Plot the Points
Coordinate geometry, also called Cartesian geometry, uses a coordinate system to plot points in a plane. A plane is a flat surface that, hypothetically, extends into infinity. The plane contains both an x-axis (horizontal) and y-axis (vertical), which divides the plane into four quadrants. When plotting points, your plane will extend only within the range of numbers of which you will be working.
On the exam, you will be tasked with working linear equations (distance between points, slope of a line, and slope-intercept form) and quadratic equations (quadratic formula). Yes, you will be applying your algebra skills to geometry!
Navigating the Coordinate System
On the plane below, there are coordinates of (-4, -2) and (3,3). Each set is referred to as an ordered pair. The placement of the points are easy to read. The point where the x-axis and y-axis intersect, is called the origin and the origin’s coordinates are (0,0). To plot points, determine the placement on the x-axis first, then the y-axis. For (-4,-2), move to the left of the origin until you reach -4. From the -4 point move down until you reach the line representing -2 on the y-axis. Do the same for (3,3) and connect the two ordered pairs by drawing a line.
Linear Equations
As the name implies, these equations deal with straight lines only. Other than being a straight line, the other distinction of linear equations is the line will only cross the x-axis once. This is important to note when we get to the quadratic equation.
Distance Between Points
For this formula, you will subtract the x coordinates from each other and the y coordinates from each other. Each of those differences are squared and then added together. Finally, find the square root of the sum to get the distance between points.
Let’s use the coordinates shown above. When choosing which coordinates will be x₁ ,y₁ , x₂ , and y₂ , it doesn’t matter which set you choose as long as you are consistent. For example, don’t set up x as (3 – (-4)), then y as (-2 – 3). Stay consistent when setting up your formula.
Note: Technically, we could have broken down the square root of 74 further, but since it is not a perfect square, you don’t have to worry about breaking it down further for the exam or if a calculator is allowed for your exam you can calculate the answer to the decimal.
Slope of a Line
The slope of the line is quite literally the direction of the line. The formula is the rise over run. Subtract x₁ from x₂ and y₁ from y₂ like you did for the distance between points formula. Next, divide the difference of y by the difference of x. That is it! You now have the slope of the line.
Slope-Intercept Form
The slope-intercept form tells you everything you need to know about the line. It contains the coordinates, slope, and y-intercept.
y = mx + b
- y and x = coordinates
- m = slope
- b = y-intercept
Using the coordinates from the graph above, let’s plug in what we know to get the y-intercept. We found the slope of the line in the last formula, ⁵⁄₇, so plug that into m. Next, plug in a set of ordered pairs (either set) to the formula, x = -4 and y = -2 or x = 3 and y = 3. With all of the known variables plugged in, you are now ready to calculate:
-2 = ⁵⁄₇(-4) + b or 3 = ⁵⁄₇(3) + b
Work either of these two problems and the answer is b = ⁶⁄₇ so the slope-intercept formula for the graph above is: y = ⁵⁄₇x + ⁶⁄₇. This can also be written in decimal form: y = .7143x + .8571.
What if we were given a problem to plot points using the slope-intercept formula without having all the coordinates? In none are given, you can find x by subbing 0 for y, then find y by subbing 0 for x (y will always equal the y-intercept when subbing 0 for x).
You may be asked to plot several points using the slope-intercept formula. In this case, you will probably be given either x or y. Let’s check out a problem.
Find the y-coordinates using the following information. To do this, you plug the x-coordinate given into the slope-intercept equation to find the corresponding y-coordinate. The numbers in red represent the answers. Note the graph beside it. See how each ordered pair plotted out formed a line?
Quadratic Equations
A quadratic equation plotted out will produce a symmetrical U-shape called a parabola. The parabola isn’t always an upright U-shape. It can be upside down or sideways (in this instance, the formula slightly differs from what we will cover). Other than its distinct shape, also note that a quadratic equation can intersect the x-axis twice, unlike a linear equation, which will only intersect the x-axis once.
The point of intersection with the x-axis is called the root. There is more than one way to determine if the parabola has one, two, or even no roots at all! For our problem-solving purposes, we will stick with the quadratic formula.
- If b² – 4ac < 0, then there are no real roots meaning that the parabola does not touch the x-axis.
- If b² – 4ac = 0, then the parabola will only touch the x-axis once so it will look like the second example of the parabolas above where the parabola is contained to one quadrant of the plane.
- If b² – 4ac > 0, then the parabola has two roots and will intersect the x-axis twice.
What does a quadratic equation look like?
No worries about confusing one with a linear equation. A quadratic equation is set up as: y = ax² + bx + c. The three variables and the square let you know right away that this is not linear. Let’s work a problem!
Solve: y = 2x² + 5x + (-3). Take the numbers representing a, b, and c in the equation and plug them into your formula. The ± means that you will both add to and subtract the square root from -b resulting in two answers when b² +4ab > 0. The two exceptions are when the square root is a negative number, b² + 4ab < 0 (look back at square roots if you don’t remember why a negative number can’t have a square root), and when b ² + 4ab =0, which produces only one root.
By solving this formula, we know that the parabola is going to intersect the x-axis twice: .5 and -3. This formula is quite tedious to work, but it is very straightforward.