Determine which pair of points has a positive slope.
A. (5, –4), (–2, 1)
B. (–10, –2), (6, 6)
C. (6, –10), (2, 10)
D. (5, –1), (–6, 6)
B. (–10, –2), (6, 6)
The ordered pairs (2, –11) and (5, –20) are solutions to which of the following equations?
A. y = 3x – 5
B. y = –3x + 5
C. y = 3x + 5
D. y = –3x – 5
B. y = -3x + 5
Which is a rule that describes the translation of a point form (4, –8) to (7, –10)?
A. (x, y) arrow(x + 3, y – 2)
B. (x, y) arrow(x + 3, y + 2)
C. (x, y) arrow(x – 3, y – 2)
D. (x, y) arrow(x – 3, y + 2)
A. (x, y) arrow (x + 3, y - 2)
∆ XYZ has vertices X(5, 6), Y(9, 12), and Z(12, 8). It is translated right 8 units and up 3 units. What are the coordinates of X’, Y’, and Z’?
A. X' (13, 9), Y' (15, 17), Z' (11, 20)
B. X' (–3, 3), Y' (1, 9), Z' (4, 5)
C. X' (8, 14), Y' (12, 20), Z' (23, 19)
D. X' (13, 9), Y' (17, 15), Z' (20, 11)
D. X' (13, 9), Y' (17, 15), Z' (20, 11)
The point M(x, y) is reflected over the x-axis. Write a translation rule to describe the original point and its reflection.
A. (x, y)arrow (x, y)
B. (x, y)arrow(–x, y)
C. (x, y)arrow(–x, –y)
D. (x, y)arrow(x, –y)
D. (x, y) arrow (x, -y)
Which table shows a proportional relationship?
A.
X 1 3 4 6
Y –11 –23 –29 –41
B.
X 1 3 4 6
Y 2 –10 –16 –28
C.
X 1 3 4 6
Y –6 –22 –30 –46
D.
X 1 3 4 6
Y –8 –24 –32 –48
D.
X | Y
--|--
1 | -8
3 | -24
4 | -32
6 | -48
All Y values are 8 times the X values, so they're proportional.
Which of the following is a graph of y = x2?
A. This is a graph of a parabola that passes through the points left parenthesis negative 2 comma 2 right parenthesis, left parenthesis 0 comma 0 right parenthesis, and left parenthesis 2 comma 2 right parenthesis.
B. This is a graph of a parabola that passes through the points left parenthesis negative 2 comma 4 right parenthesis, left parenthesis 0 comma 0 right parenthesis, and left parenthesis 2 comma 4 right parenthesis.
C. This is a graph of a parabola that passes through the points left parenthesis negative 2 comma 8 right parenthesis, left parenthesis 0 comma 0 right parenthesis, and left parenthesis 2 comma 8 right parenthesis.
D. This is a graph of a parabola that passes through the points left parenthesis negative 2 comma 6 right parenthesis, left parenthesis 0 comma 0 right parenthesis, and left parenthesis 2 comma 6 right parenthesis.
A. This is a graph of a parabola that passes through the points left parenthesis negative 2 comma 2 right parenthesis, left parenthesis 0 comma 0 right parenthesis, and left parenthesis 2 comma 2 right parenthesis.
Point A(4, 2) is translated according to the rule (x, y) right arrow (x + 1, y – 5) and then reflected across the y-axis.
a) In which quadrant of the coordinate plane is point A located?
b) What are the coordinates of translated point A’? In which quadrant of the coordinate plane is point A’ located?
c) What are the coordinates of reflected point A’’? In which quadrant of the coordinate plane is point A’’ located?
a) Point A(4, 2) is located in Quadrant I of the coordinate plane.
b) Using the given translation rule, we can find the coordinates of A’ as follows:
A' = (4+1, 2-5)
A' = (5, -3)
Point A’ is located in Quadrant IV of the coordinate plane.
c) To reflect A’ about the y-axis, we replace x with –x:
A'' = (-5, -3)
Point A’’ is located in Quadrant II of the coordinate plane.
Can the figure below tessellate a plane? Explain your answer.
A shape is formed by cutting out a smaller semicircle from a larger semicircle. The remaining closed figure resembling an arch is shown.
No, the given figure cannot tessellate a plane. To tessellate a plane, we need to have a shape that can completely cover a flat surface with no overlaps or gaps. However, the given figure has a curved boundary, which means it can never completely cover a flat surface without overlap or gaps. Therefore, it cannot be used to tessellate a plane.
Look at the shaded triangle in the drawing below. Describe in detail the transformations needed to create the rest of the square design. You can use a combination of translations, reflections, and rotations of the shaded triangle (Figure I) to create the square design.
A square is drawn on a coordinate plane with the center of the square on the origin. Two diagonals are drawn through the origin forming four congruent triangles on the inside of the square. The triangle that overlaps the positive y-axis is shaded.
To create the rest of the square design from the shaded triangle, we can use a combination of translations, reflections, and rotations. Here are the steps:
1. Reflect the shaded triangle about the y-axis. This creates a new triangle that is a mirror image of the shaded triangle, with the hypotenuse along the negative y-axis.
2. Translate the new triangle 1 unit to the right. This places the hypotenuse of the new triangle along the positive y-axis.
3. Rotate the new triangle 90 degrees counterclockwise about the origin. This places the hypotenuse of the new triangle along the positive x-axis, and the legs along the positive y-axis and positive x-axis.
4. Reflect the rotated triangle about the y-axis. This creates a new triangle that is a mirror image of the rotated triangle, with the hypotenuse along the negative x-axis.
5. Translate the new triangle 1 unit down. This places the hypotenuse of the new triangle along the negative x-axis.
6. Rotate the new triangle 180 degrees about the origin. This creates a new triangle that is a mirror image of the shaded triangle, with the hypotenuse along the positive x-axis.
7. Reflect the new triangle about the y-axis. This creates a new triangle that is a mirror image of the new triangle in step 1, with the hypotenuse along the positive y-axis.
8. Translate the new triangle 1 unit up. This places the hypotenuse of the new triangle along the top side of the square.
9. Repeat steps 1-8 for each of the other three shaded triangles in the square.
By following these steps, we can create a square design made up of 8 congruent shaded triangles that tessellate the plane.
The coordinates of an ordered pair have opposite signs. In which quadrant(s) must the ordered pair lie? Explain
If the coordinates of an ordered pair have opposite signs, then one coordinate must be positive and the other must be negative. Therefore, the ordered pair must lie in either Quadrant II or Quadrant IV.
In Quadrant II, the x-coordinate is negative and the y-coordinate is positive. In Quadrant IV, the x-coordinate is positive and the y-coordinate is negative. In both cases, the coordinates have opposite signs, which is the condition given in the problem.
For example, (-3, 4) is an ordered pair that has opposite signs. The x-coordinate is negative and the y-coordinate is positive, which means this ordered pair lies in Quadrant II. Similarly, (2, -6) is another ordered pair that has opposite signs, and it lies in Quadrant IV.