Unit 7
Answer Key
1
Unit 7 Answers
CF-1.
a) x = 1 b) x = ±1
c) x = 1
CF-4.
a) x = 2, 4
CF-5.
d) These graphs are sums and products of the parents we already have.
CF-6.
a) 5; 6; 3
b) x = 3
c) x = -2, 0, 2
CF-7.b) a) 0 or 1 b) 0, 1 or 2
d) 0, 1, 2, 3, or 4 (1 and 3 require the parabola to be tangent to the circle.)
CF-10.
The second is shifted up 5 from the first.
CF-11.
a) 2 = 1
20 10
CF-12.
all are polynomials except (c), (f), (g): c) variable in exponents, f) both x and y
squared, g) variable in denominator
CF-13.
They are not equivalent--just plug in numbers. Also the second equation can be
written y = -x + 12, which is a line, not a circle.
CF-15.
a) 4n - 23
CF-16.
a) The first grid on the resource page is for this graph.
b) where the graph curves; the parts of the graph that are "cut out"
c) roots at x = -5 and x = 2
e) In this example, the function turns three times (or goes three directions: it goes
up, then down, and then up again).
CF-18.
a) 2
CF-19.
x = -1 ± 6
a) 2 b) (-1 +
c) at x ª 1.45 and x ª -3.45
CF-21.
45° -- 21.2’; 90° -- 30’; 135° -- 21.2’; 180° -- 0; 270° - -30’
a) repeat the pattern for several cycles
b) 30'
CF-22.
circle of radius 5, centered at origin, and its interior
CF-24.
a) y = (3x) - 4 b) y = 3(x-7)
CF-25.
a) 3
b) n c) yes; if any factors are repeated
d) A: min. = 3 B: min. = 2 C: min. = 4 D: min. = 5
e) b and d
b) x =
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b) 1
19
b) 2506
7, - 7
6 , 0) and (-1 -
6 , 0)
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Unit 7
Answer Key
2
CF-26.
a) They have the same general shape, but y = x4 is flatter around the origin and
rises more quickly
b) yes
c) No--it would have to turn again.
CF-27.
a) It causes the graph to be tangent at the root
(sometimes said to "bounce off").
b) It has a factor with a power of 3.
c) It has a turn similar to that of y = x3 at the root.
CF-30.
at x = -3 ±
CF-32.
a) 2
CF-33.
a) 13, 17, 21
CF-34.
a) (x - 2)2 + (y - 6)2 = 4
CF-37.
a) Isolate x (but this won't work), graph 2 equations and find pt. of intersection
using ZOOM/TRACE features on the calculator,
graph y = 2x - x - 3 and find x-intercepts, use guess and check (interpolation).
b) The solutions are ª 2.44 and ª - 2.86.
CF-38.
a) The value that satisfied both equations simultaneously.
b) The point of intersection of the lines or planes.
CF-39.
a) exponential b) linear
CF-40.
a) y =
CF-41.
(2, 0), and ª (1.1187, -3.075)
CF-42.
x = 2 or x ª 1.1187
CF-43.
a) guess and check, graphing
CF-44.
a) nowhere
b) It has no real solution, so the graph cannot cross the x-axis.
c) We get a square root of a negative.
CF-45.
b) 4 units
CF-47.
It is not: 16 + 8 π 32 - 40
CF-48.
a) x2 + y2 = 1
CF-49.
a) Both sides of the equation are undefined; it has no solution.
5
d) 6
log x
log 7
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b) Arithmetic
b) y =
c) 4n + 1
b) (x - 3)2 + (y - 9)2 = 9
c) 0, 1, or 2 d) (3, 9); (-4.97, 1.03)
log (x + 5)
log x
c) y =
-3
log 3
log 2
c) 4
d) 2
b) p
c) 2p
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Unit 7
Answer Key
3
CF-50.
(-2, 2) and (-5, -1)
CF-51.
a) 4
b) y2 + y + 13 = 25, so (y + 4)(y - 3) = 0, and y = 3 or -4.
Therefore there are four points: (± 4, 3) and (±3, -4).
c) Without the graph, you might not get all four points.
CF-52.
(-2.998, 0.126) & (1.9575, 3.884)
CF-53.
a) 6
b) (-3,0), (-2.97,0.56), (-0.49,4.33), (3.23,4.85), (2.67,-4.95), (0.55,-4.79)
CF-54.
b) y = 2
CF-55.
a) 1
CF-56.
b) SC: y = 20x + 180, AH: y = 80(1.15)x. The are equal in about 11.8 years or
7.4 years ago.
CF-57.
a) 1
2
b) 1 c) 5 d) 1 e) 1
4
6
2
16
CF-58.
a) 1
b) 2 c) 2.5
CF-59.
a) y = 6x + k where k π -7
b) y = a(x - 4)2 + 3 where a π 0
c) y = a(x - 2)(x + 5) where a π 0
CF-60.
circles with center to (3, 5)
CF-61.
circles with radius 3
CF-62.
circles with radius 4, centers lying on a line 5 units above the x-axis
CF-64.
AC = 10 inches
CF-65.
a) The two graphs do not intersect, therefore there is no point of intersection.
CF-66.
Likely answers would be "we had no solution" or "it could not be solved". Ask
them why it could not be solved. (What algebraic problem tells them that it
cannot be solved?) The key here is to have them recognize the problem of
taking the square root of a negative.
CF-67.
x=± 3
a) square root
d) ± 2 , ±1.4142
CF-68
a) We need to take the square root of a negative.
CF-69.
a)(2i)2 = 4i2 = -4, or
b) y c) x
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d) 1
f) 4 - p
4
b) two
- 4 = 4i 2 = 2i
c) two
b) 4i
b) 6i2 = -6
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Unit 7
Answer Key
4
c) 4i2 = -4, -4(-5i) = 20i
d)(5i)2 = 25i2 = -25
CF-73.
a) -18 - 5i
b) 1 ± 2i
CF-74.
1
CF-75.
x = -8
CF-76.
Yes--both are x2 - 10x + 25.
CF-77.
i) 7
ii) 18.3
CF-78.
a) 7i
b)
CF-79.
b)
CF-80.
a) four points: (5, 0), (0, 5), (-5, 0), (0, -5)
b) (4, 3), (3, 4), (-3, 4), (-4, 3), (-4, -3), (-3, -4), (3, -4), (4, -3)
CF-81.
a) -1 and-5
b) Just one at x = -3
c) x = -3 ± 2i; It doesn't--hence the imaginary roots.
CF-82.
1 = -x + 1 means 1 = -x2 + x so x2 - x + 1 = 0. This has complex roots,
x
therefore the graphs do not intersect. x = 1 ± i 3
2
CF-83.
no
CF-84.
Answers might include all equations had 2 complex roots and the roots were the
same except for the middle sign.
CF-85.
a) 5
CF-86.
b) 2 - 7i
CF-87.
a) -21
CF-88.
a) adds 2; mult. by 3; sq. root; subtr.1
2
b) f -1(x) = x - 3 + 1; g-1(x) = 3(x + 2) - 1
2
CF-90.
a) log
b) (2, 0) is the x-intercept of the asymptote. This locator point
will not change if there is a stretch factor. (3, 0), which is the shifted x-intercept
for the function is also acceptable.
c) y = log2(x - 2) is one possibility.
iii) d
2i
a) -1
b) -1 c) yes
d) -27i
x- 2 + 3
a) -5 ± 3i
b) 7 ± 2i
c) 2 ± i 5
c) 17 d) 53 f) a - bi
c) 3 + 5i
b) -10 + 7i
(
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)
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Unit 7
Answer Key
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CF-91.
a) 1
16
CF-93.
a) y = | x |
b) absolute value graph; squaring and then square rooting will
always return a positive value
CF-93.
a) y = x(x + 3)(x - 2)
CF-94.
The second graph is a vertical stretch of the first.
CF-95.
y = (x + 3)(x + 1)(x - 2)2
b) a = 2
CF-96.
a) min.: 4th degree
c) y = 0.64x(x - 2)2 (x - 3)
d) About 181 feet.
CF-97.
a) y = -2(x + 2)2 (x - 2)
CF-98.
when (x2 - 4) = 32, or x = ± 6, and y = 1
2
CF-99.
(x + 4)2 + (y - 2)2 = 16 or (x - 4)2 + (y - 2)2 = 16
b) y = -(x + 2)2(x - 1)
c) y = -x3(x + 3)(x - 2)
a) It would stretches the graph vertically.
b) 0, 2 and 3 (2 is a double root.)
b) y = - 3 (x + 2)2(x - 1)2
4
CF-101. b ≥ 20 or b £ -20
b) h(x) = (x + 2)3 - 7; h-1(x) =
CF-102. adds 7; cube rts; subtract 2
CF-103. c) C: (-9, 4), r: 5 2
CF-104. i) 2
ii) 3
iii) 0
3
x+7 -2
d) C: (3,0), r=1
iv) 1
CF-105. b)It changes direction twice or not at all, so one end goes up and the other goes
down and it must cross the x-axis.
CF-106. 1, 2 or 3--this one actually has just 1; a cubic has to cross at least once. Zero is
not a possible number of x-intercepts.
1± i 3
a) x = 2,
, one real, two non-real b) same as (a)
2
CF-107. a) x = ± i, therefore it has no real roots and cannot cross the x-axis.
b) We are looking for: one repeated linear factor gives one real root, two
different linear factors give two real roots, the quadratic that cannot be factored
gives two non-real roots.
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v. 5.0
Unit 7
Answer Key
6
CF-108. a) three real liner factors (one repeated), therefore 2 real (1 single, 1 double) and
0 non-real roots
b) one linear and one quadratic factor, therefore 1 real and 2 complex (non-real)
roots
c) four linear factors therefore 4 real, 0 non-real roots
d) two linear and one quadratic factor, 2 real and 2 complex (non-real) roots
CF-109. e) a: 5 b: 5 c: 4 d: 6
CF-112. a) R
b) C c) C d) R e) R f) C
CF-113. b) Since the graphs do not intersect, the system has no real number solution.
CF-114. b) b2 - 4ac = 49; real
CF-115. a) repeat 1, i, -1, -i, etc.
CF-116. a) 1
c) -1
CF-117. If n is a multiple of four the value is 1, if it is one more than a multiple of four
the value is i, if it is two more than a multiple of four the value is -1, if it three
more than a multiple of four the value is -i.
CF-119. a) x: - 25 , 0, 72 ; y: 0
CF-120. a) x2 - 6x + 9
b) x: -3,
15
2
(dbl root); y: 675
c) a3 - b3
CF-124. Using inches, the smallest cut might be a square with half-inch sides, largest
would be four-inch sides.
a) The tank cannot be made; the sides will not meet evenly.
b) A very common conjecture is, the larger the cutout, the smaller the volume.
c) for inches: length = 11 - 2x, width = 8.5 - 2x; for centimeters: length = 27.9 2x, width = 21.6 - 2x
d) V = x(11 - 2x)(8.5 - 2x) in3 or V = x(27.9 - 2x)(21.6 - 2x) cm3
CF-125. b) for the model: V = 66.1 in3, h = 1.6 in, w = 5.3 in, l = 7.8 in or V = 1083 cm3, h
= 4.05 cm, l = 19.8 cm, w = 13.5 cm
CF-126. a) 0" < x < 4.25"
b) The length and width decrease as the height increases.
CF-127. about 980,000, with the cost over $19,600
CF-128. a) The new tank will be 1.2m x 5.85m x 3.98m.
b)About 367,000 balls will fill it to a depth of 0.8 m at a cost of $7355.
CF-129. (0, 0), (3, 0) and (-0.5, 0)
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Unit 7
Answer Key
7
CF-130. a) x = -2 ± 3 f)(x - 2i)(x + 2i)
b) p
CF-132. a) 2p
CF-136. a) 3
2
g)(x - (1 + i))(x +(1 - i)
d) p
c) 90˚
e) 180˚
f) 360˚
b) x = 2, x = 4, and x ª -0.767
CF-137. a = - 2 , y = - 2 x(x - 3)(x + 1)2
3
3
CF-142. y = -0.5(x - 2)(x + 4) or y = -0.5x2 - x + 4 or
y = -0.5(x + 1)2 + 4.5
CF-143. a) 1, 3
c) 2 ± 4i
d) ±10i
CF-144. the inverse operations in the reverse order
CF-145. ª (5.492, 50)
CF-146. a) (x - 9)2 + (y + 3)2 = 16
b) (x + 5)2 + y2 = 23)
CF-148. a) Quadratic Formula
b) The root part (discriminant)- more specifically, b2 - 4ac.
c) if b2 - 4ac ≥ 0
d) if b2 - 4ac < 0
CF-149. a) (2, 8), (4, 4) b) (3 + i, 6 - 2i), (3 - i, 6 + 2i)
c) (a) intersects and (b) does not
CF-150. y = 600 + 5x, y = 3(1.15)x; in 40 months.
CF-151. The function could have 3 real roots (one x-intercept must be a double root) and
2 complex roots. Or it could have one real triple root, one real double root, and
zero complex roots. Or it could have one quadruple real root, another single real
root, and no complex roots.
CF-152. a) The roots are complex numbers.
b) y = (x - 3)2 + 4 or x2 - 6x + 13
c) x = 3 ± 2i
e) maximum is 8, minimum is 4
CF-153. They are the same.
CF-154. They do not intersect in the real plane.
CF-155. The first statement is correct, but the second is false.
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