Words of “Wiz-dom”- There are three final algebra areas that you need to review: coordinate geometry, graphs of functions and logs (logarithms–not the kind for which Lincoln is well known).

## Coordinate Geometry

Coordinate Geometry is a cross between algebra and geometry.  So, I’ve included it in both the algebra and geometry parts of my course.  That way, students who are skipping around in the course will be sure to get it before they get to algebra functions which requires an understanding of coordinate geometry.

### Calculating Slopes:

#### Words of “Wiz-dom” on Basic Coordinate Geometry

What is the special shape of the lines formed by the coordinates at C, D, and F? Answer  #1

What is the perimeter of CDF? Answer  #2

What is the area of CDF? Answer  #3

What is the special shape of the lines formed by the coordinates at C, D, and A? Answer  #4

What is the perimeter of ACD? Answer  #6

What is the area of ACD? Answer  #7

## Sample Questions

14. For line PQR, P has coordinates (2,4) and R has coordinates (2,12). If the ratio of the lengths of PQ to QR is 3:1, what is y-coordinate of Q?

7.  The Wizard’s owl, Orville, flew 5 miles due east from the Castle of Wiz-dom and then turned due north and went 12 more miles. What is the shortest distance in miles back to the castle?

(A) 8.5
(B) 12
(C) 13
(D) 15
(E) 17

15. Phalange, my falconer, took Beak ‘N Talons, my favorite hunting falcon, out for some exercise. When released, he flew 300 meters west and then turned due north and flew another 400 meters. At that point, he abruptly soared 1200 meters straight up! When he reached that high point, what is the shortest flight in meters for Beak ‘N Talons back to the falconer?

(A) 500
(B) 1,200
(C) 1,250
(D) 1,300
(E) 1,500

## Graphing Functions

Words of “Wiz-dom”–Now things get a little tricky. You need to apply algebraic skills to drawing graphs on a coordinate grid.

Equation of a Circle-The standard equation is . The circle will have its center at (a,b) with a radius of r. The tricky part is the negative signs for the coordinate locations. For example, is a circle with a radius of 4 that has its center at (3,-2). Be sure that you reverse the signs of the x and y coordinates from the equation!

What would be the equation for a circle with a radius of 6 that has its center at (-6,6)? In what quadrant would the center of the circle be located? What would be the slope of the line from its center to the origin of the coordinate grid?

Answer to equation of a circle

Equation of a Parabola-The standard equation is the y-intercept (set x=0, there may be two intercepts),
what coordinate point could fall on the line (what pair of coordinates is a solution for the equation?), or
the x-intercept (set y=0 and hope you can factor it. Keep in mind that there can be two intercepts.)
What is a “parabola” anyway? Officially it’s the shape you get when you cut through a right-circular cone parallel to one of its sides. A cross-section of a concave mirror, and the reflectors of flashlights and car headlights also are examples. It looks like an arch that can be right side up, upside down or even turned on its side. The St. Louis Arch is a good example. There’s even a famous fast-food parlor that has two “golden” ones! Of course, no self-respecting wizard ever eats there. (They don’t allow me to bring my dragon.)

• If the value of a is positive the parabola opens up
• If the value of a is negative the parabola opens down
• The greater the value of a, the greater the slope and the narrower the parabola
• The maximum/minimum is the high/low point of the parabola (vertex)
• The x-coordinate of the vertex equals -b/2a
• The y-coordinate of the vertex equals (b2-4ac)/4a
• ƒ(x) = a(x-h)2 +k is referred to as the vertex form where h and k are the x- and y-coordinates respectively.

Using the equation y=x2 + 4x + 3, determine: y-intercept, the y coordinate of (1,y) and x-intercept.

Answers to x and y intercepts and the y coordinate

Equation of an Ellipse-The standard equation is The most important thing to know is how to use all these values to determine attributes of the ellipse. Let’s use an example: The x intercepts will be at the square root of a2 or BOTH positive and negative 3 (y will be 0) and the y-intercepts will be at positive and negative 2 (the square root of b2). The foci (the two interior focal points) will be on the x-axis at (0,c) and (0,-c) where (Notice it is a2 MINUS b2.) In this case, they are (0, ). Keep in mind the major (long) axis of the ellipse is 2a (or 6) and the minor (short) axis is 2b (or 4).

## Sample Questions

1. In a standard (x,y) coordinate plane, what is the area of the smallest rectangle that will surround a figure defined by ?
A. 9
B. 20
C. 80
D. 200
E. 400

2. In the standard (x,y) coordinate plane, how many times does the line for the equation intercept the positive x axis?
A. 0
B. 1
C. 2
D. 3
E. more than 3 times

3. In the standard (x,y) coordinate plane, how far is the center of the figure defined by from the origin?
A. 3
B. 4
C. 5
D. 6
E. 7

4. What point on the graph of has an x-coordinate of -3?
A. (-3,2)
B.
C. (-3,3)
D. (-3,4)
E. (-3,27)

5. The graph of which of the following equations is perpendicular to the graph of ?
A. y+x=7
B. 2y-2x=1
C. 2y=x+1
D. y=x+.5
E. 2y=2x+2

6. In the standard (x,y) coordinate plane, if the y- coordinate is 3 more than four times the x-coordinate, the slope of the line is:
A. -4
B. -3
C. -3/4
D. 3
E. 4

7. In the standard (x,y) coordinate plane, which of the following equations could pass through the origin and (-3,-4)
A. B. C. D. E. ## Logs

By definition, if and only if In English (Pillar II: Restating) what this means is that the log of a number raised to a power (d) is equal to c. For example, , because . Since there are three variables involved, the test writer can give you any two of them and ask you to calculate the third.

## Sample Questions

1. If , then y=?
A. 0
B. 3
C. 5
D. 15
E. 25

2. If , then ?
A. 3
B. 4
C. 9
D. 81
E. 243

3. If k, m and n are positive integers,  , then could equal