trace Cursorp. 79

While tracing, Y is calculated from X, so the cursor is always exactly on the function: Y=Yn(X). If the function is undefined at that X the Y value is blank. If the cursor goes past the top or bottom of the screen, the coordinates at the bottom keep updating correctly even though the cursor is out of sight.

Moving the Trace Cursor from Function to Functionp. 79

Press ▼︎ or ▲︎ to jump to another function. The cursor goes through the selected functions in Y= order and keeps the same X value. With ExprOn, the expression shown at the top updates to the new function.

Moving the Trace Cursor to Any Valid X Valuep. 79

While tracing, just start typing a number. As soon as you enter the first digit, an X= prompt with your number appears at the bottom left. You can type a whole expression there. The value must lie inside the current window. Press ENTER to move the cursor to that X.

Panning to the Left or Rightp. 80

If you trace a function off the left or right edge of the screen, the viewing window slides along with the cursor. Xmin and Xmax are updated to the new window.

Quick Zoomp. 80

While tracing, press ENTER to recenter the viewing window on the cursor. This works even when the cursor is above or below the visible screen, so it is also a way to pan up and down. The window keeps its size, and you stay in TRACE.

Leaving and Returning to TRACEp. 80

If you leave TRACE and come back, the trace cursor is where you left it, unless Smart Graph has had to replot the graph in the meantime.

Using TRACE in a Programp. 80

On a blank line of the program editor press TRACE to paste the Trace instruction. When the program reaches it, the graph is displayed and the user traces freely. When done, press ENTER to resume the program.

ZOOM Menup. 80

Press ZOOM to change the viewing window quickly in several ways. Every ZOOM instruction can also be used in a program. The menu has two tabs: ZOOM and MEMORY (ZOOM MEMORY).

Zoom Cursorp. 81

After you choose 1:ZBox, 2:Zoom In or 3:Zoom Out, the graph cursor becomes the zoom cursor, a smaller version of the free-moving cursor.

ZBoxp. 81

1. Choose 1:ZBox from the ZOOM menu. The zoom cursor appears at the center of the screen.
2. Move the cursor to one corner of the box and press ENTER. After you move away, a small square dot marks that first corner.
3. Press the arrow keys. The box grows or shrinks as you move the cursor to the opposite corner.
4. Press ENTER to replot the graph in the box.

To draw another box inside the new graph, repeat steps 2 to 4. Press CLEAR to cancel ZBox before you press ENTER on the second corner, or press it once more to leave ZBox altogether.

Zoom In, Zoom Outp. 82

Zoom In magnifies the part of the graph around the cursor, and Zoom Out shows a larger part, centered on the cursor. How much the picture changes is set by the zoom factors XFact and YFact (see ZOOM FACTORS).

1. Check XFact and YFact and change them if needed.
2. Choose 2:Zoom In from the ZOOM menu. The zoom cursor appears.
3. Move the cursor to the point that should be the center of the new window.
4. Press ENTER. The window variables are updated and the functions are replotted around the cursor.
5. To zoom again at the same point, press ENTER. To zoom at a new point, move the cursor there first and press ENTER.

To zoom out, choose 3:Zoom Out and repeat steps 3 to 5. Press CLEAR to cancel either one.

ZDecimalp. 82

ZSquarep. 82

ZStandardp. 83

ZTrigp. 83

ZIntegerp. 83

ZoomStatp. 83

ZoomFitp. 83

ZQuadrant1p. 83

ZFrac1/2, ZFrac1/3, ZFrac1/4, ZFrac1/5, ZFrac1/8, ZFrac1/10p. 84

The six ZFrac settings replot at once with a window chosen so that tracing moves in steps of 1/n: they set ΔX and ΔY to 1/n. For ZFrac1/n the window is Xmin=−47/n, Xmax=47/n, Ymin=−31/n, Ymax=31/n with Xscl=Yscl=1.

ZOOM MEMORY Menup. 85

Press ZOOM ▶︎ to open the ZOOM MEMORY menu.

ZPreviousp. 85

ZoomStop. 85

ZoomRclp. 86

ZOOM FACTORSp. 86

The zoom factors XFact and YFact are positive numbers (not necessarily whole numbers) of at least 1. They set how much Zoom In magnifies and Zoom Out shrinks the graph around the cursor. The defaults are both 4.

ZOOM FACTORS
XFact=4
YFact=4

To change a factor, either type a new number (the old value is cleared when you type the first digit) or put the cursor on a digit and type over it, or press DEL to delete it.

Using ZOOM MEMORY Menu Items from the Home Screen or a Programp. 86

From the home screen or a program you can store a value directly into any of the user-defined zoom variables: press STO►, then VARS and 2:Zoom, and pick the variable (ZXmin and friends, with further menus for the T/θ and U/V/W ones). In a program you can also paste the ZoomSto and ZoomRcl instructions from the ZOOM MEMORY menu.

CALCULATE Menup. 87

Press 2ndCALC to open the CALCULATE menu. Its items analyze the functions on the current graph. They need a graph on screen, and the menu draws it for you if needed.

valuep. 87

value evaluates the selected functions at an X you choose. After you choose 1:value, the graph appears with X= at the bottom left. Type a real value (an expression is fine) between Xmin and Xmax and press ENTER. The cursor lands on the first selected function at that X, and the coordinates are shown even if CoordOff is set.

• Press ▲︎ or ▼︎ to see the other functions at the same X.
• Press ◀︎ or ▶︎ to get the free-moving cursor back.
• If an X is showing, CLEAR erases it; if no value is showing, CLEAR cancels the operation.

zerop. 88

zero finds an x-intercept (a root) of a function, using solve(. A function can have several zeros; zero finds the one closest to your guess. How long it takes, and whether it succeeds, depends on how tight your bounds are and how good the guess is.

1. Choose 2:zero. The graph shows with Left Bound? at the bottom left.
2. Press ▲︎ or ▼︎ to put the cursor on the function whose zero you want.
3. Press ◀︎ or ▶︎ (or type a value) to pick the x for the left end of the interval, then press ENTER. A marker shows the left bound, and the prompt changes to Right Bound?. Choose the right end the same way and press ENTER. A marker shows the right bound, and the prompt changes to Guess?.
4. Move to (or type) an x near the zero, between the bounds, and press ENTER.

The cursor sits on the solution and the coordinates show, even with CoordOff. Press ▲︎ or ▼︎ to go to the same X on other functions, or ◀︎ / ▶︎ for the free-moving cursor.

minimum, maximump. 88

minimum and maximum find a low or high point of a function inside an interval, with a tolerance of 1E⁻5. The prompts and keys are the same as for zero: pick the function, then the left bound, right bound and guess. The cursor ends on the solution, with its coordinates shown and the word Minimum or Maximum at the bottom left. Press ▲︎ or ▼︎ for the same X on another function, or ◀︎ / ▶︎ for the free-moving cursor.

intersectp. 89

intersect finds the point where two functions cross, using solve(. The intersection must be visible on the screen.

1. Choose 5:intersect. The graph appears with First curve? at the bottom left.
2. Press ▼︎ or ▲︎ if needed to reach the first function, then ENTER. The prompt changes to Second curve?.
3. Move to the second function with ▼︎ or ▲︎ and press ENTER.
4. Press ▶︎ or ◀︎ to put the cursor near the intersection you want (your guess), and press ENTER.

The cursor lands on the solution, its coordinates are shown (even with CoordOff) and Intersection appears at the bottom left. Press any arrow key to get the free-moving cursor back.

dy/dxp. 89

dy/dx finds the numerical derivative (slope) of a function at a point, with step H=1E⁻³.

1. Choose 6:dy/dx. The graph appears.
2. Press ▲︎ or ▼︎ to pick the function.
3. Press ◀︎ or ▶︎ (or type a value) to choose the X, then press ENTER.

The cursor is on the point and the derivative is displayed. Press ▲︎ or ▼︎ to find the derivative of another function at the same X, or ◀︎ / ▶︎ for the free-moving cursor.

∫f(x)dxp. 90

∫f(x)dx finds the numerical integral of a function over an interval, using the fnInt( function with tolerance H=1E⁻³.

1. Choose 7:∫f(x)dx. The graph appears with Lower Limit? at the bottom left.
2. Press ▲︎ or ▼︎ to move the cursor onto the function to integrate.
3. Choose the lower limit and then the upper limit exactly as you would the left and right bounds for zero. The value of the integral is displayed and the area is shaded.