Coordinate Geometry
Coordinate Geometry
The new SAT has stepped up the emphasis on coordinate geometry, which is the study of geometric shapes on the coordinate plane. If you think the coordinate plane is a ferocious type of new jet fighter, don’t worry. We’re about to clear this all up for you.
The Coordinate Plane
The coordinate plane is where all the magic happens. It’s the space in which coordinate geometry exists. Pretty snazzy.
Every point on a coordinate plane can be mapped by using two perpendicular number lines. The x-axis defines the space from left to right. The y-axis defines the space up and down. And the two meet at a point called the origin.
Every point on the plane has two coordinates. Because it’s the center of the plane, the origin gets the coordinates (0,0). The coordinates of all other points indicate how far they are from the origin. These coordinates are written in the form (x, y). The x-coordinate is the point’s location along the x-axis (its distance either to the left or right of the origin). If the point is to the right of the origin, the x-coordinate is positive. If the point is to the left of the y-axis, the x-coordinate is negative.
The y-coordinate of a point is its location along the y-axis (either up or down from the origin). If the point is above the x-axis, its y-coordinate is positive, and if the point is below the x-axis, its y-coordinate is negative. So the point labeled (2,2) is 2 to the right and 2 above the origin. The point labeled (–7,–5) is 7 to the left and 5 below the origin.
Are We There Yet? Distance on the Coordinate Plane
The SAT tests to see if you can find the distance between any two points on the coordinate plane. It also tests to see if you can find the midpoint between two points on the test. This news should make you happy. Why? Because these are easy questions that you can answer easily as long as you know the two necessary formulas. Now we’re going to make sure you know those two formulas. Memorize them.
The Distance Between Two Points
If you know the coordinates of any two points—we’ll call them (x1,y1) and (x2,y2)—you can find their distance from each other with the aptly named distance formula:
Let’s say you were suddenly overcome by the desire to calculate the distance between the points (4,–3) and (–3,8). Just plug the coordinates into the formula:
Finding Midpoints
As for the midpoint between the two points (x1, y1) and (x2, y2), the formula to use is
In other words, the x- and y-coordinates of the midpoint are the averages of the x- and y-coordinates of the endpoints. To find the midpoint of the points (6,0) and (3,7),
Lines in the Coordinate Plane
You already know that a line is just an infinite set of points arrayed in a straight formation. But once you stick one of those “infinite set of points” into a coordinate plane, it has all sorts of properties you can analyze. And the SAT will make sure you know how to analyze ’em.
The Slope of a Line
A line’s slope is a measurement of how steeply that line climbs or falls as it moves from left to right. If you want the technical jargon, slope is a line’s vertical change divided by its horizontal change. Or, if you prefer the poetic version,
Slope is “the rise over run.”
If you’ve got two points on a line, once again (x1, y1) and (x2, y2), the slope of that line can be calculated using the following formula:
The variable most often used to represent slope is m.
So, for example, the slope of a line that contains the points (–2, –4) and (6, 1) is
Positive and Negative Slopes
The slopes of some lines are positive, the slopes of others are negative. Whether a line has a positive or negative slope is easy to tell just by looking at a graph of the line. If the line slopes uphill as you trace it from left to right, the slope is positive. If a line slopes downhill as you trace it from left to right, the slope is negative. Uphill = positive. Downhill = negative.
You can get a sense of the magnitude of the slope of a line by looking at the line’s steepness. The steeper the line, the greater the slope; the flatter the line, the smaller the slope. Note that an extremely positive slope is larger then a moderately positive slope, while an extremely negative slope is smaller then a moderately negative slope.
Check out the lines below and try to determine whether the slope of each line is negative or positive and which has the greatest slope:
Lines a and b have positive slopes, and lines c and d have negative slopes. In terms of slope magnitude, line a > b > c > d.
Slopes You Should Know by Sight
There are certain easy-to-recognize slopes that it pays to recognize by sight. Knowing how to pick them out instantly will save you precious time.
  • A horizontal line has a slope of zero. Since there is no “rise,” y2y1 = 0, m =
  • A vertical line has an undefined slope. In this case, there is no “run,” and x2x1 = 0. So, m = and any fraction with zero in its denominator is, by definition, undefined.
  • A line that makes a 45º angle with a horizontal line has a slope of either 1 or –1, depending on whether it’s going up or down from left to right. In this case, the rise equals the run: y2y1 = x2x1, or y2y1 = –(x2x1).
Of the four lines pictured below, which has a slope of 0, which has a slope of 1, which has a slope of –1, which has an undefined slope?
Line a has slope 0 because it’s horizontal. Line b has slope –1 because it slopes downward at 45º as you move from left to right. Line c has slope 1 because it slopes upward at 45º as you move from left to right. Line d has undefined slope because it is vertical.
The Slopes of Parallel and Perpendicular Lines
The slopes of parallel and perpendicular lines always have the same relationships.
  • The slopes of parallel lines are always the same. If one line has a slope of m, any line parallel to it will also have a slope of m.
  • The slopes of perpendicular lines are always the opposite reciprocals of each other. A line with slope m is perpendicular to a line with a slope of 1 /m.
In the figure below, lines q and r both have a slope of 2, so they are parallel. Line s is perpendicular to both lines q and r, so it has a slope of 1/ 2. .
Equation of a Line
Coordinate geometry is actually where algebra and geometry meet. Coordinate geometry allows you to “graph” algebraic equations. For the new SAT, you need to know the equation of a line and how to graph that equation. The equation of a line is
where m is the slope of the line, and b is the y-intercept of the line (the y-coordinate of the point where the line intersects the y-axis). As long as you know the slope of the line and the y-intercept, you can write the equation of the line.
To sketch a line whose equation you know, first plot the y-intercept, and then use the slope of the line to plot another point. Connect the two points to form your line. The figure below graphs the line y = –2x + 3.
Since the y-intercept is at 3, the line crosses the y-axis at (0, 3). And since the slope is equal to –2, the line descends two units for every one unit it moves in the positive x direction. In other words, from (0,3), the line moves one unit to the right and two units down, to point (1,1). You could graph the line using those two points.
Finding the x- and y-Intercepts of a Line
The y-intercept of a line is the y-coordinate of the point where the line intersects the y-axis. The x-intercept of a line is the x-coordinate of the point where the line intersects the x-axis. You can find either the x- or y-intercept of a line by using the slope-intercept form of the line equation.
Finding the intercepts is very straightforward. To find the y-intercept, set x = 0 and solve for y. To solve for the x-intercept, set y = 0 and solve for x. For example, if you’ve got the line equation y = –3x + 2, the y-intercept equals 2, since y = –3(0) + 2 = 2. To find the x-intercept, set y = 0 and solve:
Parabolas Attack! Quadratic Equations Invade the Coordinate Plane
When a quadratic equation is graphed on the coordinate plane, the result is a parabola, which is a giant man-eating insect. Actually, it’s a harmless, U-shaped curve that can open either upward or downward.
If the SAT covers parabolas at all, it’ll most likely do one of these things:
  1. Give you an equation and ask you to choose which graph matches the equation.
  2. Give you a graph and ask you to choose which equation matches the graph.
You can answer either of these questions as long as you can read the quadratic equation to identify the location of a parabola’s vertex and determine whether the parabola opens upward or downward. Here’s how to do it.
The equation for a parabola looks like this:
where a, b, and c are constants. By plugging a, b, or c into the correct formulas, you can figure out where the vertex is and whether the parabola opens upward or downward:
  1. The vertex of the parabola is located at point ( b/2a, c b2/4a).
  2. The parabola opens upward if a > 0, and downward if a < 0.
So, if you’re given the quadratic equation , you know that the parabola opens upward, since a > 0. And you could figure out the vertex by plugging in. The x-coordinate would be
And the y-coordinate would be
Put it all together, and you’ve got a parabola that looks something like this:
Of course, if you had a graphing calculator, you wouldn’t have to go through any of this calculation at all. This is one of the many reasons we steadfastly demand that you have a graphing calculator and know how to use it for the new SAT.
Transformations
There’s just one more bit of coordinate geometry you have to know for the new SAT: how slight changes to a function change the way that the graph of that function looks in the coordinate plane. There are two different kinds of transformations you have to know how to deal with: shifts and stretches.
Shifts
Imagine a graph. No, better yet, look at this graph:
It’s a pretty simple graph: a parabola that has a vertex at the origin. To put it into math, f(x) = x2. A shift of this graph would occur when the parabola remains exactly the same shape but is shifted over either vertically or horizontally so that the vertex no longer rests on the origin.
To get a vertical or horizontal shift, you have to do slightly different things, but each type of shift has one thing in common: addition or subtraction.
Horizontal Shifts
To get a horizontal shift, in which the graph moves either to the left or right, you use addition. But this time, you need to add within the parentheses. So, if you want to move the graph of the function f(x) two spaces to the right, you make the function f(x + 2). If you want to move it four spaces to the left, you make the function f(x – 4).
Vertical Shifts
Vertical shifts are extremely easy. If you want the image to shift up two spots, just add the number 2 to it. If you want it to shift down four spots, subtract the number 4. So, an equation of a parabola that is two spaces above the origin would look like this: f(x) + 2 = x2 + 2. And an equation that’s four spaces below would look like this f(x) – 4 = x2 – 4.
Stretches
Imagine a graph. No better yet, look at that same example we showed you before.
Stretching a graph makes it either fat or thin.
A graph stretches when a function is multiplied, whether that multiplication is 3f(x) or f(3x). If a function is multiplied by a number greater than 1, it gets taller and thinner, while if it is multiplied by a number less than 1, it gets stubbier and wider.
That’s all the SAT covers on geometry. Just one more SAT Math chapter to go. It’s got the longest name—Data, Statistics, and Probability—but it’s the shortest of the bunch. Almost there. . . .
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