PRACTICE PROBLEMS
PARK, BAE JUN
Quadratic equations
Math114
Section302 & 308
(1) Let f (x) = x2 + 6x + 11. Determin the following(if none, write “none00 ).
(a) Complete the square.
f (x) = (x + 3)2 − 9 + 11 = (x + 3)2 + 2
(b) Vertex
(−3, 2)
(c) Find the Maximum and Minimum.
Maximum : None
Minimum : 2 at x = −3
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2
PARK, BAE JUN
(d) x-intercept and y-intercept
For x-intercept,
f (x) = 0 ⇒ x2 + 6x + 11 = 0
Since b2 − 4ac = 36 − 44 = −8 < 0, there is no real solution of the equation.
∴ there is no x-intercept.
For y-intercept,
Find f (0). ⇒ f (0) = 11
∴ y-intercept is (0, 11)
Intervals over which f is (e) increasing and (f) decreasing.
Increasing on (−3, ∞)
Decreasing on (−∞, −3)
(g)Sketch the graph of f (x). Label all intercepts and the vertex.
SECTION 302 & 308
3
(2) Let f (x) = −2x2 + 12x − 16. Determin the following(if none, write “none00 ).
(a) Complete the square.
f (x) = −2(x2 − 6x + 9) + 18 − 16 = −2(x − 3)2 + 2
(b) Vertex
(3, 2)
(c) Find the Maximum and Minimum.
Maximum : 2 at x = 3
Minimum : None
(d) x-intercept and y-intercept
f (x) = 0 ⇒ −2x2 +12x−16 = 0 ⇒ x2 −6x+8 = 0 ⇒ (x−2)(x−4) = 0 ⇒ x = 2, 4
∴ x-intercept : (2, 0), (4, 0)
f (0) = −16
∴ y-intercept : (0, −16)
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PARK, BAE JUN
Intervals over which f is (e) increasing and (f) decreasing.
Increasing on (−∞, 3)
Decreasing on (3, ∞)
(g)Sketch the graph of f (x). Label all intercepts and the vertex.
SECTION 302 & 308
(3) Let f (x) = −2x2 − 12x − 11. Determin the following(if none, write “none00 ).
(a) Complete the square.
f (x) = −2(x2 + 6x + 9) + 18 − 11 = −2(x + 3)2 + 7
(b) Vertex
(−3, 7)
(c) Find the Maximum and Minimum.
Maximum : 7 at x = −3
Minimum : None
(d) x-intersecept and y-intercept
2
2
f (x) = 0 ⇒ −2x
√
√ − 12x − 11 = 0 ⇒√2x + 12x + 11√= 0
−12 ± 144 − 88
−12 ± 56
−12 ± 2 14
14
⇒x=
=
=
⇒ x = −3 ±
4
4 √
4
2
√
14
14
∴ x-intercept : (−3 +
, 0), (−3 −
, 0)
2
2
f (0) = −11
∴ y-intercept : (0, −11)
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6
PARK, BAE JUN
Intervals over which f is (e) increasing and (f) decreasing.
Increasing on (−∞, −3)
Decreasing on (−3, ∞)
(g)Sketch the graph of f (x). Label all intercepts and the vertex.
SECTION 302 & 308
(4) Let f (x) = x2 − 2x − 8. Determin the following(if none, write “none00 ).
(a) Complete the square.
f (x) = (x2 − 2x + 1) − 1 − 8 = (x − 1)2 − 9
(b) Vertex
(1, −9)
(c) Find the Maximum and Minimum.
Maximum : None
Minumum : −9 at x = 1
(d) x-intercept and y-intercept
f (x) = 0 ⇒ x2 − 2x − 8 = 0 ⇒ (x − 4)(x + 2) = 0 ⇒ x = 4, −2
∴ x-intercept : (4, 0), (−2, 0)
f (0) = −8
∴ y-intercept : (0, −8)
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PARK, BAE JUN
Intervals over which f is (e) increasing and (f) decreasing.
Increasing on (1, ∞)
Decreasing on (−∞, 1)
(g)Sketch the graph of f (x). Label all intercepts and the vertex.
SECTION 302 & 308
9
(5) Let f (x) = −3x2 + 5x − 1. Determin the following(if none, write “none00 ).
(a) Complete the square.
f (x) = −3(x2 − 35 x +
25
)
36
+
25
12
− 1 = −3(x − 56 )2 +
13
12
(b) Vertex
( 56 , 13
)
12
(c) Find the Maximum and Minimum.
Maximum :
13
12
at x =
5
6
Minimum : None
(d) x-intercept and y-intercept
f (x) = 0 ⇒√
−3x2 + 5x − 1 =√0 ⇒ 3x2 − 5x + 1 = 0
5 ± 25 − 12
5 ± 13
=
⇒x=
6
6
√
√
5 + 13
5 − 13
∴ x-intercept : (
, 0), (
, 0)
6
6
f (0) = −1
∴ y-intercept : (0, −1)
10
PARK, BAE JUN
Intervals over which f is (e) increasing and (f) decreasing.
5
Increasing on (−∞, )
6
5
Decreasing on ( , ∞)
6
(g)Sketch the graph of f (x). Label all intercepts and the vertex.
SECTION 302 & 308
11
(6) Let f (x) = 3x2 +4x−5 and g(x) = x2 −4x−1. Find the minimum value of f (x)−g(x).
f (x) − g(x) = 2x2 + 8x − 4 = 2(x2 + 4x + 4) − 8 − 4 = 2(x + 2)2 − 12
⇒ vertex is (−2, −12)
∴ f (x) − g(x) has minimum −12 at x = −2.