GRE Quiz: Coordinate Geometry Graph Interpretation

What this quiz covers

This quiz focuses on Coordinate Geometry Graph Interpretation, giving you a quick way to practice the rules, question types, and explanations that matter most for GRE.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. $-\frac{2}{5}$ (correct answer)
2. $\frac{2}{5}$
3. $-\frac{5}{2}$
4. $\frac{5}{2}$
5. $10$

Explanation: This question tests graph interpretation, requiring the slope from a line equation in standard form. To find the slope, rewrite 2x + 5y = 10 as y = -2/5 x + 2, where -2/5 is the slope. This form highlights the negative slope and y-intercept of 2. Applying the conversion confirms the slope as -2/5. Thus, the correct answer is -2/5, choice A. A common incorrect option is -5/2, from swapping numerator and denominator. Another mistake could be forgetting the negative sign, leading to 2/5.

Question 2

1. $(1,1)$
2. $(0,1)$ (correct answer)
3. $(1,-3)$
4. $(-1,3)$
5. $(2,-1)$

Explanation: This question tests coordinate geometry by asking which point satisfies a linear equation. To check if a point lies on the line 2x + y = 1, we substitute the x and y coordinates into the equation. For point (0,1): 2(0) + 1 = 0 + 1 = 1, which equals the right side, so this point lies on the line. We can verify other options don't work: for (1,1), we get 2(1) + 1 = 3 ≠ 1. The key is systematically substituting each point's coordinates and checking if the equation holds true.

Question 3

1. $y=-2x+5$
2. $y=-2x+13$ (correct answer)
3. $y=2x+13$
4. $y=-6x+9$
5. $y=-2x-13$

Explanation: This question tests coordinate geometry, involving vertical shifts in line equations. Shifting up by 4 units adds 4 to the y-intercept of y = -2x + 9, resulting in y = -2x + 13. The slope remains unchanged, only the constant term increases. This new equation reflects the parallel line above the original. Therefore, Choice B is correct. A common incorrect option is y = -2x + 5, from subtracting instead of adding. Another error might involve altering the slope, leading to options like y = 2x + 13.

Question 4

1. $-\frac{1}{3}$
2. $\frac{1}{3}$
3. $-4$
4. $4$ (correct answer)
5. $(4,0)$

Explanation: This question tests coordinate geometry, specifically identifying the y-intercept from a line equation. In the equation y = -1/3x + 4, which is in slope-intercept form y = mx + b, the constant term b represents the y-intercept. The y-intercept is 4, which is the y-value when x = 0: y = -1/3(0) + 4 = 4. This means the line crosses the y-axis at the point (0,4). Students often confuse the slope (-1/3) with the y-intercept or mistakenly write the y-intercept as a coordinate pair like (4,0).

Question 5

1. $y=-\frac{3}{2}x+3$
2. $y=-\frac{3}{2}x+9$
3. $y=-\frac{3}{2}(x-2)+6$ (correct answer)
4. $y=-\frac{3}{2}(x+2)+6$
5. $y=\frac{3}{2}x+6$

Explanation: This question tests graph interpretation, applying a horizontal shift to the line y = -3/2 x + 6. Shifting right by 2 replaces x with (x-2), yielding y = -3/2 (x-2) + 6. This maintains the slope but adjusts the intercept. Expanding gives y = -3/2 x + 9, equivalent to the form in C. Therefore, Choice C is correct. A common mistake is shifting left, leading to (x+2). Another error could be vertical shift confusion, altering the constant incorrectly.

Question 6

1. $6$ (correct answer)
2. $-6$
3. $4$
4. $-4$
5. $\frac{3}{2}$

Explanation: This question tests graph interpretation, specifically finding the y-intercept from a linear equation in standard form. The y-intercept occurs where the line crosses the y-axis, which is when x = 0. Substituting x = 0 into 3x + 2y = 12 gives: 3(0) + 2y = 12, so 2y = 12, and y = 6. Therefore, the y-intercept is 6, or the point (0,6). A common mistake is confusing the y-intercept with the x-intercept (which would be 4) or misidentifying coefficients as intercepts.

Question 7

1. A horizontal line with equation $y=2$
2. A vertical line with equation $x=2$ (correct answer)
3. A line with slope $2$ and $y$-intercept $-1$
4. A line with slope $-2$ and $y$-intercept $5$
5. A line with equation $y=x+2$

Explanation: This question tests graph interpretation, identifying a vertical line from points with same x-coordinate. Points (2,-1) and (2,5) share x=2, indicating a vertical line x=2 with undefined slope. Vertical lines are perpendicular to the x-axis. This describes the graph accurately. Thus, Choice B is correct. A common incorrect option is horizontal line y=2, confusing x and y. Another mistake might involve calculating a finite slope, leading to other options.

Question 8

1. A line with slope $5$ and $y$-intercept $2$
2. A line with slope $-2$ and $y$-intercept $5$
3. A line with slope $2$ and $y$-intercept $-5$
4. A line with slope $2$ and $y$-intercept $5$ (correct answer)
5. A line that crosses the $x$-axis at $x=5$

Explanation: This question tests graph interpretation by asking to identify key features of a linear equation. The equation y = 2x + 5 is in slope-intercept form y = mx + b, where m = 2 is the slope and b = 5 is the y-intercept. This describes a line that rises 2 units vertically for every 1 unit horizontally and crosses the y-axis at (0,5). The correct answer is a line with slope 2 and y-intercept 5. Common errors include reversing the slope and y-intercept values or confusing the y-intercept with the x-intercept.

Question 9

1. They intersect at exactly one point.
2. They are perpendicular.
3. They are parallel and distinct. (correct answer)
4. They are the same line.
5. They intersect on the $y$-axis.

Explanation: This question tests graph interpretation, comparing slopes and intercepts to determine line relationships. Both lines have slope 3/4 but different y-intercepts (1 and -5), indicating they are parallel and distinct. Parallel lines never intersect and maintain constant distance. This relationship is confirmed by the identical slopes and unequal intercepts. Thus, Choice C is correct. A common mistake is assuming they intersect at one point due to similar slopes. Another error could be thinking they are the same line if intercepts are miscompared.

Question 10

1. $-10$
2. $-6$
3. $2$ (correct answer)
4. $4$
5. $10$

Explanation: This question tests graph interpretation, specifically understanding vertical translations of linear functions. The original line y = x - 4 has y-intercept -4 (when x = 0, y = -4). When a line is shifted upward by 6 units, we add 6 to the entire equation: y = x - 4 + 6 = x + 2. The new line has equation y = x + 2, so its y-intercept is 2. This vertical shift moves every point on the line up by 6 units, including the y-intercept which moves from -4 to 2. A common error is subtracting instead of adding the shift amount.

Question 11

1. It has negative slope and crosses the $y$-axis at $4$.
2. It has positive slope and crosses the $y$-axis at $-4$.
3. It has positive slope and crosses the $y$-axis at $4$. (correct answer)
4. It has slope $4$ and crosses the $y$-axis at $\frac{2}{3}$.
5. It is a horizontal line at $y=\frac{2}{3}$.

Explanation: This question tests graph interpretation, describing slope and intercept from y = 2/3 x + 4. The positive slope 2/3 indicates rising left to right, with y-intercept at 4. This matches a line crossing y-axis at (0,4). Analysis confirms positive slope and intercept. Thus, Choice C is correct. A common incorrect option is negative slope due to sign confusion. Another mistake might involve swapping slope and intercept values.

Question 12

1. $3$
2. $-7$ (correct answer)
3. $7$
4. $-\frac{7}{3}$
5. $(0,7)$

Explanation: This question tests graph interpretation, specifically identifying the y-intercept from a line equation in slope-intercept form. The equation y = 3x - 7 is already in the form y = mx + b, where b is the y-intercept. The y-intercept occurs when x = 0, giving y = 3(0) - 7 = -7. Therefore, the y-intercept is -7, which represents the point where the line crosses the y-axis. A common mistake is confusing the slope (3) with the y-intercept or misidentifying the sign, thinking the y-intercept is positive 7.

Question 13

1. $\frac{2}{5}$ (correct answer)
2. $-\frac{2}{5}$
3. $-\frac{5}{2}$
4. $\frac{5}{2}$
5. $-2$

Explanation: This question tests graph interpretation, calculating slope from x- and y-intercepts. The points (0,-2) and (5,0) give slope m = (0 - (-2))/(5 - 0) = 2/5. This positive slope shows the line rising from left to right. Applying the formula confirms this value. Therefore, Choice A is correct. A common incorrect option is -2/5, from neglecting the double negative. Another mistake might be inverting to 5/2.

Question 14

1. $9$
2. $4$
3. $-4$ (correct answer)
4. $-\frac{9}{4}$
5. $\frac{9}{4}$

Explanation: This question tests graph interpretation, specifically identifying the slope from a line equation in slope-intercept form. The equation y = -4x + 9 follows the pattern y = mx + b, where m is the slope and b is the y-intercept. The coefficient of x is -4, so the slope is -4. This means the line decreases by 4 units vertically for every 1 unit increase horizontally. A common error is confusing the slope with the y-intercept (9) or misreading the negative sign, leading to answers like 4 or 9/4.

Question 15

1. $3$
2. $-7$ (correct answer)
3. $7$
4. $-\frac{7}{3}$
5. $-3$

Explanation: This question tests graph interpretation, focusing on identifying the y-intercept from a line's equation in slope-intercept form. The y-intercept is the value of y where the line crosses the y-axis, represented by the constant term b in y = mx + b. For the equation y = 3x - 7, the slope m is 3, and the y-intercept b is -7. This directly gives the y-intercept as -7. Therefore, the correct answer is -7, which is choice B. A common mistake is confusing the y-intercept with the slope, leading to selecting 3. Another error might involve misreading the sign, resulting in choices like 7.

Question 16

1. $y=\frac{1}{2}x+4$
2. $y=-\frac{1}{2}x+4$ (correct answer)
3. $y=-2x+4$
4. $y=2x-4$
5. $y=-\frac{1}{4}x+8$

Explanation: This question tests coordinate geometry, involving finding the equation of a line given two points. The relevant graph features are the slope and y-intercept, determined from the points (0,4) and (8,0). The slope m = (0 - 4)/(8 - 0) = -4/8 = -1/2, and since it passes through (0,4), the y-intercept is 4. Thus, the equation is y = -1/2 x + 4. This matches choice B. A common incorrect option is y = -2x + 4, which arises from not simplifying the slope fraction. Another error could be miscalculating the intercept, leading to options like y = 2x - 4.

Question 17

1. They are the same line because they have the same $y$-intercept.
2. They are perpendicular because their slopes are negative reciprocals.
3. They intersect at the point $(2,3)$.
4. They are parallel because they have the same slope. (correct answer)
5. They intersect on the $y$-axis because both have $x=0$.

Explanation: This question tests coordinate geometry by analyzing the relationship between two linear equations. Both equations y = 2x - 1 and y = 2x + 3 have the same slope (2) but different y-intercepts (-1 and 3). Lines with equal slopes are parallel, meaning they never intersect and maintain a constant distance from each other. The first line crosses the y-axis at -1, while the second crosses at 3. A common misconception is thinking lines with the same slope are the same line, but they must also have the same y-intercept to be identical.

Question 18

1. $\frac{1}{2}$
2. $-\frac{1}{2}$
3. $8$
4. $0$ (correct answer)
5. $\frac{3}{8}$

Explanation: This question tests coordinate geometry, computing slope for a line segment's endpoints. For (-1,3) and (7,3), m = (3-3)/(7 - (-1)) = 0/8 = 0, indicating a horizontal line. Horizontal lines have undefined or zero slope. Verification shows constant y-values. Therefore, Choice D is correct. A common error is miscalculating denominator as 6, leading to undefined. Another might involve inverting to 8/0, but it's zero.

Question 19

1. They intersect at exactly one point because they have different slopes.
2. They are perpendicular because their slopes are negative reciprocals.
3. They are parallel because they have the same slope and different $y$-intercepts. (correct answer)
4. They are the same line because they have the same slope.
5. They intersect on the $y$-axis because their $y$-intercepts differ by 6.

Explanation: This question tests coordinate geometry concepts about relationships between lines. Both equations y = 2x + 1 and y = 2x - 5 have the same slope (m = 2) but different y-intercepts (1 and -5 respectively). Lines with equal slopes are parallel, meaning they never intersect and maintain constant distance from each other. The 6-unit difference in y-intercepts (from 1 to -5) represents the vertical distance between the lines. Common misconceptions include thinking same slopes mean the lines are identical, or that different y-intercepts alone guarantee intersection.

Question 20

1. $-\frac{1}{2}$
2. $\frac{1}{2}$
3. $-2$ (correct answer)
4. $2$
5. $-\frac{4}{3}$

Explanation: This question tests coordinate geometry, specifically calculating the slope of a line given two points. The slope represents the rate of change of y with respect to x and is calculated using the formula m = (y2 - y1)/(x2 - x1). For the points (2,5) and (6,-3), the change in y is -3 - 5 = -8, and the change in x is 6 - 2 = 4. Thus, the slope m = -8 / 4 = -2. This confirms that the correct answer is -2, corresponding to choice C. A common incorrect option is 2, which might result from mistakenly reversing the sign in the numerator or denominator. Another error could involve confusing the order of points, leading to choices like -1/2 if miscalculating the differences.