Week 5 Day 2 (Analytic Geometry; Slope-Intercept; Algebraic Roots & Radicals; Functions)

Analytic or Coordinate Geometry p.143

The Cartesian plane, also known as the coordinate graph, has two number lines: the x-axis, which indicates horizontal position, and the y-axis, which indicates vertical position. These two axes give values in ordered pairs: the x-coordinate is also called the abscissa and comes first in the pair, and the y-coordinate is also called the ordinate and comes second in the pair.

We also divide the graph into four quadrants: I, II, III, IV.

In other words:

I: (+x, +y)

II: (-x, +y)

III: (-x, -y)

IV: (+x, -y)

Graphing Equations p. 145

When graphing an equation, we usually plug in values from {…-2, -1, 0, 1, 2…} for x and solve for the y value, and then graph the dots on the coordinate plane. A linear equation has no exponents; non-linear equations have exponents on the variables and may graph as parabolas or other conic sections.

A linear graph may look something like this:

A non-linear graph may look something like this:

 

Slope and Intercept p. 154

Linear equations have a slope and intercept points. The slope is labeled m; usually we care about the y-intercept (labeled b in a formula we will encounter shortly), but linear equations often have an x-intercept as well. The slope-intercept form of any linear equation is given in the following format: y = mx + b. Given this format and any x-value, we can solve for the y-values. Given two ordered pairs, we can solve for m and b, as well. Slope can be calculated using the ordered pairs as follows: if one ordered pair is (x₁, y₁) and the other is (x₂, y₂), then the slope or m is equal to the difference between the y-values divided by the difference between the x-values. This is also called “rise over run,” or in other words, the vertical change over the horizontal change.

Note that a positive slope (+m) will tilt to the right like a backslash: /

And a negative slope (-m) will tilt to the left like a forward slash: \

Graphing with Slope-Intercept p. 157

1. First re-write the equation in slope-intercept form if necessary.

2. Then locate the y-intercept; x will always be zero at this point (x, b).

3. Write the slope as a ratio (fraction) to locate the other points (usually two or three is enough).

4. Draw the line through these points.

 

Graphing with X- and Y-Intercepts p. 158

If you have both the x- and y-intercepts, you have two ordered pairs and can graph a line that goes through each of these points.

1. Find the x-intercept by replacing y with 0 and solving for x.

2. Find the y-intercept by replacing x with 0 and solving for y.

3. Draw a line through these points.

 

Finding an Equation of a Line p. 159

If you have any two ordered pairs or one ordered pair and the slope, you can find the equation of any line.

1. Find the slope from the two points using the slope equation for rise over run. [If the slope is given, you can go straight to step 2.]

2. Find the y-intercept by plugging in the slope and one ordered pair into y = mx + b and solving for b.

3. Write the equation in the slope-intercept format with the discovered slope and y-intercept as m and b.

 

Roots and Radicals p. 163

These were introduced in arithmetic, but now we have to see how they work in algebra. Note that when two radical signs are placed next to each other, they are multiplied.

 

Simplifying Square Roots

Note that a lack of any sign indicates a positive value, that a negative outside the radical indicates the result will have a negative value, and that a negative sign inside the radical will produce imaginary numbers for square roots. Imaginary numbers are not part of the GRE or GMAT.

 

Operations with Square Roots p. 165

Only perform operations from underneath the radical; we can only add or subtract roots when they are equal. Always simplify where possible, because this will mean you can then add or subtract as necessary. For multiplication, when both values are non-negative and under separate radicals, we can multiply and then simplify. Always simplify if you can!

 

“False” Operations (a.k.a., Functions) p. 167

The GRE and GMAT both use false operations which include strange symbols set equal to some algebraic expression and then ask you to solve an operation with some given values. The strange symbols I’ve seen include: @ # $ * §¨©ª†♫«■●▲ and other strange shapes I cannot even find in the symbol library. All of these have the same meaning as f(x) or g(x); that is to say, they are all just functions. So if you see x @ y = x ´ 3y, and you want to solve for 3 @ 4, then you simply plug 3 in for x, and 4 in for y, and then solve.

 

Remember that functions can nest together like f(g(x)) or f ° g, so you might see a pair of symbolic functions where you have to solve them from the inside out. We will cover these in the next level of GRE/GMAT Math.