Coordinate Geometry and Graph Interpretation - Quantitative Reasoning

Graph of $y=f(x)$ contains $(-2,3)$ . If $g(x)=f(x+1)-2$ , which point must be on the graph of $g$ ?
Answer: (-3, 1) Since g(-3)=f(-3+1)-2=f(-2)-2=3-2=1, the point (-3,1) must be on the graph of g. The tempting wrong answer is (-1,1), which comes from shifting right instead of left; the inside +1 shifts the graph left by 1, not right.
In the $xy$ -plane, line $\ell$ passes through the points $(2,5)$ and $(6,-3)$ . What is the slope of line $\ell$ ?
Answer: $-2$ 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.
Line $s$ passes through $(2,-1)$ and $(2,5)$ . Which of the following best describes the graph of line $s$ ?
Answer: A vertical line with equation $x=2$ 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.
In the $xy$ -plane, line $r$ passes through $(1,6)$ and $(5,2)$ . What is the $y$ -intercept of line $r$ ?
Answer: $7$ This question tests coordinate geometry, finding the y-intercept given two points. Slope between (1,6) and (5,2) is (2-6)/(5-1) = -1; equation y = -x + b, using (1,6) gives b = 7. This intercept is where x=0. Calculation confirms y=7 at x=0. Therefore, Choice C is correct. A common mistake is miscalculating slope as 1, leading to b=5. Another error could be using the wrong point, resulting in 4.
On the coordinate plane, a line is described as crossing the $y$ -axis at $(0,-2)$ and the $x$ -axis at $(5,0)$ . What is the slope of the line?
Answer: $\frac{2}{5}$ 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.
Two lines are given by $y=2x-1$ and $y=2x+3$ . Which of the following best describes the relationship between the graphs of these two lines in the coordinate plane?
Answer: They are parallel because they have the same slope. 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.
Lines $y=2x-5$ and $y=-x+1$ intersect. In which quadrant is the intersection?
Answer: Quadrant IV Set the two y-values equal: 2x - 5 = -x + 1. Solving gives 3x = 6, so x = 2. Substituting into either line gives y = -1. Since x is positive and y is negative, the point (2, -1) lies in Quadrant IV. The tempting wrong answer is Quadrant I, where both coordinates are positive, but here y is negative.
In the $xy$ -plane, a line passes through $(0,1)$ and $(3,7)$ . Which of the following points lies on this line?
Answer: $(6,13)$ This question tests coordinate geometry, determining which point lies on the line through (0,1) and (3,7). Slope is (7-1)/(3-0) = 2, equation y = 2x + 1; (6,13) satisfies 2(6) + 1 = 13. Other points like (3,1) give 7 ≠ 1. Verification shows (6,13) fits. Thus, Choice D is correct. A common error is testing without the equation, missing mismatches. Another might involve slope confusion, selecting off-line points.
A line has equation $y= -\frac{3}{2}x+6$ . If the line is shifted right by 2 units (with no vertical shift), which of the following is an equation of the new line?
Answer: $y=-\frac{3}{2}(x-2)+6$ 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.
Line $p$ has equation $y=-\frac{1}{3}x+4$ . What is the $y$ -intercept of line $p$ ?
Answer: $4$ 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).
A line has equation $y=\frac{2}{3}x+4$ . Which of the following describes the graph of the line?
Answer: It has positive slope and crosses the $y$ -axis at $4$ . 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.
Two points $C(-1,3)$ and $D(7,3)$ are endpoints of a line segment. What is the slope of the line containing segment $\overline{CD}$ ?
Answer: $0$ 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.
A line has equation $2x+5y=10$ . What is the slope of the line?
Answer: $-\frac{2}{5}$ 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.
The line with equation $3x+2y=12$ is graphed in the $xy$ -plane. What is the $y$ -intercept of the line?
Answer: $6$ 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.
A line passes through $(0,4)$ and $(8,0)$ . Which of the following is an equation of the line?
Answer: $y=-\frac{1}{2}x+4$ 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.
In the $xy$ -plane, the line $y=-3x+6$ intersects the $x$ -axis at point $P$ . What is the $x$ -coordinate of $P$ ?
Answer: $2$ This question tests coordinate geometry, focusing on finding the x-intercept from a line's equation. The x-intercept occurs where y = 0, so substitute into y = -3x + 6 to get 0 = -3x + 6, solving to x = 2. This point P is (2,0), so the x-coordinate is 2. Verification confirms this intersection. Therefore, the answer is 2, choice B. A common error is miscalculating to x = -2 by sign error. Another might involve confusing with y-intercept, leading to 6.
The line $y=x-4$ is shifted upward by $6$ units to form a new line. What is the $y$ -intercept of the new line?
Answer: $2$ 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.
Line $k$ passes through $(0,-2)$ and $(4,6)$ . What is the $y$ -intercept of line $k$ ?
Answer: $-2$ This question tests coordinate geometry, specifically finding the y-intercept when given two points on a line. The y-intercept is the y-value when x = 0, and we're already given that the line passes through (0,-2). Since this point has x-coordinate 0, its y-coordinate (-2) is the y-intercept. We can verify this by finding the line's equation: slope = (6-(-2))/(4-0) = 8/4 = 2, giving y = 2x - 2, which confirms the y-intercept is -2. Students might mistakenly calculate the slope (2) or use the wrong point's y-coordinate (6).
A line has $y$ -intercept $-3$ and passes through the point $(2,1)$ . What is the slope of the line?
Answer: $2$ This question tests coordinate geometry, determining the slope given the y-intercept and a point. With y-intercept -3, the equation is y = m x - 3; substituting (2,1) gives 1 = 2m - 3, so m = 2. This slope indicates a positive rise over run. Verification confirms the calculation. Thus, Choice B is correct. A common mistake is solving to m = -2 by sign error. Another error could involve confusing x and y, leading to 1/2.
Line $\ell$ passes through the points $(2,5)$ and $(6,-3)$ . What is the slope of line $\ell$ ?
Answer: $-2$ This question tests coordinate geometry, specifically finding the slope of a line given two points. The slope formula is m = (y₂ - y₁)/(x₂ - x₁), which measures the rate of change between two points. Applying this to points (2,5) and (6,-3), we get m = (-3 - 5)/(6 - 2) = -8/4 = -2. The slope is -2, which means the line falls 2 units vertically for every 1 unit horizontally. A common error would be to reverse the subtraction order or confuse the x and y coordinates, leading to incorrect answers like 1/2 or -1/2.

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