# Graphing Calculator:Find Symbolic Derivatives, X-Intercepts

Explore our free online Cartesian and polar graphing calculator, a powerful tool for graphing the most common types of mathematical expressions: functions, equations (including implicitly defined functions), parametric curves (also known as parametric equations), and points.

Graphing a function `f(x)` and additionally finding its x-intercepts? And what about finding symbolic derivatives for solving a calculus problem, determining where the graph is increasing or decreasing, and where itâ€™s concave up or down? Or perhaps you want to graph the function in the polar coordinate system, compare it to its Cartesian graph, and be fascinated by the step-by-step creation of polar graphs? This powerful Cartesian and polar graphing calculator is not just made for these purposes, but it allows you to graph other types of mathematical expressions: equations in two variables, and parametric equations in addition to point sets.

Here are some examples of syntax:
• f(x) = x^2sin(x) + 2x + 1 (function)
• x^3-xy+2y^2 = 5x+2y+5 (equation)
• p(t) = [sin(t), cos(t)] (parametric)
• 1,2; -2, 2/3; sin(π/3), 2^3-1 (points)
More on Syntax

Unique among graphing calculators, it provides the remarkable capability of rotating each axis independently. This advanced feature makes it the world's only graphing tool (besides other graphers developed by this site) that allows graphing in oblique coordinate systems, alongside Cartesian & polar coordinate systems.

• Unique Polar Parametric Graphing Calculator: This graphing tool, as a parametric equation graphing calculator, is the only one that can produce polar parametric graphs.
• Unmatched Animation Capability: Users can visualize the step-by-step formation of polar and parametric graphs. When graphing in the polar coordinate system, it's also the only graphing tool that shows the radial axis rotating while animating polar graphs of functions and parametric curves.
• Calculus Tools: Beyond graphing, it can also find the x-intercepts (also known as zeros or roots of a function), and even calculate symbolic derivatives of functions and parametric expressions and graph them, making this graphing calculator a powerful tool for solving problems in Calculus.

In particular, you can use this graphing calculator to:

• Graph linear functions and linear equations in point-slope form and slope-intercept form.
• Graph conic sections in the standard form such as `(x-h)^2 + (y-k)^2 = r^2`, and the general form (`Ax^2 + Bxy + Cy^2 - Dx + Ey + F = 0`), which can be a circle, ellipse, parabola, hyperbola, or some degenerate graphs.
• Graph level curves, which are in the form `F(x,y) = c`.
x y
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Label Axes

Rotate Axes

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Functions

Lines

1 x+1 2x

Semi-circles

√(9-x^2) -√(9-x^2)

Semi-ellipses

√(9-x^2/3) √(9-x^2/3)

Parabolas

x^2 0.5x^2-4x+1 -(0.5x^2-4x+1)

Semi-hyperbolas

√(x^2-4) -√(x^2-4)

Other graphs

√(4sin(2x)) √(4cos(2x))
Functions – Polar

Lines

2csc(θ) 2sec(θ) 1/(sin(θ) - cos(θ))

Circles

1 2 6sin(θ) 8cos(θ)

Spirals

θ θ/5 dom=(0, 10π) √(θ) dom=(0, 10π) 1/θ dom=(0, 10π)

Roses

4sin(3θ) 4sin(2θ) 4sin(5θ) 4sin(4θ)

Ellipses

1/(1-.8cos(θ)) 1/(1-.8sin(θ)) 1/(1+.8cos(θ)) 1/(1+.8sin(θ))

Parabolas

1/(1-sin(θ)) 1/(1+cos(θ)) 1/(1+sin(θ)) 1/(1-cos(θ))

Hyperbolas

1/(1+2cos(θ)) 4/(1+2sin(θ)) 1/(1-2cos(θ)) 4/(1-2sin(θ))

Cardioids

3+3cos(θ) 2+2sin(θ) 3-3cos(θ) 2-2sin(θ)

Limacons

2+3cos(θ) 1+2sin(θ) 2-3cos(θ) 1-2sin(θ)

Lemniscates

√(4sin(2θ)) √(4cos(2θ))

Butterfly curve

e^sin(θ)-2cos(4θ)+sin((2θ-π)/24)^5 dom=(0, 12π)
Equations

Lines

y = 1 x = 1 y = x+1 x = y+1 3x + y = 2 3x - y +5 = 4x+2y-2

Circles

x^2+y^2 = 9 (x-2)^2 + (y-2)^2 = 4

Ellipses

x^2/4 + y^2/9 = 1 x^2-xy+2y^2-x-2y-8=0

Parabolas

y=x^2 y = x^2-4x+4 2x^2-4xy+2y^2-x-2y-2=0

Hyperbolas

x^2/4 - y^2/9 = 1 2x^2-5xy-4y^2+9x+9y-16=0

Other graphs

x^2 = y^2 sin(xy) = cos(xy)
Equations — Polar
Currently, not available.
Parametric

Lines

[t, 1] dom=(-5, 5) [1,t] dom=(-5, 5) [t, 2t] dom=(-5, 5)

Circles

[4sin(t), 4cos(t)] [3sin(t)+1, 3cos(t)+1]

Ellipses

[4cos(t), 3sin(t)] [3cos(t), 4sin(t)] [4sin(t), 3cos(t)] [3sin(t), 4cos(t)]

Parabolas

[t, t^2] dom=(-4, 4) [t^2, t] dom=(-4, 4)

Hyperbolas

[3sec(t), 4tan(t)] [3tan(t), 4sec(t)]

Other parametric graphs

[5sin(t), 4cos(t)] [5sin(t), 4cos(2t)] [5sin(t), 4cos(3t)] [5sin(2t), 4cos(t)] [5sin(2t), 4cos(3t)] [5sin(2t), 4cos(5t)] [5sin(3t), 4cos(5t)] [5sin(3t), 4cos(7t)] [5sin(5t), 4cos(7t)] [5sin(7t), 4cos(9t)]

Butterfly curve

[sin(t)(e^cos(t)-2cos(4t)-sin(t/12)^5), cos(t)(e^cos(t)-2cos(4t)-sin( t/12 )^5)] dom=(0, 12π)
Parametric – Polar

Lines

[2csc(t), t] [2sec(t), t] [1/(sin(t) - cos(t)), t]

Circles

[1, t] [2, t] [6sin(t), t] [8cos(t), t]

Spirals

[t, t] [t/5, t] dom=(0, 10π) [√(t), t] dom=(0, 10π) [1/t, t] dom=(0, 10π)

Roses

[4sin(3t), t] [4sin(2t), t] [4sin(5t), t] [4sin(4t), t]

Ellipses

[1/(1-.8cos(t)), t] [1/(1-.8sin(t)), t] [1/(1+.8cos(t)), t] [1/(1+.8sin(t)), t]

Parabolas

[1/(1-sin(t)), t] [1/(1+cos(t)), t] [1/(1+sin(t)), t] [1/(1-cos(t)), t]

Hyperbolas

[1/(1+2cos(t)), t] [4/(1+2sin(t)), t] [1/(1-2cos(t)), t] [4/(1-2sin(t)), t]

Cardioids

[3+3cos(t), t] [2+2sin(t), t] [3-3cos(t), t] [2-2sin(t), t]

Limacons

[2+3cos(t), t] [1+2sin(t), t] [2-3cos(t), t] [1-2sin(t), t]

Lemniscates

[√(4sin(2t)), t] [√(4cos(2t)), t]

Other parametric graphs

[5sin(t), 4cos(t)] [5sin(t), 4cos(2t)] [5sin(t), 4cos(3t)] [5sin(2t), 4cos(t)] [5sin(2t), 4cos(3t)] [5sin(2t), 4cos(5t)] [5sin(3t), 4cos(5t)] [5sin(3t), 4cos(7t)] [5sin(5t), 4cos(7t)] [5sin(7t), 4cos(9t)]
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Graph Thickness
Angle Mode
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Done

To copy or save graphs right click on the image of a saved graph below and select "Copy image" or "Save image" from the pop-up menu.

As you type:
• pi is replaced by `π`.
• ..t is replaced by `θ`. (You can also use `x` or `t`; they are internally replaced by `θ`).
• inf (infinity) is replaced by `∞`.
More tips

MouseMatics: Find out how to use your mouse to rotate axes, change scales, and translate coordinate systems.

## Instructions for Our Graphing Calculator

This powerful graphing calculator lets you graph functions, equations, parametric curves, and point sets using either the Cartesian or polar coordinate systems. This graphing tool also allows you to rotate any axis and graph in an oblique coordinate system, where axes may not necessarily be horizontal or vertical, and can intersect at any angle. To graph, simply type an expression into any expression box, and the software will automatically detect the type of expression and generate the graph instantly by default. You can easily switch between coordinate systems by selecting/deselecting the Polar checkbox; the graphs are regenerated accordingly. The software is designed to automatically adjust the variables based on the expression type and the coordinate system, so you don't have to worry about which variable (`x`, `y`, `t`, or `θ`) to use.

Our graphing calculator is designed to be intelligent and user-friendly. When you enter an expression, it automatically detects its type:

• Function: If you use `x` as the independent variable, the function grapher will label the expression as `f(x)= ` or `f(θ)= ` depending on the coordinate system.
• Equation: If your expression includes an equal sign (`=`), the calculator switches to equation mode, and the equation grapher (also known as implicit function grapher) will display `Eq:` in the label.
• Points: If your expression uses both commas and semicolons, the calculator switches to points mode, and the points grapher will display `x,y;` or `r,θ;`, depending on the coordinate system, for the expression label.
• Parametric curve: A single comma indicates a parametric expression, switching the calculator to parametric mode. The parametric curve grapher (also known as parametric equations grapher) displays `p(t)= ` for the expression label, and replaces `x` with `t` in the expression. Deleting the comma switches back to function mode, replacing `t` with the appropriate variable.

When graphing functions or parametric expressions, if you don't specify a domain (interval), the graphing calculator automatically sets a suitable domain to ensure proper graphing. It uses dom=(`-∞`, `∞`) for functions in the Cartesian coordinate system, and dom=(`0`, `2π`) otherwise. You can change the endpoints of the interval if desired, but they must be finite for polar or parametric graphing. The calculator will automatically adjust any infinities to finite values.

### Using Polar and Parametric Graphing Animation Feature

Visualize how graphs are constructed step-by-step! Our graphing calculator offers the most powerful animation feature for function graphs in the polar coordinate system and parametric graphs in both Cartesian and polar coordinate systems.

The animation uses a step-by-step approach to draw these graphs. It shows where the graph starts and ends, and in the process, it allows you to observe whether any loops or sections of the graph are traced multiple times. You can control the speed of the animation, allowing you to see the construction process in detail. Pause/resume at any speed to tailor the animation to your pace. To activate the animation feature, press the button at the bottom of the graphing tool (if hidden, press Animate first).

• The graph animator starts the animation for the focused function or parametric curve. The animation progressively draws the graph, from the initial value to the final value of the independent variable or parameter.
• When animating in the polar coordinate system, you can optionally show or hide the rotating radial axes by checking or unchecking the Show radial axes checkbox (by default, it's checked). By showing the rotating radial axes, the polar graphing calculator animates the polar graph construction in a way unmatched by any other polar graphing tools.
• You can press to pause the animation, or Done to stop it. This also closes the animation interface. To display the animation interface again, press the Animate button.
• Use the slider to adjust the animation speed.

### Graphing Multiple Expressions

To graph multiple functions, equations, parametric curves or point sets, press the » button to show the multi-graph pane containing the expression panels, and type your expressions in expression fields on any of the available panels.

• Add or remove panels: Use the plus ( +) button to add more expression panels or the delete (×) button to remove panels if needed.
• Show/hide graphs: Select or deselect the checkbox next to an expression to show or hide its corresponding graph.

### Graph Accuracy Setting

Select an option in Graph Fineness. This controls how smooth and detailed the graph will be. Higher accuracy creates a smoother curve with more detail but takes longer to graph. Choose the level that best suits your needs.

### Labeling Axes

To label an axis, click on the icon at the top right of the canvas. Type in any number for which you want to albel the axes, and press the Label button. You can also use constants like `π`, or even constant expressions such as `1+3π/2`.

### Rotating Axes

Our graphing software offers a unique feature: axis rotation. This allows you to visualize the graph of a given function, parametric curve, equation, or point set in the oblique (non-perpendicular) coordinate system. To rotate an axis, click on the icon at the top right of the canvas, and then enter the angles by which you want to rotate an axis and press the Rotate button. The axis will rotate and the graphs will be redrawn to reflect the rotation.

### Copying & Saving Graphs

1. Click the Copy/Save graph button. This will create a copy of all the graphs on the canvas (graphing area) which will appear as an image below the graphing calculator..
2. Right-click on the image and select the appropriate option from the context menu. Depending on your device and preferences, you might be able to:
• Copy: Create a copy of the image to your clipboard for pasting elsewhere.
• Save image as... : Save the image as a file on your device in a chosen format (e.g., PNG, JPG).

### Evaluating Functions & Parametric Expressions

To evaluate a function or parametric expression, type a literal number or numerical expression in the provided box. The calculator will display the calculated value, rounded to the number of decimal places set by the slider.

### Finding X-Intercepts

Our calculator can be used as an equation solver to find the x-intercepts (also known as zeros or roots) of a function. To do this, press the Solve button. The equation solving calculator will then find the x-intercepts of the function in focus by solving the equation `f(x) = 0` and display them on the screen. Notes on finding x-intercepts

### Calculating and Graphing Symbolic Derivatives

In addition, the calculator can be used as a derivative graphing calculator. To calculate the first and second order derivatives of a function or parametric expression in focus, press the Derivative button.

After the derivatives are displayed, you can press Graph f, f' or Graph f, f', f'', which also appear on the screen, to draw the graphs of the function or parametric expression and their derivatives in a new window. You can also add the calculated derivatives to new panels by selecting them (selected by default). These panels will be appended to the bottom of the multi input pane. This will allow you to use them to find, for example, the critical points of the function by pressing the Solve button as described above. Press the OK button to close the derivative window. Find out more about first & second order derivatives.

### Interesting Curves

Graph any of the predefined expressions under the Interesting Graphs selections, located on the multi-input pane, to render some cool Cartesian and polar graphs by selecting it. For best results, you may need to select Graph Fineness as "+1" or higher.

### Settings

Press the ⚙ (gear) button to set options (if the button is hidden, first click on the icon at the top right of the canvas):

• Graph Thickness: Change graph thickness using the slider.
• Angle Mode: Select the angle mode (radians - default, degrees, grades) — you can also change angle modes by using the buttons provided on the top of the Canvas.
• Graphing Mode: If you deselect the Graph as you type option, you will have to press Graph selected expressions, which then appears at the bottom of the calculator, to update the graphs whenever you make any changes to the expressions or the coordinate plane (i.e., move the origin, rotate axes, etc.).
• Automatic Axis Rotation: You have the option to display controls that will automatically rotate the axes.
• Expression History: The graphing calculator remembers expressions between visits. You can press Reset Calculator to clear them.
Insert on the bottom of multi-input panel: