ACTIVITY 27: Quadratic Functions; (Section 3.5, pp. 259-266) Maxima and Minima.
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Transcript of ACTIVITY 27: Quadratic Functions; (Section 3.5, pp. 259-266) Maxima and Minima.
ACTIVITY 27:
Quadratic Functions; (Section 3.5, pp. 259-266) Maxima and Minima
Graphing Quadratic Functions Using the Standard Form:A quadratic function is a function f of the form
f(x) = ax2 + bx + c,where a, b, and c are real numbers and a is not 0. The graph of any quadratic function is a parabola; it can be obtained from the graph of f(x) = x2 by the methods described in Activity 26. Indeed, by completing the square a quadratic function f(x) = ax2 + bx + c can be expressed in the standard form
f(x) = a(x − h)2 + k.The graph of f is a parabola with vertex (h, k); the parabola opens upward if a > 0, or downward if a < 0.
a
bh
2fact In
Example 1:
Express the parabola y = x2 − 4x + 3 in standard form and sketch its graph. In particular, state the coordinates of its vertex and its intercepts.
342 xxy
34442 xxy
34442 xxy
342 2 xy
12 2 xy 1,2
From the graph we see that the y – intercept is:
x – intercepts are:
3,0
0,3 and 0,1
intercept-x
0y
0,3 0,1
intercept-y
0x
3,0
342 xxy
340 2 xx 130 xx
30 x 10 xx3 x1
3040 2 y
3y
Example 2:
Express the parabola y = −2x2 − x + 3 in standard form and sketch its graph. In particular, state the coordinates of its vertex and its intercepts.
xxy 32 2
xxy 32
12 2
xxy 316
1
16
1
2
12 2
xxy 316
2
16
1
2
12 2
xy8
8*3
8
1
4
12
2
xy8
241
4
12
2
xy8
25
4
12
2
8
25,
4
1V
xxy 32 2 xy8
25
4
12
2
intercept -x 0y
xxx 33220 2
xx 320 2
12 xx x 13 0
1320 xx
320 x 10 x
x23
x 2
3x1
0,2
3 0,1
intercept -y
0x
y 3002 2 3
3,0
xxy 32 2
intercept -x
0,2
3 0,1
intercept -y
3,0
8
25,
4
1V
Example 4:
Find the maximum or minimum value of the function:
2
2
77
2
7491002
7
f
maximuma has functionour 7- a thesince f
a
bx
2at occure willmaximum The
72
49
14
49
2
7
is maximum thely,Consequent
4
343
2
343100
4
343
4
686
4
400
4
743
2749100 tttf
xxxg 1500100 2 minimuma has functionour 001a thesince f
a
bx
2at occure willminimum The
1002
1500
200
1500
2
15
is minimum thely,Consequent
2
151500
2
15100
2
152
g2
22500
4
22500
112505625 5625
Example 5:
Find a function of the form f(x) = ax2 + bx + c whose graph is a parabola with vertex (−1, 2) and that passes through the point (4,16).
khxay 2
1h2k
21 2 xay
21416 2 a
21416 2 a
2516 2 a22516 a
2514 a
a25
14
2125
14 2 xy
2125
14 2 xy
Example 6 (Path of a Ball):
A ball is thrown across a playing field. Its path is given by the equation y = −0.005x2 + x + 5, where x is the distance the ball has traveled horizontally, and y is its height above ground level, both measured in feet.(a)What is the maximum height attained by the ball?
(b) How far has it traveled horizontally when it hits the ground?
a
bx
2at occure willmaximum The
005.02
1
feet 100
5100)100(005.0 2 y 510050 feet 55
5005.00 2 xx
)005.0(*2
)5)(005.0(4)1()1( 2
x01.0
1.11
01.0
1.11
01.0
0488.11
88.4
88.204feet 88.204
005.0a 1b 5c
Example 7 (Pharmaceuticals):
When a certain drug is taken orally, the concentration of the drug in the patient’s bloodstream after t minutes is given by C(t) = 0.06t − 0.0002t2, where 0 ≤ t ≤ 240 and the concentration is measured in mg/L. When is the maximum serum concentration reached, and what is that maximum concentration?
tttC 06.00002.0)( 2 tttC 3000002.0)( 2
222 1501503000002.0)( tttC
5.41500002.0)( 2 ttC
minutes 150 at t reached is onconentrati serum maximum The mg/L 4.5 is ionconcentrat maximum The
5.41503000002.0)( 22 tttC