You are designing a rectangular playground. On your scale drawing, the vertices of the rectangle are (6, 3), (6, 5), and (8, 3). What are the coordinates of the fourth vertex?
a. (4, 5)
b. (8, 5)
c. (8, 1)
d. (5, 8)
b. (8, 5)
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Which ordered pair is not a solution of y = –4x – 9 ?
A. (–9, 27)
B. (3, –21)
C. (–6, 15)
D. (5, –28)
B. (3, –21)
Which is the graph of the linear equation y = –one-fourthx + 3?
A. A line is graphed on a coordinate plane. The line passes through the points just above left-parenthesis negative 4 comma 2 right-parenthesis and left-parenthesis 4 comma 4 right-parenthesis.
B. A line is graphed on a coordinate plane. The line passes through the points just above left-parenthesis negative 6 comma 6 right-parenthesis and left-parenthesis 3 comma 3 right-parenthesis.
C. A line is graphed on a coordinate plane. The line passes through the points just above left-parenthesis negative 2 comma 6 right-parenthesis and left-parenthesis 6 comma negative 2 right-parenthesis.
D. A line is graphed on a coordinate plane. The line passes through the points just above left-parenthesis negative 4 comma 4 right-parenthesis and left-parenthesis 4 comma 2 right-parenthesis.
C. A line is graphed on a coordinate plane. The line passes through the points just above left-parenthesis negative 2 comma 6 right-parenthesis and left-parenthesis 6 comma negative 2 right-parenthesis.
Find three solutions of the equation.
y = 2x – 6
A. (–2, –10), (1, –4), (2, –1)
B. (–2, –10), (1, –4), (0, –6)
C. (0, –6), (3, –2), (–2, –10)
D. (1, –4), (0, –6), (–1, –9)
C. (0, –6), (3, –2), (–2, –10) are three solutions of the equation y = 2x – 6
Which pair of points has a negative slope?
A. (–3, 17), (2, –8)
B. (–3, 13), (3, 17)
C. (4, 26), (–2, –10)
D. (4, 14), (–2, –4)
C. (4, 26), (–2, –10)
Use integer values of x from –3 to 3 to graph the equation.
y = –|x|
A. graph A Its vertex is at left parenthesis 0 comma 0 right parenthesis. It passes through left parenthesis negative 5 comma 5 right parenthesis and through left parenthesis 5 comma 5 right parenthesis.
B. graph BIts vertex is at left parenthesis 0 comma 0 right parenthesis. It passes through left parenthesis negative 5 comma negative 5 right parenthesis and through left parenthesis 5 comma negative 5 right parenthesis.
C. graph CIts vertex is at left parenthesis 0 comma 0 right parenthesis. It passes through left parenthesis negative 2 comma negative 4 right parenthesis and through left parenthesis 2 comma negative 4 right parenthesis.
D. graph D
B. graph B Its vertex is at left parenthesis 0 comma 0 right parenthesis. It passes through left parenthesis negative 5 comma negative 5 right parenthesis and through left parenthesis 5 comma negative 5 right parenthesis.
The ordered pairs (2, –21) and (5, –45) are solutions to which of the following equations?
A. y = –8x – 5
B. y = –8x + 5
C. y = 8x – 5
D. y = 8x+ 5
A. y = –8x – 5
Which is a rule that describes the translation of a point from (–5, 4) to (–1, 2)?
A. (x, y) arrow(x – 4, y – 2)
B. (x, y) arrow(x + 4, y – 2)
C. (x, y) arrow(x + 4, y + 2)
D. (x, y) arrow(x – 4, y + 2)
D. (x, y) arrow(x – 4, y + 2)
ΔPQR has vertices P(5, –1), Q(0, 8), and R(7, 5). It is translated right 3 units and up 6 units. Find the coordinates of P', Q', and R'.
A. P'(8, 5), Q'(3, 14), R'(10, 11)
B. P'(2, 5), Q'(–3, 14), R'(4, 11)
C. P'(8, –7), Q'(3, 2), R'(10, –1)
D. P'(2, –7), Q'(–3, 2), R'(4, –1)
A. P'(8, 5), Q'(3, 14), R'(10, 11)
The point Q left parenthesis 9 comma negative 1 right parenthesis is reflected across the y-axis. Use arrow notation to describe the original point and its reflection.
A. Q left parenthesis 9 comma negative 1 right parenthesis right-arrow Q prime left parenthesis 9 comma 1 right parenthesis
B. Q left parenthesis 9 comma negative 1 right parenthesis right-arrow Q prime left parenthesis negative 9 comma 1 right parenthesis
C. Q left parenthesis 9 comma negative 1 right parenthesis right-arrow Q prime left parenthesis negative 9 comma negative 1 right parenthesis
D. Q left parenthesis 9 comma negative 1 right parenthesis right-arrow Q prime left parenthesis 9 comma negative 1 right parenthesis
B. Q left parenthesis 9 comma negative 1 right parenthesis right-arrow Q prime left parenthesis negative 9 comma 1 right parenthesis
The coordinates of three vertices of a rectangle are shown on the grid.
A coordinate graph with x and y axes labeled from negative 5 to 5 is shown with a black dots at the points negative 1 comma negative 2, negative 1 comma 4, and 3 comma 4.
Which could be the coordinates of the fourth vertex?
A. (3, –2)
B. (3, –1)
C. (–2, 3)
D. (4, –2)
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D. (4, –2)
Draw the image of the figure after a rotation of 180º around O.
An isosceles trapezoid LMNO is shown with segment LM as the shorter base and ON and the longer base.
A. Two trapezoids are graphed on a coordinate grid.Trapezoid LMNO is graphed on a coordinate grid. Trapezoid L prime M prime N prime O prime shares a single point, O, with LMNO and appears to have exactly the same shape and proportions. However, L prime M prime N prime O prime shares point O/O prime with LMNO and is upside down.
B. Two trapezoids are graphed on a coordinate grid.Trapezoid LMNO is graphed on a coordinate grid. Trapezoid L prime M prime N prime O prime shares a single point, O, with LMNO and appears to have exactly the same shape and proportions. However, L prime M prime N prime O prime is on its side, with the shorter parallel side of the trapezoid pointing right.
C. Two trapezoids are graphed on a coordinate grid.Trapezoid LMNO is graphed on a coordinate grid. Trapezoid L prime M prime N prime O prime shares a single point, O, with LMNO and appears to have exactly the same shape and proportions. L prime M prime N prime O prime is immediately to the right of LMNO.
D. Two trapezoids are graphed on a coordinate grid.Trapezoid LMNO is graphed on a coordinate grid. Trapezoid L prime M prime N prime O prime shares a single point, O, with LMNO and appears to have exactly the same shape and proportions. L prime M prime N prime O prime is immediately to the right of LMNO and on its side, with the shorter parallel side on the left.
A. Two trapezoids are graphed on a coordinate grid.Trapezoid LMNO is graphed on a coordinate grid. Trapezoid L prime M prime N prime O prime shares a single point, O, with LMNO and appears to have exactly the same shape and proportions. However, L prime M prime N prime O prime shares point O/O prime with LMNO and is upside down.
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 1 3 4 6
Y –8 –24 –32 –48
In this table, Y is always 8 times lesser than X, indicating proportional relationship as Y varies directly with X.
Identify the unit rate shown in the graph.
distance walked graph
A. one-third mile per hour
B. 3 hours per mile
C. 3 miles per hour
D. 4 miles per hour
D. 4 miles per hour
The graph below shows Cara's earnings according to the number of hours she works.
A coordinate graph is shown.
The graph has an x axis labeled 'Number of Hours Worked' with coordinates labeled from 0 to 40 in multiples of 5. The y axis is labeled 'Number of Dollars Earned' and is marked from 0 to 400 in multiples of 100. A red ray on the graph begins at 0 comma 0 and extends up and to the right through the point 35 comma 350.
What does the slope of the line represent?
A. For every $50 Cara earns, she works 10 hours.
B. For every $50 Cara earns, she works 3 hours.
C. For each hour Cara works, she earns $10.
D. For each hour Cara works, she earns $15.
C. For each hour Cara works, she earns $10.
Slope of the line represents the rate of change of earnings per hour worked and the slope of the given line is (350 - 0) / (35 - 0) = 10. It means that Cara earns $10 for every hour she works.
Which of the following is a graph of y = one-halfx2?
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.
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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.
Which equation is true for all x- and y- values in the table below?
x y
1 7
2 14
3 21
4 28
A. y = x + 7
B. y = x – 7
C. y = start fraction x over 7 end fraction
D. y = 7x
D. y = 7x.
The values of y are seven times the corresponding values of x.
Short Answer
Note: For questions 19 and 20, your teacher will grade your response to ensure you receive proper credit for your answer.
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 is located in quadrant I of the coordinate plane.
b) The translation rule (x, y) right arrow (x + 1, y – 5) gives the new coordinates of A as (5, -3), which is located in quadrant IV of the coordinate plane.
c) Reflecting (5, -3) across the y-axis results in the point (-5, -3), which is located in quadrant III of the coordinate plane. Therefore, the coordinates of reflected point A’’ are (-5, -3).
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 figure cannot tessellate a plane.
In order for a figure to tessellate a plane, it must be able to fit together with copies of itself so that there are no gaps or overlaps. The given figure is not capable of this because of the curved boundary of the arch-shaped figure. Therefore, it cannot tessellate a plane.
Essay
Note: For questions 21–22, your teacher will grade your response to ensure you receive proper credit for your answer.
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 using the shaded triangle as a template, we can apply the following transformations:
1. Translation: Move the shaded triangle two units to the right to create a new triangle (Figure II).
2. Reflection: Reflect the new triangle across the y-axis to create another triangle (Figure III).
3. Rotation: Rotate the new triangle 90 degrees clockwise about the origin to create a new triangle (Figure IV).
4. Translation: Move the fourth triangle up two units to create a new triangle (Figure V).
5. Reflection: Reflect the new triangle across the x-axis to create another triangle (Figure VI).
6. Rotation: Rotate the new triangle 90 degrees clockwise about the origin to create a new triangle (Figure VII).
7. Translation: Move the seventh triangle two units to the left to create a new triangle (Figure VIII).
8. Reflection: Reflect the new triangle across the y-axis to create another triangle (Figure IX).
Finally, we can combine the 9 triangles to form the complete square design.
In summary, the transformations needed to create the rest of the square design are translation, reflection, and rotation of the shaded triangle. These transformations are applied in a specific sequence to create the individual triangles that form the design, which can be combined to create the entire square design.
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 the ordered pair must lie in either quadrant II or quadrant IV.
This is because in quadrant II, both the x-coordinate and the y-coordinate are negative, while in quadrant IV, both the x-coordinate and the y-coordinate are positive. Therefore, in these two quadrants, the x-coordinate and y-coordinate will always have opposite signs.
In contrast, in quadrant I, both the x-coordinate and y-coordinate are positive, while in quadrant III, both the x-coordinate and y-coordinate are negative. Thus, the coordinates of an ordered pair in these two quadrants will always have the same sign.
Therefore, if the coordinates of an ordered pair have opposite signs, we can conclude that the ordered pair lies in either quadrant II or quadrant IV of the coordinate plane.