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