Algebra: Three Additional Areas

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.

Words of “Wiz-dom”-Coordinate Geometry sounds worse than it is. First, you need to compute distances on horizontal and vertical number lines and their diagonals. No problem. It’s as simple as subtracting numbers and using the Pythagorean Theorem (a2+b2=c2). Second, you have to calculate “slope.” If you can subtract and divide, you can do this too. Let’s get started with a quick review. Always be on the lookout for “hidden” right triangles.

Whenever you determine the position of a coordinate point, be sure you remember that the x position is written before the y position (x,y). Also keep in mind each quadrant has its own signs. Starting with Quadrant I at the top right, the signs are (+,+) and going counter-clockwise to Quadrant II (-,+), to Quadrant III (-,-), and finishing in Quadrant IV (+,-).

Calculating Distances:

  • For horizontal lines, subtract the x coordinates.

  • For vertical lines, subtract the y coordinates.

  • For “diagonal” lines, treat it as the hypotenuse of a right triangle.

Calculating Slopes:

“Slope” tells you how fast a line went up or down as it is drawn from left to right. There are two special cases: Zero (0) Slope means it’s a horizontal line (It doesn’t go up or down!) and “undefined” slope means it goes straight up and down. All other slopes are calculated by dividing how far the line goes up or down (change in y) divided by how far the line goes from left to right (change in x). This is sometimes called the rise (or fall) divided by the run. If the line is going up, it’s a positive slope. If it’s going down, it’s a negative slope.

What are the coordinate positions of:
A
B
C
D

What are the lengths of:
AC
CF

What are the slopes of:
AC
CD
CF

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 slope of AD? Answer  #5

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?

Answer to Question #14

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

Answer to Question #7

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

Answer to Question #15

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

You might get asked for:

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.)

Here are some additional things to remember about a quadratic equation:

  • 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

Answer to question Graphing #1

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

Answer to question Graphing #2

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

Answer to question Graphing #3

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)

Answer to question Graphing #4

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

Answer to question Graphing #5

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

Answer to question Graphing #6

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.

Answer to question Graphing #7

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

Answer to question Logs #1

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

Answer to question Logs #2

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

A. 4
B. 8
C. 32
D. 64
E. 128

Answer to question Logs #3

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