Grade 8 Math Practice Test: Transformations and Volume of Cylinders, Cones and Spheres

Left 5 subtracts 5 from the x-coordinate (3 − 5 = −2), and up 4 adds 4 to the y-coordinate (−2 + 4 = 2), so P′ is (−2, 2). (8, 2) moves right instead of left, and (−2, −6) moves down instead of up.

A reflection across the x-axis keeps the x-coordinate and changes the sign of the y-coordinate: (x, y) → (x, −y), so the image is (4, −7). (−4, 7) is a reflection across the y-axis.

A 90° clockwise rotation about the origin maps (x, y) to (y, −x), so (2, 5) becomes (5, −2). (−5, 2) is the result of a 90° counterclockwise rotation, and (−2, −5) is a 180° rotation.

A 180° rotation about the origin changes the sign of both coordinates: (x, y) → (−x, −y), so (−3, 4) becomes (3, −4). (−3, −4) and (3, 4) change only one sign, which is a reflection.

A dilation centered at the origin multiplies both coordinates by the scale factor: (2 × 3, −1 × 3) = (6, −3). (5, 2) adds 3 instead of multiplying, and (6, −1) multiplies only the x-coordinate.

Rotations, reflections and translations keep all lengths and angles, so their images are congruent. A dilation with a scale factor of 2 doubles every length, so the image is similar to the original but not congruent.

Reflecting across the y-axis changes the sign of x: (−5, 2) → (5, 2). Moving 3 units down subtracts 3 from y: (5, 2 − 3) = (5, −1). (−5, −1) skips the reflection, and (5, 5) moves up instead of down.

The scale factor is image length ÷ original length: 9 ÷ 6 = 12 ÷ 8 = 15 ÷ 10 = 1.5. 3 is the difference 9 − 6, which is not how a dilation works, and 23 divides in the wrong order.

Each vertex of P, (1, 1), (4, 1) and (1, 3), matches a vertex of Q with the opposite x-coordinate: (−1, 1), (−4, 1) and (−1, 3). That is a reflection across the y-axis. A translation 5 units left would put the right angle at (−4, 1), so the triangle would point the wrong way.

The radius is 6 ÷ 2 = 3 cm. V = πr²h = 3.14 × 3² × 10 = 3.14 × 90 = 282.6 cubic centimeters. 1,130.4 uses the diameter as the radius, and 94.2 forgets to square the radius.

V = 13πr²h = 13 × π × 16 × 9 = 48π cubic centimeters. 144π is the volume of a cylinder with the same base and height (it forgets the 13), and 12π forgets to square the radius.

V = 43πr³ = 43 × π × 27 = 36π cubic inches. 12π squares the radius instead of cubing it, and 108π forgets to divide by 3.

The radius is 6 cm. V = 43 × 3.14 × 6³ = 43 × 3.14 × 216 = 904.32 cubic centimeters. 7,234.56 uses the diameter as the radius, 678.24 leaves out the 43, and 452.16 is the surface area 4πr².

V = πr²h, so 200π = π × 25 × h and h = 200 ÷ 25 = 8 feet. 40 divides by the radius instead of the radius squared, and 2 squares the diameter (10² = 100) instead of the radius.

Cone: 13 × π × 9 × 8 = 24π. Hemisphere: half of 43 × π × 27 = 18π. Total: 24π + 18π = 42π cubic centimeters. 60π uses a whole sphere (36π) on top, and 90π forgets the 13 in the cone formula (72π + 18π).