The Coordinate Plane, Graphs & Number Patterns

The Coordinate Plane, Graphs & Number Patterns

Coordinate geometry appears on the HSPT in a deliberately elementary form: plotting and naming points, identifying quadrants, measuring straight-line horizontal and vertical distances, reading graphs, and extracting the rule from a table of values. No calculator is needed for any of it, and the arithmetic stays small - which makes this some of the most reliably earnable material on the Mathematics subtest.

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Ordered Pairs

Every point is written (x, y): the horizontal coordinate first, the vertical coordinate second.

• (4, 2) means 4 units right of the origin and 2 units up.
• (2, 4) means 2 units right and 4 units up - a different point entirely.

The origin is (0, 0). A point on the x-axis has y = 0; a point on the y-axis has x = 0.

The reversal trap: (a, b) and (b, a) name the same point only when a equals b. HSPT items regularly offer both orderings as choices.

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The Four Quadrants

Quadrants are numbered counterclockwise beginning in the upper right.

The sign pattern alone identifies the quadrant, so a question asking where (-8, 2) lies is answered in about a second: negative then positive means Quadrant II. Points sitting on an axis belong to no quadrant.

            y
            |
   II       |       I
 (-, +)     |     (+, +)
------------+------------ x
   III      |       IV
 (-, -)     |     (+, -)
            |

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Horizontal and Vertical Distance

When two points share a coordinate, the distance between them is the absolute difference of the other coordinate.

• Horizontal segment: (-3, 4) to (6, 4) - the y values match, so the length is the absolute value of 6 minus (-3), which is 9.
• Vertical segment: (5, -2) to (5, 7) - the x values match, so the length is the absolute value of 7 minus (-2), which is 9.

Subtracting a negative is where errors appear. From -3 to 6 you travel 3 units to reach zero and 6 more beyond it, for 9 total - the arithmetic and the picture agree.

Diagonal Distance

For a slanted segment, drop a right triangle and use the Pythagorean theorem. From (1, 2) to (5, 5): the horizontal leg is 4, the vertical leg is 3, and the hypotenuse is 5 - the familiar 3-4-5 triple. HSPT diagonal-distance items are almost always built on such triples so that no calculator is required.

Midpoint

The midpoint of a segment is the average of the endpoints, coordinate by coordinate.

• Midpoint of (2, 3) and (8, 11): x is (2 + 8)/2 = 5 and y is (3 + 11)/2 = 7, so the midpoint is (5, 7).

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Reading Graphs

Three habits prevent most graph errors:

1. Read the scale before the shape. Axes that start above zero exaggerate differences, and one gridline may represent 5, 10, or 250 units.
2. Read the labels and the key. On a pictograph, one symbol frequently stands for 10 or 100 items rather than one.
3. Translate steepness correctly. On a line graph, a rising segment means increase, a falling segment means decrease, and a flat segment means no change - not zero quantity.

Worked Example

A line graph shows a company's monthly sales: January 40, February 55, March 55, April 30 (in thousands of dollars). Between which consecutive months was the change greatest in magnitude?

Changes are +15, 0, and -25. The largest magnitude is the March-to-April drop of 25. A question asking for the greatest increase would instead be January to February.

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Function Tables and Number Patterns

A function-table item gives paired values and asks for the rule or for a missing entry.

• y increases by 3 each time x increases by 1, so the rule has the form y = 3x + b.
• Substitute the first pair: 5 = 3(1) + b gives b = 2, so y = 3x + 2.
• Verify on another row: 3(4) + 2 = 14. Correct.
• Predict: for x = 5, y = 3(5) + 2 = 17.

Verify before predicting. A rule fitted to a single row will often satisfy one of the wrong answers. Test it against at least two other rows - it costs three seconds and removes the trap.

When the Rule Is Not Linear

If the differences are not constant, check ratios and powers.

Here y doubles each step, so the rule is y = 2 raised to the x power, and the next value is 32. The diagnostic order is the same one used on Quantitative number series: differences first, then ratios, then powers.

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Traps Worth Rehearsing

• Reversed coordinates. Plotting (3, 7) at 7 right and 3 up.
• Quadrant miscount. Numbering clockwise instead of counterclockwise, which swaps Quadrants II and IV.
• Sign loss in distance. Computing 6 minus (-3) as 3 rather than 9.
• Unlabeled scale. Assuming each gridline equals one unit when the axis says otherwise.
• Rule fitted to one row. Accepting y = x + 4 from the pair (1, 5) without checking (2, 8).