1
MATH 105 Test 1 (Sec: 1.1 - 1.6) Spring 2015
Sec Number:
Name: KEY
1. (4 pts)Write the equation of a circle that has a center at the point (6, −4) and a radius of 14
units.
196 = (x − (6))2 + (y − (−4))2
2. (6 pts) Find the domain of the following functions. Write your answer using interval notation,
as well as in set notation.
a) f (x) = x3 −15x
b) g(x) =
6x + 2
12 − x
c) h(x) =
√
10 − x
For f :
x ∈ (−∞, ∞) or {x| − ∞ < x < ∞}
For g :
x ∈ (−∞, 12) ∪ (12, ∞) or {x|x 6= 12}
For h :
x ∈ (−∞, 10] or {x| − ∞ < x ≤ 10}
3. (10 pts) Find the equation of a line containing the points (8, 12) and (17, 8). Write your
answer in slope-intercept form to get full credit. Let m denote the slop of the line.
m=
y2 − y1
8 − (12)
−4
4
=
=
=−
x2 − x1
17 − 8
9
9
Next use point slope form of a line to get the linear equation:
4
(y − y1 ) = m(x − x1 ) =⇒ y − (12) = − (x − (8))
9
Simplify to slope-intercept form:
=⇒
=⇒
1
Using SAGE seed 4030
4
y − (12) = − (x − (8))
9
4
32
y =− x− −
+ (12)
9
9
4
140
y =− x+
9
9
4. (6 pts) Determine if the following pairs of lines are parallel, perpendicular, or neither.
• The lines y = 8x + 5 and y = −8x − 12 . neither
• The lines y = 2x − 8 and y = −2x + 8. neither
• The lines y = 2x + 7 and y = 2x − 53 . parallel
• The lines y = 2 and x =
−5
8 .
perpendicular
• The lines y = −4x + π and y = 14 x − 4. perpendicular
• The lines y = 6x + 3 and y = −7x + 4. neither
5. (10 pts) Find the intercepts and graph the line: 3y + 6x = 3
Set x = 0 and solve for y to get the y-intercept,
3y = 3 =⇒ y =
10
3
3
Set y = 0 and solve for x to get the x-intercept,
5
3
6x = 3 =⇒ x =
6
x-intercept =
y-intercept =
-10
3
6
−2
5
10
-5
3
3
-10
What is the slope of the given line?
Slope =
-5
3y + 6x = 3 =⇒ y =
−6
x+1
3
6. (10 pts) James Bond invested $10000, a portion earning a simple interest rate of 4 percent
per year and the rest earning a rate of 7 percent per year. After one year the total interest
earned on these investments was $535.0. How much money did he invest at each rate?
Let x be the amount invested at 4 percent. Then 10000 − x is the amount invested at 7.
0.04x + 0.07(10000 − x) = 535.0
−0.03x = 535.0 − 700.0
x = 5500.0
James invested $4500.0 at 7 percent and $5500.0 at 4 percent.
7. (10 pts) Determine the distance between the points (7, 6) and (−5, −3). Idea: use Pythagorean
theorem (a2 + b2 = c2 ) to find the length of the hypotenuse.
(x1 − x2 )2 + (y1 − y2 )2
d =
1
2
((7) − (−5))2 + ((6) − (−3))2
1
= (12)2 + (9)2 2
√
=
225 ≈ 15.0000000000000
=
1
2
8. (4 pts)Find the equation of a line that is perpendicular to the line y =
1
5x
+ 8 that has a
y-intercept of −2.
y=
−5
x + (−2) = −5x + (−2)
1
9. (6 pts) For each of the following plots determine whether the graph is that of a function.
Answer the question by writing “Is a Function” or “Not a Function” on the appropriate line.
A
B
C
• A.
Is a Function
• B.
Is a Function
• C.
Not a Function