Why it matters on the HSPT
The test is taken in the first months of 8th grade, before many students have started algebra, so it only asks for the beginnings of coordinate geometry: naming a point, naming a quadrant, counting a slope on a grid, and matching a drawn line to a short equation. There are only one or two of these questions, but they are worth the same point as any other, and they take under a minute once you can count rise and run. This lesson gives you exactly what the test asks and nothing more.
The rules
Ordered pairs. A point is written as two numbers in parentheses, the x-coordinate then the y-coordinate. The x-coordinate says how far to move right or left from the origin, and the y-coordinate says how far to move up or down. So (3, -2) is 3 to the right and 2 down. The origin is (0, 0), where the two axes cross. A point on the x-axis has a y-coordinate of 0, like (5, 0), and a point on the y-axis has an x-coordinate of 0, like (0, -4).
The four quadrants. The axes cut the plane into four regions, numbered with Roman numerals counterclockwise from the upper right: I is upper right, II is upper left, III is lower left, IV is lower right. The signs tell you the quadrant without looking. Both positive is I. Negative x and positive y is II. Both negative is III. Positive x and negative y is IV. A point on an axis is not in any quadrant.
Slope is rise over run. Pick any two points where the line crosses grid corners. Count how many units the line rises going from the left point to the right point, then count how many units it runs to the right. Slope is rise divided by run. A line that goes up 3 for every 2 to the right has slope 3/2. If the line goes down as you move right, the rise is negative and so is the slope. A horizontal line has slope 0. The steeper the line, the bigger the size of the slope.
Slope from two points. If you are given the points instead of a picture, subtract: slope equals the difference in the y-values divided by the difference in the x-values, taken in the same order. From (1, 2) to (4, 8), the y-values change by 6 and the x-values by 3, so the slope is 2. Watch the signs when a coordinate is negative: from (-2, 3) to (5, 4) the run is 5 minus negative 2, which is 7, and the slope is 1/7.
Slope-intercept form. Every line on the test can be written as y = mx + b. The number m in front of x is the slope, and b is the y-intercept, the y-value where the line crosses the y-axis. In y = 2x - 3 the slope is 2 and the line crosses the y-axis at -3. If the equation has no number added or subtracted, b is 0 and the line passes through the origin.
Reading a line off a graph. First find b: the point where the line crosses the vertical axis. Then find m by counting rise and run between two grid corners. Write y = mx + b. To check, pick any other point on the line and put its coordinates into the equation; both sides should match.
Moving a line up or down. Shifting a line up adds to the y-intercept and leaves the slope alone. Move y = 3x - 4 up 3 units and it becomes y = 3x - 1. Move it down 5 and it becomes y = 3x - 9. The test asks this in words: "the line is shifted up 3 units."
Worked examples
Example 1
In which quadrant is the point (-6, 2)?
- A.I
- B.II
- C.III
- D.IV
Show answer
Answer: B: II
The x-coordinate is negative, so the point is to the left of the y-axis. The y-coordinate is positive, so it is above the x-axis. Left and up is the upper left, quadrant II. Choice D is the trap for reading the numbers as right-then-down.
Example 2
What is the slope of the line shown?
- A.1/2
- B.2
- C.-1/2
- D.-2
Show answer
Answer: D: -2
The line goes through the origin and through (1, -2). Moving 1 unit to the right, it drops 2 units, so the rise is -2 and the run is 1. The slope is -2 divided by 1, which is -2. Any line that falls from left to right has a negative slope, which rules out A and B at once. Choice C is the trap for dividing the run by the rise.
Example 3
Which equation describes the line shown?
- A.y = 2x + 1
- B.y = 2x - 1
- C.y = x + 2
- D.y = -2x + 1
Show answer
Answer: A: y = 2x + 1
The line crosses the y-axis at 1, so b is 1. That rules out B. From (0, 1) to (1, 3) the line rises 2 and runs 1, so the slope is 2, which rules out C. The line rises, so the slope is positive, which rules out D. The equation is y = 2x + 1. Check with the point (1, 3): 2 times 1 plus 1 is 3.
Example 4
What is the slope of the line through (2, 5) and (6, 13)?
- A.1/2
- B.2
- C.4
- D.8
Show answer
Answer: B: 2
The y-values go from 5 to 13, a rise of 8. The x-values go from 2 to 6, a run of 4. Slope is 8 over 4, which is 2. Choice D, 8, is the rise alone, and choice C, 4, is the run alone.
Example 5
The line y = 4x - 2 is shifted up 5 units. Which is the equation of the new line?
- A.y = 9x - 2
- B.y = 4x + 3
- C.y = 4x - 7
- D.y = 4x + 5
Show answer
Answer: B: y = 4x + 3
Moving a line up changes only where it crosses the y-axis. The intercept goes from -2 to -2 plus 5, which is 3, and the slope stays 4. The equation is y = 4x + 3. Choice A adds the 5 to the slope, choice C shifts the line down, and choice D forgets the intercept the line started with.
Example 6
Which point lies on the x-axis?
- A.(0, 3)
- B.(3, 3)
- C.(-3, 0)
- D.(3, -3)
Show answer
Answer: C: (-3, 0)
A point on the x-axis has a y-coordinate of 0 and any x-coordinate at all. (-3, 0) has y equal to 0, so it sits on the x-axis, 3 units to the left of the origin. (0, 3) is on the y-axis, not the x-axis; that is the trap. (3, 3) is in quadrant I and (3, -3) is in quadrant IV.
Common mistakes
Reading the pair backward. The first number is always horizontal. Say "over, then up" to yourself each time you plot or read a point.
Losing the sign of a slope. Before you count anything, look at the direction. Rising to the right means positive, falling to the right means negative. Use that to eliminate half the choices, then count.
Dividing the wrong way. Slope is rise over run, vertical change on top. A line that climbs 1 for every 2 across has slope 1/2, not 2.
Reading the intercept from the wrong axis. The b in y = mx + b is where the line crosses the y-axis, the vertical one. Where it crosses the x-axis is not part of the equation.
Changing the slope on a shift. Moving a line up or down never tilts it. Only the number at the end of the equation changes.
Practice
Coordinate questions sit inside the pre-algebra format, and grid drawings also appear in basic geometry items that ask for the length of a side drawn on a grid. If subtracting negatives in the slope formula slows you down, review the integers lesson first. Tables of inputs and outputs, which are a close cousin of lines, are covered in the functions, sets, and counting lesson.