Reading Coordinates - SHSAT Math

The coordinate plane shown has a non-standard scale. Using the coordinate plane shown, what are the coordinates of point $K$ ?
Answer: $(15, 10)$ Each gridline on the x-axis represents 5 units, and each gridline on the y-axis represents 2 units. Point $K$ is 3 gridlines right of the y-axis (3 × 5 = 15) and 5 gridlines above the x-axis (5 × 2 = 10), giving (15, 10). Choice A counts gridlines without applying the scale. Choice B swaps the scales of the two axes. Choice D reverses the coordinates.
On a map grid, one unit represents 1 mile. A campsite is located at (-2, 5). How far north of the x-axis is the campsite?
Answer: 5 miles north When you see coordinate geometry questions, remember that the coordinate plane gives you precise location information. The x-coordinate tells you horizontal position (east-west), while the y-coordinate tells you vertical position (north-south). For the campsite at (-2, 5), you need to focus on what "north of the x-axis" means. The x-axis is the horizontal line where $y = 0$ . Any point north of this line has a positive y-coordinate, and the distance north equals the y-coordinate value. Since the campsite has coordinates (-2, 5), the y-coordinate is 5. This means the campsite is 5 units north of the x-axis. Given that each unit represents 1 mile, the campsite is 5 miles north of the x-axis, making choice A correct. Choice B (2 miles north) incorrectly uses the x-coordinate instead of the y-coordinate. The x-coordinate of -2 tells you about east-west position, not north-south distance. Choice C (5 miles east) uses the correct number but confuses direction—it mistakes the y-coordinate for eastward movement. Choice D (2 miles west) makes two errors: it uses the wrong coordinate and ignores that we're measuring distance north, not describing directional movement west. Remember this key pattern: distance north or south of the x-axis always depends on the y-coordinate, while distance east or west of the y-axis depends on the x-coordinate. When measuring distance from an axis, you're looking for how far away the point is, which means taking the absolute value if needed.
A point P' is reflected across the y-axis from point P(-8, 3). What are the coordinates of P'?
Answer: $(8,\,3)$ When you encounter reflection problems, focus on understanding how each type of reflection transforms coordinates systematically. Reflecting a point across the y-axis follows a specific rule: the y-coordinate stays the same, while the x-coordinate changes to its opposite (negative becomes positive, positive becomes negative). Think of it as "flipping" the point horizontally across the vertical y-axis line. Starting with point P(-8, 3), when we reflect it across the y-axis: - The x-coordinate changes from -8 to +8 - The y-coordinate remains 3 - Therefore, P' = (8, 3) Looking at the wrong answers: Choice B (-8, -3) represents reflection across the x-axis instead, where the x-coordinate stays the same and the y-coordinate changes sign. Choice C (3, 8) incorrectly swaps the x and y coordinates, which would happen in a reflection across the line y = x. Choice D (-3, 8) combines multiple errors—both swapping coordinates and changing the wrong sign. The key pattern to remember: reflection across the y-axis affects only the x-coordinate (it becomes its opposite), while reflection across the x-axis affects only the y-coordinate. On the SHSAT, coordinate geometry problems often test whether you can distinguish between different types of transformations, so practice identifying the axis of reflection and applying the correct rule to avoid mixing up these transformations.
On a coordinate grid, a point has an x-coordinate equal to the y-coordinate of $(-5,7)$ and a y-coordinate equal to the x-coordinate of $(-5,7)$ . What are the coordinates of this new point?
Answer: $(7,\,-5)$ This question tests your understanding of coordinate pairs and how to extract and use the x and y coordinates from given points. When working with coordinate pairs, remember that they're always written as $(x, y)$ - the first number is the x-coordinate (horizontal position) and the second number is the y-coordinate (vertical position). In the point $(-5, 7)$ , the x-coordinate is $-5$ and the y-coordinate is $7$ . The question asks you to create a new point where the coordinates are swapped: the new x-coordinate equals the original y-coordinate (which is $7$ ), and the new y-coordinate equals the original x-coordinate (which is $-5$ ). Therefore, the new point is $(7, -5)$ , which is choice A. Looking at the wrong answers: Choice B gives you $(-5, 7)$ , which is just the original point unchanged - this would happen if you misread the question and thought no transformation was needed. Choice C shows $(-7, 5)$ , which incorrectly changes the signs of both coordinates while swapping them. Choice D gives you $(5, -7)$ , which changes the sign of the original x-coordinate but keeps the y-coordinate's sign when it should be flipped. Strategy tip: When dealing with coordinate transformations, always identify what each coordinate represents first, then carefully apply the transformation step by step. Write out "original x = ?" and "original y = ?" to avoid mixing up the values during the swap.
In the coordinate plane shown, parallelogram $ABCD$ has three labeled vertices. Based on the coordinate plane shown, what are the coordinates of the unlabeled fourth vertex $D$ ?
Answer: $(7, 5)$ Reading directly from the coordinate plane, the fourth vertex $D$ is plotted at $(7, 5)$ . Choice B swaps the x- and y-coordinates. Choice C represents a miscounting of grid squares. Choice D would place the point at the wrong location on the graph.
Based on the coordinate plane shown, what is the distance between points $X$ and $Y$ ?
Answer: $\sqrt{85}$ Reading the graph, X = (-3, 5) and Y = (4, -1). Distance = $\sqrt{(4-(-3))^2 + (-1-5)^2} = \sqrt{49 + 36} = \sqrt{85}$ . Choice A is just the horizontal distance. Choice B is the sum |Δx| + |Δy| (taxicab distance). Choice D comes from misreading Y as (4, -2): $\sqrt{49+49}$ ... actually $\sqrt{49+64}=\sqrt{113}$ from reading X as (-3, 6).
Using the coordinate plane shown, a line passes through the two marked points. What is the slope of the line?
Answer: $-\dfrac{3}{5}$ Reading the two marked points: (-3, 4) and (2, 1). Slope = $\frac{1-4}{2-(-3)} = \frac{-3}{5}$ . Choice B inverts rise and run. Choice C drops the negative sign (line slants down left-to-right, so slope is negative). Choice D both inverts and drops the sign.
On a coordinate grid, point S is 3 units to the left of the origin and 6 units above it. What are the coordinates of S?
Answer: $(-3,\,6)$ When working with coordinate grids, you need to understand how position relates to coordinates. The coordinate system uses an ordered pair $(x, y)$ where the first number represents horizontal position and the second represents vertical position. To find point S, let's translate the given information systematically. "3 units to the left of the origin" means you move 3 units in the negative x-direction from $(0, 0)$ , giving you an x-coordinate of $-3$ . "6 units above the origin" means you move 6 units in the positive y-direction, giving you a y-coordinate of $+6$ . Therefore, point S is at $(-3, 6)$ . Looking at the answer choices: Choice A gives $(-3, 6)$ , which correctly represents 3 units left (negative x) and 6 units up (positive y). Choice B shows $(3, 6)$ , which would place the point 3 units to the right of the origin instead of left. Choice C gives $(-6, 3)$ , which switches the distances—this would be 6 units left and 3 units up. Choice D shows $(-3, -6)$ , which has the correct x-coordinate but places the point 6 units below the origin instead of above it. Remember this key pattern: "left" and "below" correspond to negative coordinates, while "right" and "above" correspond to positive coordinates. Always check that your x-coordinate (first number) matches the horizontal direction and your y-coordinate (second number) matches the vertical direction described in the problem.
Based on the coordinate plane shown, point $Q$ lies on the segment connecting $(-4, -2)$ and $(8, 4)$ at the location indicated. What are the coordinates of point $Q$ ?
Answer: $(4, 2)$ Reading directly from the graph, point Q is plotted at $(4, 2)$ . Students who assume Q is the midpoint of the segment's endpoints would wrongly compute $(2, 1)$ (choice B). Choice A is the origin, where the segment crosses the axes. Choice D is the reflection of the midpoint through the origin.
Based on the coordinate plane shown, point $W$ is reflected over the x-axis to create point $W'$ . What are the coordinates of $W'$ ?
Answer: $(-6, 4)$ Reading from the graph, W = (-6, -4). Reflecting over the x-axis negates the y-coordinate: W' = (-6, 4). Choice B reflects over both axes (or the origin). Choice C is the reflection over the y-axis. Choice D is the original point W itself.
In the coordinate plane, a point W is plotted at the intersection of $x=-2$ and $y=\tfrac{5}{2}.$ What ordered pair represents W?
Answer: $(-2,\,2.5)$ When you encounter questions about plotting points from line intersections, you're working with the fundamental concept that every point in the coordinate plane has exactly one x-coordinate and one y-coordinate, written as an ordered pair $(x, y)$ . The question tells you that point W is at the intersection of $x = -2$ and $y = \frac{5}{2}$ . This means W has an x-coordinate of $-2$ and a y-coordinate of $\frac{5}{2}$ . Since $\frac{5}{2} = 2.5$ , the ordered pair is $(-2, 2.5)$ . Looking at the wrong answers: Choice B gives $(2.5, -2)$ , which reverses the coordinates—this puts the y-value in the x-position and makes the x-coordinate positive instead of negative. Choice C shows $(-2.5, 2)$ , which incorrectly makes the x-coordinate $-2.5$ instead of $-2$ and converts $\frac{5}{2}$ to $2$ instead of $2.5$ . Choice D gives $(-2, -2.5)$ , which has the correct x-coordinate but makes the y-coordinate negative when it should be positive. Remember that in ordered pairs, the first number is always the x-coordinate and the second is always the y-coordinate. When you see equations like $x = -2$ or $y = \frac{5}{2}$ , these directly give you the coordinates. Also, be careful with fraction-to-decimal conversions: $\frac{5}{2} = 2.5$ , not $2$ or any other value.
Based on the coordinate plane shown, four points are labeled. Which of the labeled points lies on the graph of $y = x^2 - 3$ ?
Answer: Point $D$ Reading the points: A = (1, 1), B = (-2, 4), C = (0, 3), D = (3, 6). Testing $y = x^2 - 3$ : A: 1² - 3 = -2 ≠ 1 ✗; B: (-2)² - 3 = 4 - 3 = 1 ≠ 4 ✗; C: 0² - 3 = -3 ≠ 3 ✗; D: 3² - 3 = 9 - 3 = 6 ✓. Choice A confuses with y = x². Choice B confuses with y = x². Choice C misreads the sign on the constant.
Point F lies exactly halfway between (0, 12) and (0, -4) on the y-axis. What are the coordinates of F?
Answer: $(0,\,4)$ When you see a problem asking for the midpoint between two points, you're looking for the point that sits exactly halfway between them. Since both given points $(0, 12)$ and $(0, -4)$ lie on the y-axis (they both have x-coordinates of 0), point F will also be on the y-axis with an x-coordinate of 0. To find the y-coordinate of the midpoint, you add the two y-coordinates and divide by 2: $\frac{12 + (-4)}{2} = \frac{8}{2} = 4$ . Therefore, point F is at $(0, 4)$ . You can verify this makes sense: from $(0, -4)$ to $(0, 4)$ is a distance of 8 units, and from $(0, 4)$ to $(0, 12)$ is also 8 units, confirming F is exactly halfway between them. Looking at the wrong answers: Choice B gives $(0, 8)$ , which would be the result if you mistakenly used only the positive y-values or made an arithmetic error. Choice C gives $(0, -8)$ , which might occur if you subtracted instead of adding, or confused the direction. Choice D gives $(4, 0)$ , which incorrectly places the point on the x-axis instead of the y-axis—this happens when students mix up coordinates or don't notice that both original points share the same x-coordinate. Remember the midpoint formula: for any two points, average their x-coordinates and average their y-coordinates separately. When points share the same x- or y-coordinate, the midpoint will too.
In the coordinate plane shown, four points are plotted. Using the coordinate plane shown, which point satisfies the equation $2x - y = 5$ ?
Answer: Point $H$ Reading the coordinates from the graph: F = (1, 3), G = (3, 2), H = (4, 3), J = (-1, -3). Testing each point in the equation $2x - y = 5$ : Point F: $2(1) - 3 = -1 \neq 5$ . Point G: $2(3) - 2 = 4 \neq 5$ . Point H: $2(4) - 3 = 5 \checkmark$ . Point J: $2(-1) - (-3) = 1 \neq 5$ . Only point H satisfies the equation.
A point G is at the same x-coordinate as H(5, -2) and the same y-coordinate as J(-3, 7). What are the coordinates of G?
Answer: $(5,\,7)$ When you encounter coordinate problems, remember that every point is written as $(x, y)$ where the first number is the x-coordinate (horizontal position) and the second is the y-coordinate (vertical position). To find point G, you need to identify which coordinate it shares with each given point. Since G has the same x-coordinate as H(5, -2), G's x-coordinate is 5. Since G has the same y-coordinate as J(-3, 7), G's y-coordinate is 7. Therefore, G is at $(5, 7)$ . Looking at the answer choices: Choice A gives $(5, 7)$ , which correctly takes the x-coordinate from H and the y-coordinate from J. This is our answer. Choice B gives $(-3, -2)$ , which incorrectly takes the x-coordinate from J and the y-coordinate from H — the opposite of what the problem asks for. Choice C gives $(7, 5)$ , which swaps the coordinates we found. This represents a common mistake where students mix up which number goes in which position. Choice D gives $(-2, 5)$ , which takes the y-coordinate from H (instead of the x-coordinate) and places the correct x-coordinate in the y-position. Study tip: When working with coordinates, always double-check which point contributes which coordinate. Write out "x from ___" and "y from ___" to avoid mixing them up. Remember that coordinates are always written in alphabetical order: x comes before y, just like in the alphabet.
Refer to the coordinate plane shown. Point $P$ lies exactly halfway between two gridlines in both directions. What are the coordinates of point $P$ ?
Answer: $(2.5, 3.5)$ Point P is between x = 2 and x = 3 (so x = 2.5) and between y = 3 and y = 4 (so y = 3.5), giving $(2.5, 3.5)$ . Choice A rounds both coordinates down. Choice C rounds both up. Choice D has the coordinates swapped.
Refer to the coordinate plane shown. A triangle has vertices at points $R$ , $S$ , and $T$ . What are the coordinates of the vertex that lies in Quadrant III?
Answer: $(-3, -4)$ Quadrant III contains points where both x and y are negative. Reading the graph, R = (-5, 2) is in Quadrant II, S = (3, -4) is in Quadrant IV, and T = (-3, -4) is in Quadrant III. Choice A is in Quadrant II. Choice B is in Quadrant IV. Choice D swaps the x- and y-coordinates of point T.
Refer to the coordinate plane shown. Point $M$ is the midpoint of segment $\overline{PQ}$ , where $P$ and $Q$ are plotted on the graph. What are the coordinates of point $M$ ?
Answer: $(1, 2)$ Reading P = (-3, -2) and Q = (5, 6) from the graph. The midpoint is $M = \left(\frac{-3+5}{2}, \frac{-2+6}{2}\right) = (1, 2)$ . Choice B swaps x and y. Choice C uses subtraction instead of addition for x. Choice D averages x-values incorrectly or counts the origin.
The coordinate plane shown contains four points labeled $A$ , $B$ , $C$ , and $D$ . Using the coordinate plane shown, which point has coordinates whose sum $x + y$ is the greatest?
Answer: Point $B$ Reading the coordinates: A = (-2, 6), B = (5, 3), C = (6, 1), D = (-3, 8). The sums are: A: 4, B: 8, C: 7, D: 5. Point B has the greatest sum (8). Students who pick D likely chose the point that appears highest on the graph (largest y alone). Students who pick C chose the rightmost point (largest x alone). Students who pick A confused the coordinates and read y before x.
A point Y is on the x-axis 9 units to the left of the y-axis. What are the coordinates of Y?
Answer: $(-9,\,0)$ When you encounter questions about points on coordinate axes, remember that the coordinate plane has specific rules: points on the x-axis always have a y-coordinate of 0, and points on the y-axis always have an x-coordinate of 0. Since point Y is on the x-axis, its y-coordinate must be 0. The phrase "9 units to the left of the y-axis" tells you about the x-coordinate. Moving left from the y-axis means moving in the negative x-direction. Since the y-axis is the line where x = 0, moving 9 units to the left gives you x = -9. Therefore, point Y has coordinates $(-9, 0)$ , making choice A correct. Let's examine why the other options are wrong: Choice B, $(0, -9)$ , represents a point on the y-axis (since x = 0), not the x-axis. This point would be 9 units below the x-axis. Choice C, $(9, 0)$ , is on the x-axis but 9 units to the right of the y-axis, not to the left. This reflects a common sign error. Choice D, $(0, 9)$ , is also on the y-axis, 9 units above the x-axis, which doesn't match the given description at all. Study tip: Remember that "left" always means negative x-direction, and "right" means positive x-direction. Also, any point described as being "on the x-axis" automatically has y-coordinate 0, while points "on the y-axis" have x-coordinate 0.

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