Coordinate Geometry
Coordinate Geometry: interactive exploration and practice activities.
Inhaltsverzeichnis
- Points
- The Coordinate Plane (Cartesian Plane)
- Cryptography and Coordinates
- Pick's Theorem
- Lines
- Line Through Two Points and Slope
- Systems of Linear Equations - Practice
- Parabolas
- Quadratic Equations, Inequalities and Parabolas
- Quadratic Equations: the Geometry of the Relationships Between the Coefficients and the Solutions
- Coefficients of a Parabola - Practice
- The Geometry of the Coefficients of a Quadratic Function
- Parabolas in Vertex Form - Practice
- Parabola - Write the Equation - Practice
- How to: Parabola Through 3 Points
- Orthogonal parabolas - Exploration
- Explore! From 3D to 2D: Parabola as Projection of a Circle
- Parabola - An Optical property
- Parabola as Envelope
- Circles
- Circle and Coordinate Geometry - Lesson+Practice
- Ellipses
- The Gardener's Ellipse
- Ellipse as locus of points - Lesson
- Ellipse - An Optical Property
- Ellipse as Envelope of Segments
- Ellipse as Envelope
- Conics
- "All-in-One" Loci: Ellipse, Hyperbola and Circle - Lesson
- Conics and Euclidean Invariants
- Ellipse, Hyperbola and Circle as Envelopes
- Transformations
- Matrices and Linear Transformations
- Transformations of Triangles and Matrices
The Coordinate Plane (Cartesian Plane)
Discover and explore the connection between the position of a point in the Cartesian plane and its coordinates, then answer the questions that follow.
Practice Zone
Point [math]F[/math] lies in the first quadrant, point [math]S[/math] lies in the second quadrant, point [math]T[/math] lies in the third quadrant and point [math]Q[/math] lies in the fourth quadrant.[br]Write the possible coordinates of these points.
Points [math]M[/math] and [math]N[/math] have the same [i]y[/i]-coordinate, equal to [math]3.14[/math].[br]How would you describe their position in the coordinate plane?
What is the [i]x[/i]-coordinate of a point on the [i]y[/i]-axis?[br]What is the [i]y[/i]-coordinate of a point on the [i]x[/i]-axis?
Write the coordinates of the points shown in the graph below.[br][br][img]data:image/png;base64,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[/img]
Line Through Two Points and Slope
Graph and Slope of a straight line
The graph of a function of the form [math]y=mx+b[/math] is a [i]line[/i]. [br]This is why all the functions of this form are called [i]linear functions[/i].[br][br]If we know the coordinates of two points of the line, [math]P_1=\left(x_1,y_1\right)[/math] and [math]P_2=\left(x_2,y_2\right)[/math], we can calculate the [i]slope[/i] [math]m[/math] of the line: [math]m=\frac{y_2-y_1}{x_2-x_1}[/math]. [br]The slope is a constant value: however you choose two points on the line, the value of [i]m[/i] is always the same.
Try it yourself...
In the app below, move points [math]A[/math] and [math]B[/math], then determine the value of the slope [math]m[/math] of your line [math]AB[/math].[br]Select [i]CHECK[/i] to get a feedback for your answer and view the solution of this exercise.[br]
When things go wrong algebraically...
If you have the equation of a line [math]y=mx+b[/math] and / or the coordinates of two of its points, [math]P_1=\left(x_1,y_1\right)[/math] and [math]P_2=\left(x_2,y_2\right)[/math], you can calculate: [br]- the value [math]b[/math] of the [i]y[/i]-intercept[br]- the value [math]m[/math] of the slope, using the formula [math]m=\frac{y_2-y_1}{x_2-x_1}[/math].[br][br]Move points [i]A[/i] and [i]B[/i] in the app above, and align them vertically.[br]You will discover which is the algebraic issue that is generated by such a configuration.
... and geometrically
Move points [i]A[/i] and [i]B[/i] in the app above, and align them vertically.[br]Observe the graph of the line.[br]Is this the graph of a [i]linear function[/i]?[br]Explain your conjectures.
Quadratic Equations, Inequalities and Parabolas
Solving a Quadratic Inequality Graphically
The graph of a [i][color=#1e84cc][b]quadratic function[/b][/color][/i] [math]y=ax^2+bx+c[/math] in a Cartesian coordinate system is a [i][color=#1e84cc][b]parabola[/b][/color][/i].[br][br]To [i][b][color=#1e84cc]solve[/color][/b][/i] a [i][b][color=#1e84cc]quadratic inequality[/color][/b][/i] [math]ax^2+bx+c>0[/math] or [math]ax^2+bx+c<0[/math] [i][b][color=#1e84cc]graphically[/color][/b][/i], draw the corresponding parabola, then:[br][list][*]The[color=#1e84cc] [i][b]x-coordinates[/b][/i][/color] of the points (if they exist) where the [i][b][color=#1e84cc]parabola intersects[/color][/b][/i] the[i][b] [color=#1e84cc]x-axis[/color][/b][/i] are the [i][b][color=#1e84cc]solution[/color][/b][/i] of the[color=#1e84cc] [i][b]equation[/b][/i][/color] [math]ax^2+bx+c=0[/math] (zeros of the equation)[/*][/list][list][*]The [color=#1e84cc][i][b]x-coordinates[/b][/i][/color] of the points (if they exist) where the [i][b][color=#1e84cc]parabola[/color][/b][/i] is[color=#1e84cc] [/color][i][b][color=#1e84cc]above[/color] [/b][/i]the [i][b][color=#1e84cc]x-axis[/color][/b][/i] are [i][b][color=#1e84cc]solution[/color] [/b][/i]of the [i][b][color=#1e84cc]inequality[/color][/b][/i] [math]ax^2+bx+c>0[/math][/*][/list][list][*]The[color=#1e84cc] [i][b]x-coordinates[/b][/i][/color] of the points (if they exist) where the [i][b][color=#1e84cc]parabola[/color][/b][/i] is [i][b][color=#1e84cc]below[/color][/b][/i] the [i][b][color=#1e84cc]x-axis[/color][/b][/i] are [b][i][color=#1e84cc]solution [/color][/i][/b]of the[color=#1e84cc] [i][b]inequality[/b][/i][/color] [math]ax^2+bx+c<0[/math][br][/*][/list]
Practice Zone
Find the solutions of a quadratic equation or inequality by exploring the graph of the corresponding parabola.[br][br]Use the input box to enter different quadratic expressions and the drop down list to select the equation or inequality form to solve. [br][br]Use the mouse wheel or the predefined gestures for mobile devices to zoom in/out and view details in the [i]Graphics View[/i].[br][br]
You Are the Teacher, Today!
Today's assignment is the inequality [math]x^2-49<0[/math]. [br][br]Alice solves it like this: [math]x^2<49\rightarrow x<\pm7[/math].[br][br]Bob solves [math]x^2-49=0[/math] first, and gets [math]x=\pm7[/math]. Then he graphs the parabola corresponding to the given equation and finds the solution [img]data:image/png;base64,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[/img].[br][br]Chuck uses ChatGPT 2 and finds that the solution is [math]-7\le x\le7[/math].[br][br]Alice says that her method is faster than Bob's, because it doesn't require sketching a graph.[br][br]Grade your students' solutions, and explain the reasons for your grading.
Hamletic Doubt...
Below you can see the solution of one of the following inequalities. [br]Choose the correct one.[br][img]data:image/png;base64,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[/img]
Circle and Coordinate Geometry - Lesson+Practice
Discover how to write the equation of a circle given its center and radius, and how to find center and radius given the general equation of a circle.[br][br]Explore the applet, then answer the questions displayed below.
Select the option "Circle - given center and radius" in the app above, then drag the center C at (0,0).[br]Use the slider to display different circles with center at (0,0).[br]What is the equation of a circle with center at (0,0) and radius [i]r[/i]?
A circle with center [i]C[/i]=(0,0) passes through [i]P[/i]=(2,3). What is the equation of the circle?
Given [i]A[/i]=(-8,-3) and [i]B[/i]=(2, 5), write the equation of the circle with diameter [i]AB[/i].
Select the option "Circle - general equation" in the applet above and use the sliders to explore the equation and the corresponding graph.[br][br]When [math]a=0[/math]...[br]
When [math]b=0[/math]...
When [math]a=b=0[/math]...
When [math]c=0[/math]...
Does the equation [math]x^2+y^2+10x+12y+61=0[/math] represent a circle?[br]If so, determine its center and radius.[br]If not, explain why this is not the equation of a circle.
The Gardener's Ellipse
How can a gardener draw a perfectly elliptical contour for a flowerbed?[br][br]Three stakes and a rope are enough: two stakes are fixed at the position of the foci of the flowerbed, and the ends of the rope are attached to them. [br][br]At this point, the gardener can mark the contour of the elliptical flowerbed using the third stake to hold the rope, moving around the fixed stakes while ensuring that the rope is always taut.[br][br][i]Use the app below to create your virtual flowerbed using the gardener's method: the slider [math]\ell[/math] allows you to change the length of the rope, and the other slider sets the semi-distance of the foci (that is, half the distance between them). [br][i]Drag point P or the tip of the stake in the 3D View to create the flowerbed.[/i][br][br]The 3D view can be rotated by dragging it with the mouse or using the predefined gestures for touch screens, and the buttons allow you to restore the 3D view to the original settings and delete the traces.[br][/i]
What do you notice about the distances from [math]P[/math] to [math]F_1[/math] and from [math]P[/math] to [math]F_2[/math]?
Which conic section do you obtain when the two foci coincide?
"All-in-One" Loci: Ellipse, Hyperbola and Circle - Lesson
Move the foci [color=#9900ff][i]F[/i][sub]1[/sub][/color]and[color=#9900ff] [i]F[/i][sub]2[/sub][/color] that define the position of the focal axis of the conic, set the length of the [i]major axis[/i] of the conic using the [color=#1551b5][i]mAxis[/i][/color] slider, then move point [i][color=#6aa84f]P[/color][/i] along the circle to build the locus.[br][br]To create the construction:[br][list][*]Plot 2 points,[color=#9900ff][i] F[sub]1[/sub][/i][/color][color=#ff0000] [/color]and [i][color=#9900ff]F[/color][/i][sub][i][color=#9900ff]2[/color][/i] [/sub]that will be the foci of the conic.[br][/*][*]Draw the circle with center [color=#9900ff][i]F[/i][sub]1[/sub][/color] and [color=#1e84cc]radius [i]r[/i] = [i]mAxis[/i][/color][/*][*]Create a point [color=#6aa84f][i]P[/i][/color] on the circle, then draw the perpendicular bisector of segment [i]P[/i][i]F[/i][sub]2[/sub][/*][*]Line [i]P[/i][color=#ff0000][color=#000000][i]F[/i][/color][sub][color=#000000]1[/color] [/sub][/color]intersects the perpendicular bisector at a point [color=#6aa84f][i]L[/i][/color], which is a point of the conic[/*][/list][br]Point [color=#6aa84f][i]P[/i][/color], moving on the circle, creates the locus of points of [color=#38761D][i]L[/i][/color], that is:[br][br]- an [i]ellipse [/i]if [color=#9900ff][i]F[/i][sub]2[/sub][/color] is [i]inside [/i]the circle [i] distance between foci[/i] < [i]axis length[/i] → [i]e[/i] < 1[br][br]- an [i]hyperbola [/i]if [color=#9900ff][i]F[/i][sub]2[/sub][/color][color=#ff0000][sub] [/sub][/color]is [i]outside [/i]the circle [i] distance between foci[/i] > [i]axis length[/i] → [i]e[/i] > 1[br][br]- a [i]circle[/i] if [color=#9900ff][i]F[/i][sub]1[/sub] ≡[/color][color=#9900ff][i]F[/i][sub]2[/sub] [/color] [i] distance between foci[/i] = 0 → [i]e[/i] = 0[br][br]([i]e[/i] = [i]eccentricity[/i])
Explore the locus
After creating the locus, draw triangle [i]PLF[sub]2[/sub][/i] and answer the following questions:
What type of triangle is [i]PLF[sub]2[/sub][/i]?[br]Explain.
Write the canonical definition (as locus) of the conic that you see in the construction.
Use the properties of triangle [i]PLF[/i][sub]2[/sub] to show that the graph in the construction matches the canonical definition.
Matrices and Linear Transformations
Select an object to transform (point, line, circle, function, unit square), then select a transformation.[br][br]You can explore the transformations by dragging the initial objects that are not constrained by a definition. These objects are displayed in green.[br][br]The transformed objects are displayed in red, and cannot be dragged, since they are dependent on the initial objects.[br][br]You can apply some [color=#2980b9][i]predefined transformations[/i][/color]: [color=#2980b9][i]reflections[/i][/color] about the axes, [color=#2980b9][i]shears[/i][/color], [color=#2980b9][i]dilations [/i][/color]and [color=#2980b9][i]rotations[/i][/color].[br][br]Or you can select a [color=#2980b9][i]Custom[/i][/color] transformation, and define yourself the matrix of the transformation [math]T=\begin{bmatrix} t_{11} & t_{12} \\ t_{21} & t_{22} \end{bmatrix} [/math] using the displayed sliders.
How to Compute Transformed Points Using Matrices?
You can use matrices to compute the coordinates of transformed points from a given set, under a transformation represented by a matrix [math]T[/math].[br][br]If the coordinates of the given points are [math]\left(x_1,y_1\right),\left(x_2,y_2\right)[/math] and [math]\left(x_3,y_3\right)[/math], then the coordinate matrix is [math]M=\begin{bmatrix} x_{1} & x_{2} & x_{3} \\ y_{1} & y_{2} & y_{3} \end{bmatrix} [/math], and the coordinate matrix of the transformed points is obtained by [math]M'=T\cdot M[/math].
Explore...
Practice Zone
Consider the unit square in the app above. If you consider the coordinates of its vertices, listed anticlockwise starting from the origin, you can form the coordinate matrix of the vertices: [math]M=\begin{bmatrix} 0 & 1 & 1 & 0\\ 0 & 0 & 1& 1 \end{bmatrix} [/math].[br][br]Now select a [i]Custom[/i] transformation, and set the two elements of the main diagonal of the transformation matrix [math]T[/math] equal to 2, and the remaining elements equal to 0.[br]What transformation have you obtained?[br]Compare the areas of the unit square and the transformed square.[br][br]If you set the transformation matrix equal to [math]T=\begin{bmatrix} 3 & 0 \\ 0 & 2 \end{bmatrix} [/math], what would be the area of the transformed figure?[br][br]Verify your conjecture by computing the coordinates of the vertices of the transformed figure, and determining its area.[br]
Reflection Brings Reflections...
The list of transformations available in the app doesn't include [i]reflection about the origin[/i].[br]Can you obtain this transformation using one of the available options?[br]Explain your answer in detail.