How to Draw a Circle on a Plane in Geogebra

Curves

Apart from plotting the graph of a part, GeoGebra can also plot a curve that is

• defined by an equation in \(x\) and \(y\),
• the locus of a bespeak satisfying some status,
• a parametric curve.

Click in the lower left corner to turn off the animation. Redefine the function.

Implicit functions

GeoGebra distinguishes between explicit functions and functions that are defined implicitly past equations in \(10\) and \(y\).

An explicit function is written by using brackets ‐ using \(f(x)\)-notation.

An implicit function tin can be written equally \(y=x^ii\). In this case \(y\) can be expressed explicitly in terms of \(x\) and the curve shown, could likewise be shown equally the graph of the function \(f(x) = x^two\). An implicit function still, can also have several \(y\)-values corresponding to a \(x\)-value. An implicit function is hence a more general concept than a function. An equation describes a relation between \(x\) and \(y\), and it is not ever the case that \(y\) can be expressed explicitly in terms of \(10\). The equation \(x^2+y^2=1\) defines a bend that is the unit circle. All points \((x, y)\) satisfying the equation lie on the curve. The curve is implicitly divers past the equation.

GeoGebra categorizes a curve defined by an equation as a Line, a Conic, or as an Implicit curve.

GeoGebra can create implicit curves defined past equations, if the equations are written using polynomials in \(x\) and \(y\). If you, for instance, write y = sin(ten) GeoGebra will classify this equally part, a function that will exist given a name by GeoGebra.

Commands that take a function as argument, do not work on conic sections or lines. If a is defined by the equation y=x^two, you cannot utilise the control TurningPoint on a. In order to use commands for functions, write it as a function, i.e. use \(f(ten)\)-notation!

Locus

If you have a point satisfying some condition, then the set of all such points is a locus.

To make a locus in GeoGebra, y'all need a point \(B\) that depends either on another point \(A\) or a slider. The point \(A\) must prevarication on a bend or a coordinate axis. Using the tool Locus y'all can and then create the locus past first clicking on \(B\) and so on \(A\) or the slider.

Polar coordinates

In GeoGebra you can use polar coordinates when defining a bespeak. When using polar coordinates, a semicolon is used equally delimiter, as in A=(r;α). Past letting the angle get its value from a slider \(t\), and by defining a function \(r(x)\), it is possible to draw the bend traced out by a indicate A=(r(t);t). Use the tool Locus, click on the betoken and so the slider. Under the preferences for Graphics, under the Filigree tab, information technology is possible to testify the grid in polar class.

Polar coordinates can be converted to Cartesian coordinates by letting \(ten=r\cos(t)\) and \(y=r\sin(t)\). This can be used for drawing the curves equally parametric curves instead of a locus.

Parametric curves

If the \(x\)- and the \(y\)-coordinate both depend on a parameter \(t\), they describe a parametric bend with parameter \(t\). Each bespeak on the curve can be described by the coordinates \((ten(t),y(t))\). By letting \(t\) starting time at a value \(beginning\) and finish at a value \(finish\) you tin describe the curve \(one thousand\) by entering:

k = Curve( 10(t), y(t), t, first, stop)

in the input bar.

You can ever convert a regular function to a parametric role. Every bit an case, the graph of \(f(10)=ten^ii,-10\leq x \leq 10\) tin can be drawn as a parametric bend by writing the lawmaking:

k = Curve(t, t^ii, t, -10, 10)

Past letting a slider correspond the end-value, it is possible to visualize how a graph is drawn. Create a slider tmax and define the curve as:

g = Bend(t, t^2, t, -10, tmax)

Transformations and curves

The simply transformation yous can use on the graph of a regular role is Translate by Vector.

Apart from translating a parametric curve, you can besides transform it by using the transformations Reverberate about Line and Rotate around Point.

Exercises

Exercise 1

Make conic sections

Create four sliders \(a\), \(b\), \(h\) and \(thousand\).

In analytic geometry a parabola tin can exist defined past the equation

\[y = a(ten-h)^ii +k,\]

an ellipse by the equation

\[\frac{(x-h)^2}{a^2} + \frac{(y-k)^ii}{b^2} = 1, \]

and a hyperbola past the equation

\[\frac{(10-h)^2}{a^2} - \frac{(y-one thousand)^ii}{b^2} = i.\]

Create the three conic section using these equations.

What upshot do the values of \(h\) and \(1000\) accept on the three curves?

What event practice the values of \(a\) and \(b\) have on the ellipse and on the hyperbola?

Exercise ii

Dominicus, earth and moon

Create a slider \(R\) representing the radius of the trajectory of the globe moving around the sun.

Create a slider \(r\) representing the radius of the trajectory of the moon moving around the globe.

Create a slider \(m\) taking integer values representing the number of months in a yr.

Place a point \(S\) (the lord's day) at the origin.

Version i ‐ using degrees and rotations

Create a slider \(\blastoff\) taking angle values representing the angle of the earth.

Identify a point \(E\) (the earth) on the positive \(x\)-axis. The altitude between \(S\) and \(E\) should be \(R\). Use the tool Rotate around Bespeak to rotate \(E\) effectually \(S\) the angle \(\alpha\). The rotated indicate volition be called \(E'\). Hide the point \(Eastward\).

Create a point \(M\) (the moon) by writing M = E'+(r, 0). Now \(M\) should be rotated around \(Eastward'\) past an bending depending on the number of months \(m\) and the bending \(\blastoff\). Discover this angle and create the rotated point \(M'\). Hide \(One thousand\).

Use the tool Locus. First click on \(Grand'\) and then on the slider \(\blastoff\).

Version 2 ‐ using radians and trigonometry

Create a slider \(a\) with values between \(0\) and \(2\pi\) representing the angle of the earth.

Let the point \(E\) represent the earth. Discover the \(10\)- and \(y\)-coordinate of \(E\) in terms of appropriate sliders. Enter the expression for the bespeak in the input bar.

Let \(M\) represent the moon. Find the \(x\)- and \(y\)-coordinate of \(M\) in terms of appropriate sliders. Enter the expression for the betoken in the input bar.

The expressions used for the coordinates of the moon, tin can also exist used to make a parametric bend showing the path of the moon. Create this bend. Let the parameter outset with the value \(0\) and stop with the value \(a\).

Comment: It is possible to prove images instead of points. The images in the worksheet below are from NASA's image gallery.

Click in the lower left corner to turn off the animation.

Exercise 3

Domain and range for inverse trigonometric functions

If \(f^{-1}\) is the inverse role of a function \(f\), then the graph of \(f^{-1}\) is the reflection of the graph of \(f\) in the line \(y = x\).

In order to reflect a graph, it must be fabricated as a parametric curve.

• Ascertain the function f(x) = sin(x). Hibernate the graph.

• Identify two points \(A\) and \(B\) on the \(x\)-axis. The points represent the terminate points of the domain of the function.

  Y'all tin show the \(10\)- or \(y\)-coordinate of a point as a label. Write %x or %y as Explanation in the properties window.

• Create the parametric curve to the part \(f(x)= \sin(x), x(A) \le 10 \le ten(B)\).

• Reflect the curve in \(y=x\). You can either use the tool Reflect about line or use the command Reverberate( <Object>, <Line> ).

• Link \(f\) to an input box using the tool Input Box so you easily can redefine the role.

• Drag the points. Study the intervals for which the reflected bend could be the graph of a function. In those cases, what is the domain and what is the range of the inverse function?

Exercise 4

Make a Lissajous bend

A Lissajous curve appears from a periodic motility along the \(x\)-axis and one forth the \(y\)-axis.

A Lissajous curve tin be described by an equation

\[ \begin{cases} ten & = A\sin (at+\delta)\\ y &= B\sin (bt) \end{cases} \]

An easier equation is used in the worksheet beneath.

Animated Lissajous curve. When \(a=1\) it's a circumvolve.

Visualize a Lissajous curve in such a way that the ii periodic motions along the coordinate axes are also shown.

You can besides make damped Lissajous curves, meet Damped Lissajous Curves for more data.

further info:

Wikipedia: Algebraic bend

Lissajous curves tin be fatigued by harmonographs

by Malin Christersson nether a Creative Eatables Attribution-Noncommercial-Share Alike two.v Sweden License

www.malinc.se

kelleherquan1940.blogspot.com

Source: http://www.malinc.se/math/geogebra/curvesen.php

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