Function & Equation Grapher |Implicit Function Grapher
As a function grapher, this online plotter graphs functions like
f(x) = x^2+x−6.
As an equation grapher, it allows for plotting general equations in two variables, such as
2y^2+xy = x^2+2y.
Since implicitly defined functions are expressed by equations like
2x+3y = 6 (solving for y, in terms of x, we get y = -2/3x + 2, an explicit function of x), this plotter also serves as an implicit function grapher.
A (one-variable) function grapher is limited to equations of the form y = f(x). To graph a function, you simply enter the right-hand side of the 'equation' into the provided expression box.
In contrast, an equation grapher is more versatile than a function grapher. It can graph any equation of the form of G(x,y) = F(x,y). It can also graph functions since a function f(x) is also represented by the equation y = f(x), which is a special form, where G(x,y) = y and F(x,y) = f(x).
Being a more versatile graphing tool, an equation plotter can handle implicitly defined functions because implicit functions are inherently defined by an equation. Other expressions that an equation grapher can handle include the equation of a line in both general form (ax+by = c) and point-slope form (y-y₁ = m(x-x₁)), quadratic equation—whose graphs are conic sections (circles, ellipses, parabolas, and hyperbolas)— and level curves, for examples.
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sin(πx) = cos(πy)
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Functions
Lines
1x+12x
Semi-circles
√(9-x^2)-√(9-x^2)
Semi-ellipses
√(9-x^2/3)√(9-x^2/3)
Parabolas
x^20.5x^2-4x+1-(0.5x^2-4x+1)
Semi-hyperbolas
√(x^2-4)-√(x^2-4)
Other graphs
√(4sin(2x))√(4cos(2x))
Equations
Lines
y = 1x = 1y = x+1x = y+13x + y = 23x - 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 = 1x^2-xy+2y^2-x-2y-8=0
Parabolas
y=x^2y = x^2-4x+42x^2-4xy+2y^2-x-2y-2=0
Hyperbolas
x^2/4 - y^2/9 = 12x^2-5xy-4y^2+9x+9y-16=0
Other graphs
x^2 = y^2sin(xy) = cos(xy)
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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.
Tips - as you type:
pi is replaced by π.
inf (infinity) is replaced by ∞.
Note: Our graphing software allows you to use constant expressions, such as π or 1+√(2) wherever you can use a literal number.
Instructions for Function Grapher and Equation Grapher
It's easy to use our function and equation grapher; type in a functionf(x) or an equation, for example 3xy-2y = x^2+4y in any expression box. The grapher graphs as you type (default).
When graphing functions in the Cartesian coordinate system, the grapher plots the function on the interval (domain) dom=(-∞,∞) by default if no interval is already specified. You can change the endpoints of the interval if desired. You can also plot polar graphs, to visualize functions in the polar coordinate system.
Note: The quickest way to type dom=(-∞,∞) is to delete the domain entirely, including dom=.
Function & equation plotter: Function and equation graphs in rectangular Cartesian coordinate system.Oblique function & equation plotter: Function and equation graphs in oblique Cartesian coordinate system.
Graphing Multiple Functions and Equations
To graph multiple functions and equations, press the » icon to show the multi-graph pane containing expression panels. Type in your expression in any box.
Add or delete panels with the + or × buttons if necessary.
Select or deselect checkboxes to show or hide graphs.
Remark: To graph piecewise defined functions type in each piece with the corresponding subinterval as a single function.
Graph Accuracy Setting
The quality of the resulting graph is controlled by the Graph Fineness setting. A higher graph fineness results in a more accurate graph, but it also takes longer to graph the equation.
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 label 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
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..
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
To evaluate a function, type a number or expression in the provided box.
The function graphing calculator will display the calculated value, rounded to the number of decimal places set by the slider.
Interesting Curves
Interesting curves: Graph any of the expressions under Interesting Graphs. 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):
Change graphthickness using the slider.
Select the angle mode (radians - default, degrees, grades).
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.).
You have the option to display controls that will automatically rotate the axes.
The function and equation graphing calculator remembers the expressions so that you can use them in future visits. You can press Reset Calculator to clear them.
Function vs Equation
Many online resources incorrectly use "function" and "equation" interchangeably. However, these terms are not identical.
For simplicity, first let's consider functions and equations involving only one variable; the general case is similar.
A function of one variable, x, is a rule that assigns a unique output value, f(x), to each input value, x. The symbol f represents the function's name.
Formally, a function f(x) is a set of ordered pairs (x,f(x)) where no two pairs have the same first element.
An equation, on the other hand, is actually a “question” about when equality holds between two expressions, usually function expressions. Unlike functions, which are "assignment" or "declarative expressions", equations are "interrogative expressions" (notice the analogy to the difference between declarative and interrogative sentences in grammar), typically asking for solutions or graphing. In eirher case, an equation asks for the values of x that make the two expressions equal. Specifically, an equation of the form f(x) = g(x) asks for the values of x, where the equality holds, which requires solving the equation.
For example, if f(x) = x^2 + 2x and g(x) = 3x - 6, the equation f(x) = g(x) asks for the solution which we can find by solving the equation x^2 + 2x = 3x + 6 (the solutions are x = -2 abd x = 3).
Remark: The expression y = f(x) is certainly an equation in two variables x and y. We can also say y is a function of x (with x being the independent variable, and y the dependent variable). So, in this special case, it does not realy matter whether we regard y = f(x) as an equation, or a function assigned to y. That's because, as an equation, it has infinitely many solutions (x,y) = (x,f(x)), all lie on the graph of the functionf(x), and vice versa. Therefore, we can regard it either way.
How Our Equation Grapher Works
Our equation grapher grapher employs an advanced algorithm. This algorithm starts by investigating rows of pixels on the canvas to find the zeros of f(x,y)-g(x,y) for each value of y, employing Newton's method. It then uses implicit differentiation to draw tiny tangent lines at those values that satisfy the equation. This process effectively builds the graph.
Functions & Equations in Oblique Cartesian Coordinate System
Our function and equation grapher’s ability to graph in oblique coordinate systems relies on our graphing software’s unique capability to plot points in oblique coordinate systems.
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