Coordinate Geometry and Slope (Upper Level only)
The two topics E3n tests at Upper Level and not at Middle Level: the coordinate plane, slope every way it is asked, and the equations of lines.
10 topics · 43 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
The Coordinate Plane and Plotting Points
- How it worksHow does the ordered pair (3, -5) tell you to travel right first and then down?
- Worked exampleFind where (4, 2), (-3, 2) and (0, -6) land once you plot them on the same grid.
- Common mistakeWhy isn't the point (2, 5) the same place on the grid as the point (5, 2)?
- Going furtherWhat happens to a point when you swap its two coordinates, drawn as a reflection across a diagonal?
Your turneasy1 of 3
How far apart are the points (7, 2) and (-7, 2) on the grid?Quadrants, Axes and Signs
- How it worksHow do the four quadrants get numbered counterclockwise starting from the upper right corner?
- Worked exampleFind which quadrant holds each of (-7, -2), (8, -1) and (-4, 6).
- Common mistakeWhy doesn't a point sitting on the x-axis belong to any quadrant at all?
Your turneasy1 of 3
On item 17, which quadrant holds the point (-7, -9)?Slope as Rise over Run
- How it worksHow does a rise-over-run triangle keep the same fraction no matter where you slide it along the line?
- How it worksHow does the slope number change as the very same line is tilted from nearly flat toward steep?
- Worked exampleFind the slope of a line that climbs 6 units for every 4 units it travels to the right.
- Worked exampleWalk me through reading a slope straight off a graph by counting grid squares up and across.
- Common mistakeWhy isn't a bigger rise on its own enough to prove one line is steeper than another?
Your turneasy1 of 3
A line climbs 9 units for every 6 units it travels to the right. What is its slope?Slope from Two Points
- How it worksHow does the order you pick the two points in cancel out and leave the same slope either way?
- Worked exampleFind the slope of the line through (2, 3) and (8, 15).
- Worked exampleFind the slope through (-4, 7) and (2, -5), and say which way that line tilts.
- Common mistakeWhy doesn't run over rise give you the slope just as well as rise over run?
Your turneasy1 of 3
Write an expression, in terms of x, for the slope of the line through (2, 3) and (x, 24).Positive, Negative, Zero and Undefined Slope
- How it worksHow does a run of zero break the slope formula the moment a line stands straight up?
- Worked exampleFind the slopes of the line through (1, 5) and (9, 5) and the line through (3, 2) and (3, -6).
- Common mistakeWhy isn't no slope another way of saying a slope of zero?
- Going furtherWhat does the sign of a slope tell you about whether the line climbs or falls from left to right?
Your turneasy1 of 3
What is the slope of the line through (1, 5) and (9, 30)?Slope-Intercept Form
- How it worksHow does changing b slide the whole line up or down without tilting it even slightly?
- How it worksHow does changing m tilt the line while it stays pinned in place at its y-intercept?
- Worked exampleFind the slope and the y-intercept of y = -3x + 7 and sketch that line.
- Worked exampleWalk me through graphing y = (2/3)x - 4 starting at the intercept and stepping by rise over run.
- Common mistakeWhy isn't the b in y = mx + b the place where the line crosses the x-axis?
Your turneasy1 of 3
A line is given by y = 7x + 7. What is y when x = 8?Equations of Lines in Standard Form
- How it worksHow does setting x to zero and then y to zero hand you both intercepts in a few seconds?
- Worked exampleFind both intercepts of 4x - 5y = 20 and draw the line that runs through them.
- Common mistakeWhy doesn't the number in front of x in 2x + 3y = 12 give you the slope directly?
- Worked exampleWalk me through rearranging 3x + 2y = 8 into slope-intercept form.
Your turneasy1 of 3
Write an expression, in terms of c, for the x-intercept of the line 7x - 5y = c.Writing the Equation of a Line
- How it worksHow does point-slope form build a line's equation out of one slope and any single point on it?
- Worked exampleFind the equation of the line passing through (1, 4) and (5, 16).
- Worked exampleFind the equation of the line with slope -2 that passes through (3, -1).
- Going furtherWhat changes about writing the equation when you are handed a drawn graph instead of coordinates?
Your turnmedium1 of 3
A line passes through (1, 4) and (5, 23). Write its equation, giving y in terms of x.Parallel and Perpendicular Slopes
- How it worksHow does rotating one line until it sits 90 degrees from another drive the product of the slopes to -1?
- How it worksHow does a negative reciprocal swap rise with run and then flip the sign?
- Worked exampleFind the slope of any line perpendicular to y = (3/4)x - 2.
- Worked exampleWalk me through deciding whether the line through (0, 1) and (2, 5) runs parallel to y = 2x + 9.
- Common mistakeWhy isn't a line of slope -2 perpendicular to a line of slope 2?
Your turneasy1 of 3
Write an expression, in terms of a, for the slope of any line perpendicular to y = (a/7)x - 2.Distance and Midpoint on the Grid
- How it worksHow does the distance formula hide the Pythagorean theorem inside a single square root?
- Worked exampleFind the distance between the points (2, 1) and (7, 13).
- Worked exampleFind the midpoint of the segment joining (-3, 8) to (9, -2).
- Common mistakeWhy doesn't averaging the coordinates give you the distance the way it gives you the midpoint?
- Ties togetherHow does the distance between two plotted points connect to the 3-4-5 right triangle?
Your turneasy1 of 3
Write an expression, in terms of y, for the distance between (3, 4) and (12, y).AP®, SAT® and PSAT/NMSQT® are trademarks registered by the College Board (PSAT/NMSQT® jointly with the National Merit Scholarship Corporation). ACT® is a registered trademark of ACT Education Corp. GRE® is a registered trademark of ETS. LSAT® is a registered trademark of the Law School Admission Council. MCAT® is a registered trademark of the Association of American Medical Colleges. ISEE® and SSAT® are registered trademarks of their owners. None of these organizations is affiliated with, or endorses, Jentoo.