Algebra II / Mr. Hansen

Name: _____________________________

March 2000 Practice Test on Chapter 9

 

Show all work
For full credit, show all work. You may use your calculator in any way that you wish, including study aids stored into your calculator’s memory, but you must show sufficient work on your paper to demonstrate clearly that you understand the concepts. For example, if you assert that the semimajor axis of an ellipse is 7, you should show the algebraic transformations needed to put the equation into "x – h, y – k" form showing the number 7 in the appropriate position.

What to study
The following descriptions should be studied thoroughly (memorized, if you possibly can) and will not appear on your test sheet:

1. For any quadratic relation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, the discriminant is B2 – 4AC. If the discriminant is negative, the relation is an ellipse (or circle if A = C). If the discriminant is positive, the relation is a hyperbola. Finally, if the discriminant is zero, the relation is a parabola. Memory aid: "NEC, PH, ZP." NEC is a Japanese company that makes cell phones, theater TVs, computers, etc. PH (pH) is a measure of acidity, in case you’ve had any chemistry. Finally, ZP (zap) is what you get by process of elimination.

2. Equation of circle with center (h, k) and radius r:
(x – h)2 + (y – k)2 = r2

3. Equation of ellipse with center (h, k), x-radius rx, and y-radius ry:

Note: Semimajor axis a is the larger of rx and ry; semiminor axis b is the smaller value. Focal radius c is found from c2 = a2 – b2. Foci are always located on the major axis.

4. Equation of hyperbola with center (h, k), x-radius rx, and y-radius ry:

Note: The first form gives branches that open left and right. (Memory aid: x positively reminds you of left and right.) The second form gives branches that open up and down. (Memory aid: y positively reminds you of up and down.) With hyperbolas, a is always rx, and b is always ry. Focal radius c is easier than for ellipses: c2 = a2 + b2. Asymptotes are the two lines passing through the center with slopes ± b/a.

5. Equation of parabola with vertex (h, k):
x – h = a(y – k)2 or y – k = a(x – h)2
Note: a is a constant that determines how "lean" the parabola is. Large values of a make a skinny parabola, while small values (close to 0) make a fat parabola. You may have to determine a by plugging in another known point on the parabola.

The first form [ x – h = a(y – k)2 ] opens right if a > 0, left if a < 0. (Memory aid: since (y – k)2 is always positive, x takes its cue from the sign of a. Rightward values of x tend to be positive, and leftward values tend to be negative.) The second form [ y – k = a(x – h)2 ], which we studied last fall, opens up if a > 0, down if a < 0. (Memory aid: since (x – h)2 is always positive, y takes its cue from the sign of a. Upward values of y tend to be positive, and downward values tend to be negative.)

Cued formulas
The following formulas will be printed on your test sheet. Of course, since they are missing the explanations shown above, they will be of no use to you unless you have studied the list above very carefully. Here is exactly what you will see on your test sheet:
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
B2 – 4AC
(x – h)2 + (y – k)2 = r2
, c2 = a2 – b2
, c2 = a2 + b2, asymptote slopes ± b/a
x – h = a(y – k)2 or y – k = a(x – h)2

Skill areas
You are responsible for the following skill areas, which will not be listed on the test sheet:
1. Given an equation of a quadratic relation, be able to identify the type of conic section, transform the equation into the proper "x – h, y – k" form, and make a sketch that shows any or all of the salient features: vertices, foci, x-intercepts (if any), y-intercepts (if any), x-radius, and y-radius. For hyperbolas, be able to use the x-radius and y-radius as aids in sketching asymptotes. You must be able to compute and interpret the discriminant. Note that you are not required to plot directrices. (The directrix of a conic section is a line having special properties. For example, a parabola can be defined as the set of points in a plane that are equidistant from a focus F and a directrix.)

2. Given a sketch of a quadratic relation, be able to produce its equation. In most cases, the "x – h, y – k" form will be sufficient. However, you should be able to produce the Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 form as well.

3. Given a system of quadratic and/or linear relations in x and y, be able to find the solution set algebraically as well as graphically. Remember that your answer must be expressed as Æ if there are no points of intersection, or otherwise as a set of ordered pairs.

Additional problems
If you need more practice problems on any of the skill areas listed above, please locate suitable problems in the textbook. For example, the review problems at the end of Chapter 9 are an excellent source of practice problems. If you need additional guidance, please leave a voice mail message at (703) 599-6624 (available 24 hours).

1.

Use the discriminant to identify each of the following conic sections.

 

(a)
2xy = 17

 

(b)
177x2 + 3497y2 + 51998x = 714,898

 

(c)
34x2 – 34y2 = 915

 

(d)
34x2 + 34y2 = 915

 

(e)
2x = 3y2

 

(f)
2x = 3y2 – 6y + 7

 

(g)
2y = 3x2

 

(h)
2y2 = 3x2

 

2.
Sketch (c), (d), and (f) from the previous question. Rough sketches are perfectly acceptable. For (c), label the center, foci, and y-intercepts, and sketch the asymptotes as dotted lines. For (d), label the center, x-radius, and y-intercepts. For (f), label the vertex and x-intercepts.

 

3.
Fill in the blanks below with the name of the proper conic section. No term is used more than once.

 

(a)
The set of points in a plane that are equidistant from a point F and a line (directrix) is called a(n) __________ .

 

(b)
The set of points in a plane that are equidistant from a single fixed point is called a(n) __________ .

 

(c)
The set of points P in a plane such that PF1 + PF2 (for foci F1 and F2) is a constant is called a(n) __________ .

 

(d)
The set of points P in a plane such that |PF1 – PF2| (for foci F1 and F2) is a constant is called a(n) __________ .

 

4.
Write the equation for each of the following relations. For part (a) only, write the equation in Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 form; for the other parts, any form is acceptable.

 

(a)

High and low points marked are at (–2, –3) and (–2, 7). Major axis is vertical, and focal radius is 4.25.

 

(b)

Asymptotes have slopes of 3 and –3, and they cross at the point (8, 0). The x-intercepts are at (5.5, 0) and (10.5, 0).

 

(c)

The x-intercept is at 5, and the y-intercepts are at –4 and 8.

 

(d)

Center is (2/3, –1); x-radius and y-radius are both 8/3.

 
5.
Use your calculator to plot problem 4(d) accurately. Raise your hand when you are ready to show it to your instructor. Do not worry if the last pixel at the left and right edge are skipped.

 
6.
Solve the following system graphically (recommended) or algebraically:

16x2 + 9y2 – 128x + 18y < 311
–2x + y ³ 18

 
7.
Solve the following system algebraically. (You may use your graphing calculator to check your answer.)

x2 + y2 + 6x = 16
2x2 – 3y2 = 24