Exercises for Coordinate Systems and Graphs

Exercise 1: Calculating the Length of a Line Segment Using the Pythagorean Theorem
Task 1:
Calculate the distance between the points A = (2; 3) and C = (4; 6).
Hint: Construct a right triangle ABC with A = (2; 3), B = (4; 3), and C = (4; 6).

Task 2:
Calculate the distance between A = (1; 5) and C = (6; 3).

Task 3:
Provide the coordinates of the vertices of a triangle ABC whose hypotenuse is equal to

1. 5
2. sqrt(3)
3. sqrt(13)

(Note: Multiple answers are possible for each sub-question.)

Task 4:
Explain whether the Pythagorean theorem equation can be applied to the following triangles ABC.

1. A = (-1; 4), B = (2; 4), and C = (2; 6)
2. A = (-2; 3), B = (1; 4), and C = (1; 6)
3. A = (-3; -1), B = (0; -1), and C = (0; 5)

Exercise 2: Area and Perimeter of Plane Figures
The coordinate system can also be used for exercises involving the calculation of area and perimeter of plane figures.

1. Construct a square ABCD with an area of 16. The square is positioned in all four quadrants. Write down the coordinates of A, B, C, and D.
2. Construct a rhombus ABCD with an area of 25, where point A is located at (-2; 1).
3. Construct a square ABCD with point A at (-3; -2), point B at (1; -2), point C at (1; ..), and point D at (..; ..). Write down the coordinates of point C and D.

Exercise 3: Identifying the Quadrilateral in a Coordinate System
Using the coordinates of the vertices of a quadrilateral, one can determine the type of quadrilateral, such as a square or rhombus. Note: If a braille reader is very good in using coordinates, you may try to practice with description of images (instead of using tactile images).

Identify the type of quadrilateral for

1. ABCD, where A = (1; 1), B = (1; 5), C = (3; 5), and D = (3; 1)
2. JKLM, where J = (1; 3), K = (5; 1), L = (8; 1), and M = (4; 3)
3. PQRS, where P = (3; 0), Q = (6; 2), R = (5; 2), and S = (2; 0)
4. WXYZ, where W = (1; 1), X = (0; 3), Y = (4; 1), and Z = (3; 1)

Exercise 4: A Coordinate System With Adjustable Axes
This exercise utilizes coordinate system with adjustable axes. Instead of using a coordinate system with adjustable axes, sighted students may opt to sketch a coordinate system.

Preparation teacher: Make two tactiles drawing on a blank sheet of paper: one paper with only one dot, one paper with two dots. You may use a drawing with or without a grid.

Task 1:
Use paper with one dot and

1. properly position the axes of the coordinate system to create A = (1; 1)
2. properly position the axes of the coordinate system to create B = (-1; 4)

Use paper with two dots and

3. properly position the axes of the coordinate system to create A = (1; 1) and B = (-1; -1)
4. properly position the axes of the coordinate system to create C = (0; 0) and D = (-1; -1)

Answer:
Task 2:
Preparation for teacher: Make a tactile drawing of a parabola and click it on the board. Assume that the shape of the parabola is the same as the one with equation y = x^2. Furthermore, assume that the axes use the same scale. If you want to use a grid, use a grid that fits with the scale on the axes.

1. Make a parabola of the function y = x^2.
2. What translations do you need to apply to transform the graph of the parabola y = x^2 into the graph of the parabola y = (x ‐ 1)^2?
3. Make a parabola with the vertex located on (3; 1) What is the equation of the parabola?
4. Make a mountain parabola with the vertex on (‐2; 4). What is the equation?
5. Make a parabola with the vertex in the third quadrant. One of the zeros is located at (‐3; 0). What is the equation?

Exercise 5: A Stripped Coordinate System
Note: If a braille reader is able to make a mental picture of the coordinate system, you may try to practice with the description of a coordinate system instead of using a tactile drawing.

Task 1:
Assume that both axes have the same scale.

1. Is the point in figure 1 located at (3; ‐6) or (‐6; 3)?
2. Using a description: a point is located in the fourth quadrant. Is this point located at (3; ‐6) or (‐6; 3)?
3. Line l, in figure 2, passes through (1; 2) and (4; 2). What is the equation of line l?
4. Line m is parallel to line l and passes through (1; ‐1). What is the equation of line m?

Task 2:
The graph depicted in figure 3 is represented by the expression y = ‐2x + 6. Determine the step size along the axes.



Task 3:
Assume that both axes in each coordinate system have the same scale.







1. What is the correct expression for graph 1: y = 2x ‐ 2 or y = 2x + 2? Explain your answer.
2. What is the correct expression for graph 2: y = (x ‐ 1)^2 or y = x^2 + 1? Explain your answer.
3. What is the correct expression for graph 3: y = ‐(x + 1)^2 or y = ‐(x ‐ 1)^2? Explain your answer.

Task 4:
The figure below displays two parabolas. The expression for the parabola with the dotted line is y = x^2. What is the expression for the red parabola y = 2x^2 or y = 1/2 x^2? Explain your answer.

Exercise 6: Transformation of a Parabola
Introduction On the TouchingMaths+ website, you can find Excel worksheets that the student can use to complete this exercise. This workbook contains worksheets about the quadratic formula, the vertex form and the factored form.

Task 1:
Determine values for a, b, and c such that the parabola

1. has the same shape as the parabola of y = 2x^2 + 3x + 4
2. has a minimum for x = ‐1/2
3. has one zero point

Task 2:

1. Determine values for a, b, and c such that the parabola has one zero point.
2. Determine values for a, b, and c such that the tangent to the vertex equals y = 3.

Vertex form
The vertex form is expressed with y = c(x – a)^2 + b.
Task 1:
Determine the values for a, b, and c that result in a parabola

1. with the same shape as the parabola of y = ‐3(x ‐ 2)^2 + 4
2. with a vertex that is located at (2; 6)

Task 2:
Find values for a, b, and c that yield a parabola with zeros at x = ‐3 and x = 5.

Factored Form
The factored form is represented as y = c(x ‐ a)(x ‐ b).
Task 1:

1. Determine values for a, b, and c such that the parabola of y = 2(x ‐ 2)(x + 3) is translated horizontally by 2 units to the left.
2. Determine values for a, b, and c such that the parabola has zeros for x = ‐3 and x = 5 and has the same shape as y = ‐3x^2.

Exercise 7: Body Graphs
Body graphs offer students a means to visually represent mathematical functions through physical movements. These figures depict students imitating the graphs of various functions using their arms. For example, one figure shows a puppet raising both hands with bent elbows, symbolizing the graph of y = x^2, while moving the arms closer together represents y = 2x^2. It is important to complete the graph on the braille reader’s back. Otherwise, the graph will appear as if it consists of two arms that are not connected. This exercise promotes inclusivity in education by enhancing students’ – meaning all students’ – understanding of the relationship between graphs and the corresponding functions.