MATH A105 Intermediate Algebra
Exam 4
Instructions
1. Do NOT write your answers on these sheets. Nothing written on the test papers will be graded.
2. Do NOT write your name on any of your answer sheets.
3. Please begin each section of questions on a new sheet of paper.
4. Do not write problems side by side.
5. Do not staple test papers.
6. Limited credit will be given for incomplete or incorrect justification.
Questions
1. Solving
x
(a) (6) 2x
4x
+2y
+y
+5y
+2z
−z
+4z
=
=
=
2.
0.
3.
−2(
x
−2x
+2x
+2y
−4y
+y
−3y
+2z
−4z
−z
−5z
)
=
=
=
=
−2 (2)
−4
0
−4
−4(
x
−4x
+4x
+2y
−8y
+5y
−3y
+2z
−8z
+4z
−4z
)
=
=
=
=
−4 (2)
−8
3
−5
−3y
+3y
−3y
−5z
+5z
−4z
z
)
=
=
=
=
−1 (−4)
4
−5
−1
−3y
y
x +2(+3)
x
−5(−)
=
=
=
=
−4
3
2
−2
−1(
+2(−)
(−2, 3, −1)
x
(b) (6) 3x
6x
+y
−2y
+y
+z
+4z
+7z
=
3.
=
4.
= 12.
−3(
x
−3x
+3x
+y
−3y
−2y
−5y
+z
−3z
+4z
+z
)
=
=
=
=
−3 (3)
−9
4
−5
−6(
x
−6x
+6x
+y
−6y
+y
−5y
+z
−6z
+7z
+z
)
=
=
=
=
−6 (3)
−18
12
−6
−5y
+5y
−5y
+z
−z
+z
0
)
= −1 (−5)
=
5
=
−6
=
−1.
−1(
1
No solution.
(c) (4) 3x2 + 14x − 5 = 0.
3x2 + 14x − 5
=
0.
(3x − 1)(x + 5)
=
0.
3x − 1
=
0.
x =
x+5
=
x =
1/3.
0.
−5.
(d) (4) 4x2 − 12x + 10 = 0.
x
=
=
=
=
=
−(−12) ±
12 ±
√
p
(−12)2 − 4(4)(10)
2(4)
144 − 160
√8
12 ± −16
8
12 ± 4i
8
3±i
.
2
2
2. Graphing
Graph each of the following using the techniques from class. No more than two points may be plotted for a
parabola and no more than five for a circle.
(a) (5) x = −2(y + 3)2 + 5.
-10
-5
5
-1
-2
-3
-4
-5
-6
2
2
(b) (4) (x − 7) + (y + 3) = 4.
-1
-2
-3
-4
6
7
8
9
(c) (10) y = 3x2 − 24x + 43.
y
=
3x2 − 24x + 43
=
3(x2 − 8x) + 43
=
3(x2 − 8x + 16 − 16) + 43
=
3([x − 4]2 − 16) + 43
=
3[x − 4]2 − 48 + 43
=
3[x − 4]2 − 5.
40
30
20
10
2
4
3
6
8
3. Other Fun
(a) Interpret
i. The lift equation can be written as L(v) = ρKv 2 . If the density of air (ρ) is increased (e.g., from 1 to
2) what does that do for lift curve? Note K is unimportant to this question.
ρ scales the curve. The larger ρ is the steeper the lift curve (faster increase of lift).
2
ii. Centripetal force F is given by F = mv
R . If the radius R of the circle is increased (e.g., from 1 to 2)
what does this do to the centripetal force curve? Note m is unimportant to this question.
R scales the centripetal force curve. The larger R is the less steep the force curve. Bigger circles at
the same speed produce less force.
iii. The height of a bullet shot straight up in the air is given by h(t) = 250, 016 − (t − 125)2 . What does
the negative in front of the square expression do to the curve?
The negative flips the curve upside down.
iv. In the bullet height function what does the 250, 016 do to the curve?
That is the highest height (shifted the top up to that point).
(b) Determine whether each set varies linearly, quadratically, exponentially, or otherwise. (2 each)
i. Curve 1
1
2
3
4
5
x
y 2/7 13/21 20/21
9/7 34/21 This curve is linear.
∆y
7/21
7/21 7/21 7/21
ii. Curve 2
x
1
2
3
4
5
y 2/7 13/21 34/21
23/7 118/21
This curve is quadratic.
∆y
7/21 21/21 35/21 49/21
∆∆y
14/21 14/21 14/21
iii. Curve 3
1
2
3
4
5
x
y 2/7
1/7
2/21
1/14
2/35
∆y
−1/7 −1/21 −1/42 −1/70 This curve is something else.
∆∆y
2/21
1/42 1/105
÷y
1/2
2/3
3/4
4/5
iv. Curve 4
1
2
3
4
5
x
y 2/7 3/7 9/14 27/28 81/56 This curve is exponential.
÷y
3/2
3/2
3/2
3/2
4