Coordinate Geometry and Graphs Lesson Test

Answer: D

Correct answer: Quadrant II. The x-coordinate -4 means 4 units left and the y-coordinate 7 means 7 units up. Left and up is the upper-left region, Quadrant II, with signs (-, +).

• A: Quadrant I needs both coordinates positive, but x = -4 is negative.
• B: Quadrant IV has the pattern (+, -); this point has the opposite pattern.
• C: Quadrant III needs both coordinates negative, but y = 7 is positive.
• D (correct): x is negative and y is positive, the (-, +) sign pattern of Quadrant II.

Answer: A

Correct answer: (5, 3). Across the x-axis, (x, y) becomes (x, -y). So (5, -3) becomes (5, 3).

• A (correct): Reflecting across the x-axis keeps x and changes the sign of y.
• B: This changes the sign of x, which is a reflection across the y-axis.
• C: The coordinates were swapped instead of changing the sign of y.
• D: Both signs were changed; that is a reflection through the origin, not across the x-axis.

Answer: D

Correct answer: 0. Every point on

has the same y-value, so the line is horizontal. Its rise is 0 for any run, so the slope is 0.

• A: Undefined slope belongs to vertical lines such as x = -2, not horizontal ones.
• B: A slope of 1 would mean the line climbs one unit for each unit to the right, but this line never climbs.
• C: The -2 is the height where the line sits, not its steepness.
• D (correct): A line of the form y = constant is horizontal, so its rise is 0 and its slope is 0.

Answer: B

Correct answer: (7, 3). Average each coordinate separately:

The midpoint is (7, 3).

• A: These are the differences of the coordinates, which measure change, not the halfway point.
• B (correct): Averaging gives (4 + 10)/2 = 7 and (-2 + 8)/2 = 3.
• C: These are the sums of the coordinates; they still need to be divided by 2.
• D: These are half the differences of the coordinates, not the averages.

Answer: A

Correct answer: (0, 7). The equation is already in the form y = mx + b with

and b = 7. Setting x = 0 gives y = 7, so the line crosses the y-axis at (0, 7).

• A (correct): In y = mx + b, b = 7 is the y-intercept, so the line crosses at (0, 7).
• B: The coordinates are reversed; a point on the y-axis must have x = 0.
• C: The sign of the constant was changed; the intercept is +7.
• D: -3 is the slope (the coefficient of x), not the y-intercept.

Answer: B

Correct answer: (2, 2). Moving right adds to x and moving down subtracts from y. The new point is

• A: Moving right adds to x, but here 4 was subtracted.
• B (correct): Right adds 4 to x and down subtracts 3 from y: (-2 + 4, 5 - 3) = (2, 2).
• C: Both moves were applied in the wrong direction (left and up).
• D: Moving down subtracts from y, but here 3 was added.

Answer: C

Correct answer: 10 units. The horizontal change is

and the vertical change is

which has length 8.

• A: This is the sum of the squared changes; the square root was never taken.
• B: This adds the horizontal and vertical changes instead of using the Pythagorean theorem.
• C (correct): The changes are 6 and 8, a 6-8-10 right triangle, so the distance is 10.
• D: This subtracts the two changes, which does not give a straight-line distance.

Answer: A

Correct answer: -2. Use change in y over change in x, subtracting in the same order.

• A (correct): The change in y is -7 - 5 = -12 and the change in x is 4 - (-2) = 6, so the slope is -12/6 = -2.
• B: The sign was lost; y decreases as x increases, so the slope must be negative.
• C: This is the change in x over the change in y, which is upside down.
• D: The fraction is upside down and the sign is wrong.

Answer: B

Correct answer: 2/3. Subtract 4x from both sides:

Divide by -6:

x - 2. The slope is 2/3.

• A: This is the reciprocal of the slope.
• B (correct): Solving for y gives y = (2/3)x - 2, so the slope is 2/3.
• C: The coefficient of x in standard form is not the slope until the equation is solved for y.
• D: This treats the y-coefficient as +6 instead of -6; dividing by -6 makes the slope positive.

Answer: B

Correct answer: (10, 0). At the x-intercept y = 0, so

and x = 10. The line crosses the x-axis at (10, 0).

• A: This is the y-intercept, found by setting x = 0.
• B (correct): Setting y = 0 gives 3x = 30, so x = 10.
• C: This divides 30 by the coefficient of y instead of the coefficient of x.
• D: The value 10 is right, but an x-intercept lies on the x-axis, so y must be 0.

Answer: D

Correct answer: Dashed line; shade the side away from the origin. Because > does not include equality, the boundary

is drawn dashed. Testing (0, 0) gives

or 0 > 3, which is false, so shade the side that does not contain the origin.

• A: The boundary should be dashed for >, and the origin fails the test 0 > 3.
• B: A solid line is used only for the symbols that include equality.
• C: The dashed line is right, but (0, 0) fails the test, so the origin side is not shaded.
• D (correct): The symbol > calls for a dashed line, and testing (0, 0) gives 0 > 3, which is false.

Answer: D

Correct answer: 5/2. Solve for y:

x + 2 and the slope is -2/5. A perpendicular slope is the negative reciprocal: 5/2, since (-2/5)

• A: This is the slope of the given line itself, which gives a parallel line.
• B: This changes the sign but forgets to take the reciprocal.
• C: This flips the fraction but forgets to change the sign.
• D (correct): The given slope is -2/5, and its negative reciprocal is 5/2.

Answer: A

Correct answer: (3, 11). Set the right sides equal:

Add x and subtract 2: 4x = 12, so x = 3. Then

check with the other line:

The lines meet at (3, 11).

• A (correct): Setting 3x + 2 = -x + 14 gives 4x = 12, so x = 3 and y = 3(3) + 2 = 11.
• B: Moving -x to the left side should give 4x, not 2x; the sign slip leads to x = 6.
• C: The x- and y-values are swapped.
• D: This ignores the -x term when solving, giving 3x = 12 and x = 4.

Answer: C

Correct answer: (8, -5). Double the midpoint and subtract the known endpoint, one coordinate at a time.

Check: the average of (-4, 3) and (8, -5) is (2, -1).

• A: This adds P to 2M instead of subtracting it.
• B: This is the midpoint of P and M, not the far endpoint.
• C (correct): Q = 2M - P: x = 2(2) - (-4) = 8 and y = 2(-1) - 3 = -5.
• D: This is M - P, the step from P to M, but it was not added on to M.

Answer: D

Correct answer:

Use point-slope form:

Distribute:

Add 1:

Check:

• A: This distributes -2(x - 3) as -2x - 6 instead of -2x + 6.
• B: This uses y + 1 instead of y - 1, so the 1 is subtracted instead of added.
• C: This uses the point's y-value as the y-intercept, but the point is not on the y-axis.
• D (correct): Point-slope form gives y - 1 = -2(x - 3) = -2x + 6, so y = -2x + 7.

Answer: D

Correct answer:

Solve for y:

and the slope is -3. Parallel lines share a slope, so

is parallel.

• A: This uses the coefficient of x from standard form as the slope.
• B: A slope of 1/3 is the negative reciprocal of -3, which makes the line perpendicular, not parallel.
• C: The size is right but the sign is wrong; the given line falls, so its slope is negative.
• D (correct): Solving 6x + 2y = 5 for y gives slope -3, and this line also has slope -3 with a different intercept.

Answer: A

Correct answer: 48 square units. The width is the difference of x-values:

The height is the difference of y-values:

Area = 8 times 6 = 48 square units.

• A (correct): Width 3 - (-5) = 8 and height 5 - (-1) = 6 give an area of 8 times 6 = 48.
• B: This takes half of width times height, which is the area of a triangle, not a rectangle.
• C: This adds the width and height instead of multiplying them.
• D: This is the perimeter, 2(8 + 6), not the area.

Answer: C

Correct answer: No solution. Solve the second equation for y:

Both lines have slope 2 but different y-intercepts (3 and -3), so they are parallel and never meet. The system has no solution.

• A: One solution needs different slopes, but both lines have slope 2.
• B: Two distinct straight lines can never cross in exactly two points.
• C (correct): The second equation becomes y = 2x - 3, which has the same slope but a different intercept, so the lines are parallel.
• D: That happens only when both equations describe the same line, but the intercepts 3 and -3 differ.

Answer: B

Correct answer: (-9, 2). Reflecting across the y-axis changes the sign of x: (4, -7) becomes (-4, -7). Moving 5 left subtracts 5 from x and moving 9 up adds 9 to y:

• A: This reflects across the x-axis, giving (4, 7), before applying the translation.
• B (correct): Reflecting gives (-4, -7); then (-4 - 5, -7 + 9) = (-9, 2).
• C: Moving left subtracts from x, but here 5 was added to -4.
• D: Moving up adds to y, but here 9 was subtracted from -7.

Answer: B

Correct answer: (4, 11). A point is on the circle when its distance from the center equals the radius. For (4, 11), the changes are

The other points give sums of squares of 52, 68, and 200, none equal to 100.

• A: The changes are 10 and 10, so the distance is $\sqrt{200}\approx 14.1$ and the point is outside.
• B (correct): The changes from the center are 6 and 8, and $\sqrt{36+64}=10$, which equals the radius.
• C: The changes 8 and 2 add to 10, but the distance is $\sqrt{68}\approx 8.2$, so the point is inside.
• D: The changes 6 and 4 add to 10, but the distance is $\sqrt{52}\approx 7.2$, so the point is inside.

Answer: C

Correct answer: 40 units. The horizontal leg is

and the vertical leg is

These fit the 8-15-17 pattern, so the hypotenuse is 17. Perimeter =

units.

• A: This is the area, (1/2)(8)(15), not the perimeter.
• B: This adds only the two legs and leaves out the hypotenuse.
• C (correct): The legs are 8 and 15 and the hypotenuse is 17, so the perimeter is 8 + 15 + 17 = 40.
• D: This multiplies the legs, which is neither the perimeter nor the area.

Answer: C

Correct answer: (4.5, 0). Slope:

Substitute (-3, 10):

so b = 6 and

x + 6.

Set y = 0: (4/3)x = 6, so x = 6 times

The x-intercept is (4.5, 0).

• A: This is the y-intercept, where the line crosses the y-axis instead.
• B: The sign was lost when solving; with a negative slope and positive y-intercept, the line crosses the x-axis to the right of the origin.
• C (correct): The line is y = (-4/3)x + 6; setting y = 0 gives (4/3)x = 6, so x = 4.5.
• D: This multiplies 6 by 4/3 instead of dividing 6 by 4/3.

Answer: A

Correct answer:

·x + 7. Solve

for y:

x - 2, so the slope is 3/4 and the perpendicular slope is -4/3.

Point-slope form:

x + 8, so

x + 7. Check: (-4/3)

• A (correct): The given slope is 3/4, the perpendicular slope is -4/3, and y + 1 = (-4/3)(x - 6) simplifies to this line.
• B: This uses y - 1 instead of y + 1 for the point (6, -1), so the 1 is added instead of subtracted.
• C: This keeps the original slope 3/4, which gives a parallel line instead of a perpendicular one.
• D: This takes the reciprocal but forgets to make the perpendicular slope negative.

Answer: C

Correct answer: (2, 5). Reflecting across y = x swaps the coordinates: (-5, 2) becomes (2, -5). Reflecting across the x-axis keeps x and changes the sign of y: (2, -5) becomes (2, 5).

• A: This is only the first reflection; the x-axis reflection was skipped.
• B: The reflections were done in the wrong order: x-axis first gives (-5, -2), then swapping gives (-2, -5).
• C (correct): Reflecting across y = x swaps to (2, -5); reflecting across the x-axis then changes the sign of y to give (2, 5).
• D: This is only the x-axis reflection; the swap across y = x was skipped.

Answer: B

Correct answer: (-1, -3). Average each coordinate separately:

The midpoint is (-1, -3).

• A: This halves the differences of the coordinates, which gives half the run and rise, not the halfway point.
• B (correct): Averaging the x-values gives -1 and averaging the y-values gives -3.
• C: This adds the coordinates but forgets to divide each sum by 2.
• D: This has the right numbers in the wrong order; x must be listed first.

Answer: D

Correct answer: On the y-axis. The first number, 0, means do not move left or right. The second, -5, means move 5 units down. The point lies on the y-axis and belongs to no quadrant.

• A: Quadrant III needs both coordinates negative; a zero coordinate puts a point on an axis, not in a quadrant.
• B: Quadrant IV needs a positive x-coordinate, but here x is 0.
• C: Points on the x-axis have y equal to 0; this point has x equal to 0 instead.
• D (correct): An x-coordinate of 0 means no movement left or right, so the point sits on the y-axis, 5 units below the origin.

Answer: C

Correct answer: 13 units. Horizontal change:

Vertical change:

• A: This adds the horizontal and vertical changes; the distance is the hypotenuse, not their sum.
• B: This is the distance squared; the square root must still be taken.
• C (correct): The changes are 5 and 12, a 5-12-13 right triangle, so the distance is 13.
• D: This finds the x-change as 2 - 3 = -1, dropping the negative sign of -3; the x-change is 5.

Answer: C

Correct answer: -3/4. Subtract in the same order on top and bottom.

A quick sketch agrees: the line falls from left to right.

• A: This subtracts in one order on top and the opposite order on the bottom, losing the negative sign.
• B: This divides the run by the rise; slope is rise over run.
• C (correct): The rise is 2 - 8 = -6 and the run is 5 - (-3) = 8, so the slope is -6/8, or -3/4.
• D: This adds the coordinates instead of subtracting them.

Answer: A

Correct answer: (-4, 0). On the x-axis, y = 0. Solve

so the line crosses at (-4, 0).

• A (correct): Setting y = 0 gives 3x = -12, so x = -4.
• B: This is the y-intercept, where x equals 0, not where the line meets the x-axis.
• C: This loses the negative sign when moving 12 to the other side of the equation.
• D: This finds the correct value -4 but puts it in the y position; x-intercepts have the form (x, 0).

Answer: A

Correct answer: (-6, 2). Across the y-axis, (x, y) becomes (-x, y). So (6, 2) becomes (-6, 2), the same height but on the other side of the y-axis.

• A (correct): Reflecting across the y-axis changes the sign of x and keeps y.
• B: This changes the sign of y, which is a reflection across the x-axis.
• C: This changes both signs, which is the match through the origin, not a single reflection.
• D: This swaps the coordinates, which is a reflection across the line y = x.

Answer: C

Correct answer: (0, -7). On the y-axis, x = 0. Then

The line crosses at (0, -7).

• A: This drops the negative sign on the y-term when dividing by -2.
• B: This is the x-intercept, found by setting y = 0 instead of x = 0.
• C (correct): Setting x = 0 gives -2y = 14, so y = -7.
• D: This reads the constant 14 as the intercept without dividing by the y-coefficient.

Answer: A

Correct answer:

·x + 5. Parallel lines have equal slopes and different y-intercepts. Keep

and use b = 5.

• A (correct): Parallel lines share the slope 2/3, and the new intercept is 5.
• B: This uses the negative reciprocal slope, which makes the roads perpendicular, not parallel.
• C: This flips the fraction; parallel lines keep exactly the same slope.
• D: This changes the sign of the slope, so the new road would fall while the old one rises.

Answer: A

Correct answer:

·x + 9/2. Midpoint:

Slope of AB:

so the perpendicular slope is -3/2.

Substitute (1, 3):

• A (correct): The midpoint is (1, 3) and the perpendicular slope is -3/2; then 3 = -3/2 + b gives b = 9/2.
• B: This passes through the midpoint but uses the segment's own slope, 2/3, so it lies along AB instead of crossing it at a right angle.
• C: This flips the slope to 3/2 but forgets to make it negative, so the product of slopes is 1, not -1.
• D: This has the right slope but is drawn through endpoint A instead of the midpoint.

Answer: D

Correct answer:

Slope first:

Then substitute (2, 1) into

The line is

• A: This has the right slope but uses the y-value of (2, 1) as the intercept, though that point is not on the y-axis.
• B: This divides the change in x by the change in y, giving the reciprocal of the slope.
• C: This gives the slope the wrong sign; the y-values rise as x increases, so the slope is positive.
• D (correct): The slope is 8/4 = 2, and 1 = 2(2) + b gives b = -3; (6, 9) checks since 12 - 3 = 9.

Answer: D

Correct answer: 12 units. Both points have y = 4, so they lie on a horizontal line. The distance is the difference of the x-values:

units.

• A: This subtracts 7 from 5, dropping the negative sign of -7.
• B: This adds the two y-values, but the points lie on the same horizontal line, so y does not change.
• C: This subtracts in the opposite order and keeps the sign, but a distance is never negative.
• D (correct): The points share y = 4, so subtract the x-values: 5 - (-7) = 12.

Answer: A

Correct answer: (4, 3). Add the two equations so y cancels: 3x = 12, so x = 4. Substitute into

so y = 3.

Check in the first equation:

The lines meet at (4, 3).

• A (correct): Adding the equations gives 3x = 12, so x = 4, and then 4 - y = 1 gives y = 3.
• B: This has the correct numbers in swapped order; (3, 4) fails both equations.
• C: This finds x = 4 but then solves x - y = 1 as y = x + 1; the correct step is y = x - 1.
• D: This point lies on 2x + y = 11 but not on x - y = 1, since 2 - 7 is -5.

Answer: A

Correct answer: 169π square meters. The radius is the distance from (-1, 2) to (4, 14). The changes are 5 and 12, a 5-12-13 triangle, so r = 13.

• A (correct): The radius is the distance from the center to the edge point, 13, and pi times 13 squared is 169 pi.
• B: This multiplies pi by the radius without squaring it.
• C: This is 2 pi r, the circumference, which is a length, not an area.
• D: This adds the changes 5 and 12 to get 17 as the radius instead of using the distance formula.

Answer: B

Correct answer: 30 units. Width:

Height:

Perimeter =

units.

• A: This is 9 times 6, the area, not the distance around.
• B (correct): The width is 3 - (-6) = 9 and the height is 4 - (-2) = 6, so the perimeter is 2(9 + 6) = 30.
• C: This finds the width as 3 - 6 = 3 by dropping the negative sign, then uses 2(3 + 6).
• D: This adds one width and one height but forgets that a rectangle has two of each.

Answer: C

Correct answer: (-5, 8). Symmetry about the origin matches (x, y) with (-x, -y). Changing both signs of (5, -8) gives (-5, 8).

• A: This changes only the x sign, which is symmetry about the y-axis.
• B: This changes only the y sign, which is symmetry about the x-axis.
• C (correct): Symmetry about the origin pairs (x, y) with (-x, -y), so both signs change.
• D: This swaps the coordinates and changes a sign, which is a turn, not origin symmetry.

Answer: B

Correct answer: (4, 7). Use

one coordinate at a time:

Check: the average of (-6, 1) and (4, 7) is (-1, 4).

• A: This averages the start point and the midpoint, which finds a point a quarter of the way along the pipe.
• B (correct): Doubling the midpoint gives (-2, 8), and subtracting the start point gives (4, 7).
• C: This adds the start point to twice the midpoint instead of subtracting it.
• D: This gives the change from the start to the midpoint, not the far endpoint.

Answer: B

Correct answer: Undefined. The equation

is a vertical line. Between any two of its points the run is 0, so rise over run cannot be computed and the slope is undefined.

• A: A slope of 0 belongs to a horizontal line; x = -3 is vertical.
• B (correct): Every point has x = -3, so the run is 0, and division by zero is impossible.
• C: This reads the constant as the slope; -3 only tells where the vertical line crosses the x-axis.
• D: The line does not rise one unit for each unit of run; it does not run at all.

Answer: D

Correct answer: -4. Rewrite in slope-intercept order:

The slope m is the coefficient of x, which is -4.

• A: This is the y-intercept, the constant term, not the slope.
• B: This drops the negative sign in front of 4x.
• C: This mixes the intercept with the sign of the x-term.
• D (correct): The coefficient of x, including its sign, is the slope; the line is y = -4x + 9.

Answer: B

Correct answer: Infinitely many. Rewrite the first equation:

Both equations have slope 2 and intercept -3, so they are the same line and every point on it is a solution.

• A: Lines cross once only when their slopes differ; here both slopes are 2.
• B (correct): Solving 6x - 3y = 9 for y gives y = 2x - 3, so the two equations describe the same line.
• C: Equal slopes alone are not enough for no solution; the intercepts must also differ, and here both are -3.
• D: Two straight lines can never meet in exactly two points.

Answer: C

Correct answer: A flat fee charged before any hours are worked. The y-intercept is the output when the input is zero. With h = 0,

so $90 is the flat fee, and $65 is the added cost per hour.

• A: The cost per hour is the slope, 65, the number multiplied by h.
• B: One hour costs 65 plus 90, or 155, not 90.
• C (correct): When h is 0, C is 90, so 90 is the charge just for showing up.
• D: Hours are the input h, not a fixed number in the equation.

Answer: C

Correct answer: 54 square units. Two vertices share

giving a horizontal leg of

Two share

giving a vertical leg of

• A: This multiplies the legs but forgets the one half.
• B: This uses the 15 unit slanted side as the height; the height must be perpendicular to the base.
• C (correct): The horizontal leg is 9 and the vertical leg is 12, and half of 9 times 12 is 54.
• D: This takes half of 12 times 15, again using the slanted side instead of a leg.

Answer: D

Correct answer: 36. The rises between points are 1, 3 and 5, so the rate of change is not constant and no single slope fits. Use the equation directly: y = 6 squared = 36.

• A: This extends a straight line using the last step's rise of 5 per unit, but the rate keeps changing on a curve.
• B: This draws a straight line from (0, 0) to (3, 9) with slope 3 and extends it, which a curve does not follow.
• C: This doubles x instead of squaring it.
• D (correct): The rule y = x squared gives 6 times 6, or 36.

Answer: D

Correct answer: (2, 1). Test each point in the inequality. For (2, 1):

and 1 < 4 is true. The boundary

is dashed because the sign is strict, so points on it do not count.

• A: Substituting gives 0 < -2, which is false, so the origin is on the unshaded side.
• B: Substituting gives 3 < 1, which is false.
• C: This point lies on the boundary, giving 4 < 4, which is false; the dashed line itself is not included.
• D (correct): Substituting gives 1 < 6 - 2, or 1 < 4, which is true.

Answer: B

Correct answer: (20, -6). Read each axis separately: x = 4 times 5 = 20, and y = 3 times 2 = 6 units down, so

The point is (20, -6).

• A: This counts grid lines as single units without checking the axis scales.
• B (correct): Four x-lines at 5 units each give 20, and three y-lines down at 2 units each give -6.
• C: This uses the y-axis scale for x and the x-axis scale for y.
• D: This reads the scales correctly but forgets that below the origin means a negative y-value.

Answer: A

Correct answer: 1. Perpendicular slopes multiply to -1, so the new slope is -1/3. Set up rise over run from (1, 2) to (4, k).

So

and k = 1.

• A (correct): The needed slope is -1/3, and (k - 2)/3 = -1/3 gives k - 2 = -1, so k = 1.
• B: This uses slope 3, which would make the lines parallel, not perpendicular.
• C: This uses slope 1/3, the reciprocal without the negative sign.
• D: This uses slope -3, changing the sign but not taking the reciprocal.