Description:
Coordinate Geometry questions ask you to determine values and relationships based on
graphical data. Most of these questions relate to the equation of a line along with
finding the slope, distance, and midpoint between two points.
Approach:
Follow the four steps below to master Coordinate Geometry questions.
-
Memorize the equation of a line along with the equations for slope, distance, and
midpoint.
-
Draw a picture as accurately as possible and don't be afraid to estimate.
-
Use your graphing calculator whenever possible.
-
Plug in points (x, y)
to match equations to the correct graphs.
1) Memorize the Equation of a line along with the
equations for slope, distance, and midpoint
Coordinate Geometry equations are not provided at the beginning of the math sections, so
you must memorize and know how to use them!
2) The Equation of a Line
The equation of a line is given by y
= mx
+ b
(slope intercept form), where m is the slope of the line and
b is the place where the
line crosses the y-axis (y-intercept).
2-1. Practice:
What is the equation of a line with a slope of 3 and a
y-intercept of -2?
-
y = 3x + 2
-
y = 2x + 3
-
y = -3x - 2
-
y = 3x - 2
-
y = 2x - 3
3) Slope
Slope is the rise (change in the y
values) over the run (change in the x
values) for a set of two points, and is given by the constant m in the equation y = mx
+ b.
Perpendicular
lines have slopes that are opposite reciprocals of each other. If a line has a slope of 4, a
line perpendicular to that line has a slope of -1/4.
Parallel lines
have equal slopes but can have different y-intercepts.
The reflection
of a line is not necessarily perpendicular to it; a line's reflection has an opposite slope and
y-intercept to its reflection. Consequently,
the reflection of the line y = 2x + 3 is
y = -2x - 3
|
Equation
|
Picture
|
![[Image: Equation for the slope of a line]](/MindFish/resized-image.ashx/__size/550x0/__key/CommunityServer.Wikis.Components.Files/sat_5F00_preparation/2185.slope.png)
|
![[Image: Diagram of the slope of a line]](/MindFish/resized-image.ashx/__size/550x0/__key/CommunityServer.Wikis.Components.Files/sat_5F00_preparation/6862.180px_2D00_Slope_5F00_picture_5F00_svg.png)
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3-1. Practice:
What is the slope of a line that is perpendicular to the line
x = 3y - 2?
-
-3
-
-1/3
-
1/3
-
3
-
6
What is the slope of a line that is the reflection of the line x = 3y - 2
over the x-axis?
-
-3
-
-1/3
-
1/3
-
3
-
6
4) Distance
Distance formula is the Pythagorean Theorem applied to two points. Draw a right triangle with
the two points forming the ends of the hypotenuse.
![[Image: Formula for the distance of a line]](/MindFish/resized-image.ashx/__size/550x0/__key/CommunityServer.Wikis.Components.Files/sat_5F00_preparation/1018.Distance.png)
![[Image: Diagram for the distance of a line]](/MindFish/resized-image.ashx/__size/550x0/__key/CommunityServer.Wikis.Components.Files/sat_5F00_preparation/6303.DitanceFormula.jpg)
4-1. Practice:
What is the distance between the point (3, 4) and the point (6, 8)?
-
2
-
3
-
4
-
5
-
6
5) Midpoint
The midpoint is the point equidistant to the two other points in the question. If you forget
the equation below, you can always estimate the midpoint with a good graphical drawing.
![[Image: Formula for the midpoint of a line]](/MindFish/resized-image.ashx/__size/550x0/__key/CommunityServer.Wikis.Components.Files/sat_5F00_preparation/6708.Midpoint.png)
![[Image: Diagram for the midpoint of a line]](/MindFish/resized-image.ashx/__size/550x0/__key/CommunityServer.Wikis.Components.Files/sat_5F00_preparation/7853.MidpointFormula.jpg)
5-1. Practice:
What is the midpoint of the points (4, 4) and (6, 8)?
-
(2, 4)
-
(1, 2)
-
(5, 6)
-
(5, 12)
-
(10, 12)
6) The Graphing Calculator
Master the graphing functionality of your calculator. Remember that you must solve
your equation in terms of y in order to enter it into your calculator.
Do not try to graph equations with y2
in them as you will only see portions of these graphs. Make sure you understand how
to change the window for your graph, as well as how to trace your graph for values
and find intercepts.
6-1. Practice:
Use your calculator to determine where the equation y = 6x
+ 3 intersects with the equation y = -2x - 5
-
(-3, -2)
-
(-1, -3)
-
(3, 6)
-
(-2, 6)
-
(-3, -1)
7) Plug In Points When Asked to Pick the Correct Graph
Frequently, challenging coordinate geometry questions can be solved by matching
points on the graphs to the equation in question. Pick an x value
and plug it into the equation in question to solve for the resulting y value.
Then use this (x, y) point to confirm or eliminate answer choices.
7-1. Practice:
Determine the equation for the parabola below:
![[Image: Diagram of a parabola]](/MindFish/resized-image.ashx/__size/550x0/__key/CommunityServer.Wikis.Components.Files/sat_5F00_preparation/2804.Parabola.jpg)
-
y = (x - 2)2 + 9
-
y = (x + 2)2 - 9
-
y = (x - 2)2 - 9
-
y = x2 - 9
-
y = x2 - 2
