71. Celsius/Fahrenheit. A formula for converting Celsius de-grees to Fahrenheit dede-grees is given by the linear function
F⫽ ⫹ 32
Determine to the nearest degree the Celsius range in tem-perature that corresponds to the Fahrenheit range of 60°F to 80°F.
72. Celsius/Fahrenheit. A formula for converting Fahrenheit degrees to Celsius degrees is given by the linear function
C⫽ (F ⫺ 32)
Determine to the nearest degree the Fahrenheit range in temperature that corresponds to a Celsius range of 20°C to 30°C.
★73. Earth Science. In 1984, the Soviets led the world in drilling the deepest hole in the Earth’s crust—more than 12 kilometers deep. They found that below 3 kilometers the temperature T increased 2.5°C for each additional 100 meters of depth.
(A) If the temperature at 3 kilometers is 30°C and x is the depth of the hole in kilometers, write an equation using x that will give the temperature T in the hole at any depth beyond 3 kilometers.
(B) What would the temperature be at 15 kilometers? [The temperature limit for their drilling equipment was about 300°C.]
5 9 9 5C
(C) At what interval of depths will the temperature be between 200°C and 300°C, inclusive?
★74. Aeronautics. Because air is not as dense at high altitudes, planes require a higher ground speed to become airborne. A rule of thumb is 3% more ground speed per 1,000 feet of elevation, assuming no wind and no change in air tempera-ture. (Compute numerical answers to 3 significant digits.) (A) Let
Vs⫽ Takeoff ground speed at sea level for a particular plane (in miles per hour)
A⫽ Altitude above sea level (in thousands of feet) V⫽ Takeoff ground speed at altitude A for the same
plane (in miles per hour)
Write a formula relating these three quantities. (B) What takeoff ground speed would be required at Lake
Tahoe airport (6,400 feet), if takeoff ground speed at San Francisco airport (sea level) is 120 miles per hour?
(C) If a landing strip at a Colorado Rockies hunting lodge (8,500 feet) requires a takeoff ground speed of 125 miles per hour, what would be the takeoff ground speed in Los Angeles (sea level)?
(D) If the takeoff ground speed at sea level is 135 miles per hour and the takeoff ground speed at a mountain resort is 155 miles per hour, what is the altitude of the mountain resort in thousands of feet?
Section 2-3 Quadratic Functions
Quadratic FunctionsCompleting the Square
Properties of Quadratic Functions and Their Graphs Applications
Quadratic Functions
The graph of the square function, h(x) ⫽ x2
, is shown in Figure 1. Notice that the graph is symmetric with respect to the y axis and that (0, 0) is the lowest point on the graph. Let’s explore the effect of applying a sequence of basic transfor-mations to the graph of h. (A brief review of Section 1-5 might prove helpful at this point.) h(x) 5 ⫺5 5 x FIGURE 1 Square function h(x) ⫽ x2 .
E x p l o r e / D i s c u s s
1
Indicate how the graph of each function is related to the graph of h(x) ⫽ x2
. Discuss the symmetry of the graphs and find the highest or lowest point, whichever exists, on each graph.
(A) f (x) ⫽ (x ⫺ 3)2⫺ 7 ⫽ x2 ⫺ 6x ⫹ 2
(B) g(x) ⫽ 0.5(x ⫹ 2)2 ⫹ 3 ⫽ 0.5x2 ⫹ 2x ⫹ 5
(C) m(x) ⫽ ⫺(x ⫺ 4)2 ⫹ 8 ⫽ ⫺x2⫹ 8x ⫺ 8
(D) n(x) ⫽ ⫺3(x ⫹ 1)2⫺ 1 ⫽ ⫺3x2 ⫺ 6x ⫺ 4
Graphing the functions in Explore/Discuss 1 produces figures similar in shape to the graph of the square function in Figure 1. These figures are called parabo-las. The functions that produced these parabolas are examples of the important class of quadratic functions, which we now define.
QUADRATIC FUNCTIONS
If a, b, and c are real numbers with a ⫽ 0, then the function f (x) ⫽ ax2 ⫹ bx ⫹ c
is a quadratic function and its graph is a parabola.*
Since the expression ax2⫹ bx ⫹ c represents a real number for all real
num-ber replacements of x,
the domain of a quadratic function is the set of all real numbers.
We will discuss methods for determining the range of a quadratic function later in this section. Typical graphs of quadratic functions are illustrated in Figure 2.
(a) f(x) ⫽ x2⫺ 9 (b) g(x) ⫽ 2x2⫺ 15x ⫹ 30 (c) h(x) ⫽ ⫺0.3x2⫺ x ⫹ 4
Completing the Square
In Explore/Discuss 1 we wrote each function as two different, but equivalent, expressions. For example,
f (x) ⫽ (x ⫺ 3)2 ⫺ 7 ⫽ x2⫺ 6x ⫹ 2 ⫺10 ⫺10 10 10 ⫺10 ⫺10 10 10 ⫺10 ⫺10 10 10 FIGURE 2 Graphs of quadratic functions.
*A more general definition of a parabola that is independent of any coordinate system is given in Section 7-1.
E x p l o r e / D i s c u s s
2
It is easy to verify that these two expressions are equivalent by expanding the first expression. The first expression is more useful than the second for analyzing the graph of f. If we are given only the second expression, how can we determine the first? It turns out that this is a routine process, called completing the square, that is another useful tool to be added to our mathematical toolbox.
Replace ? in each of the following with a number that makes the equation valid.
(A) (x⫹ 1)2⫽ x2 ⫹ 2x ⫹ ?
(B) (x⫹ 2)2⫽ x2 ⫹ 4x ⫹ ?
(C) (x⫹ 3)2⫽ x2 ⫹ 6x ⫹ ? (D) (x⫹ 4)2⫽ x2 ⫹ 8x ⫹ ?
Replace ? in each of the following with a number that makes the expres-sion a perfect square of the form (x⫹ h)2
.
(E) x2 ⫹ 10x ⫹ ? (F) x2 ⫹ 12x ⫹ ? (G) x2⫹ bx ⫹ ?
Given the quadratic expression x2⫹ bx
what must be added to this expression to make it a perfect square? To find out, consider the square of the following expression:
We see that the third term on the right side of the equation is the square of one-half the coefficient of x in the second term on the right; that is, m2
is the square of (2m). This observation leads to the following rule:
COMPLETING THE SQUARE
To complete the square of the quadratic expression x2 ⫹ bx
add the square of one-half the coefficient of x; that is, add
or
The resulting expression can be factored as a perfect square:
x2⫹ bx ⫹
冢
b 2冣
2 ⫽冢
x⫹b 2冣
2 b2 4冢
b 2冣
2 1 2m2is the square of one-half the
coefficient of x.
Completing the Square
Complete the square for each of the following:
(A) x2 ⫺ 3x (B) x2 ⫺ 6bx
S o l u t i o n s (A) x2 ⫺ 3x
x2 ⫺ 3x ⫹ ⫽ Add that is,
(B) x2 ⫺ 6bx
x2 ⫺ 6bx ⫹ 9b2 ⫽ (x ⫺ 3b)2 Add that is, 9b2.
Complete the square for each of the following: (A) x2 ⫺ 5x
(B) x2 ⫹ 4mx
It is important to note that the rule for completing the square applies to only quadratic expressions in which the coefficient of x2
is 1. This causes little trou-ble, however, as you will see.
Properties of Quadratic Functions and Their Graphs
We now use the process of completing the square to transform the quadratic function f (x) ⫽ ax2⫹ bx ⫹ c
into the standard form f (x) ⫽ a(x ⫺ h)2 ⫹ k
Many important features of the graph of a quadratic function can be determined by examining the standard form. We begin with a specific example and then gen-eralize the results.
Consider the quadratic function given by
f (x) ⫽ 2x2⫺ 8x ⫹ 4 (1)
We use completing the square to transform this function into standard form: f (x) ⫽ 2x2⫺ 8x ⫹ 4
⫽ 2(x2 ⫺ 4x) ⫹ 4 ⫽ 2(x2 ⫺ 4x ⫹ ?) ⫹ 4
⫽ 2(x2 ⫺ 4x ⴙ 4) ⫹ 4 ⴚ 8 We add 4 to complete the square inside the parentheses. But because of the 2 outside the parentheses, we have actually added 8, so we must subtract 8.
⫽ 2(x ⫺ 2)2 ⫺ 4 The transformation is complete and can be checked by expanding. Factor the coefficient of x2 out of
the first two terms. M A T C H E D P R O B L E M
1
冢
⫺6b 2冣
2 ; 9 4.冢
⫺3 2冣
2 ;冢
x⫺3 2冣
2 9 4 E X A M P L E1
Thus, the standard form is
f (x) ⫽ 2(x ⫺ 2)2⫺ 4 (2)
If x ⫽ 2, then 2(x ⫺ 2)2 ⫽ 0 and f(2) ⫽ ⫺4. For any other value of x, the
positive number 2(x ⫺ 2)2is added to ⫺4, making f(x) larger. Therefore,
f (2) ⫽ ⫺4
is the minimum value of f (x) for all x—a very important result! Furthermore, if we choose any two values of x that are equidistant from x ⫽ 2, we will obtain the same value for the function. For example, x ⫽ 1 and x ⫽ 3 are each one unit from x ⫽ 2 and their functional values are
f (1) ⫽ 2(⫺1)2⫺ 4 ⫽ ⫺2
f (3) ⫽ 2(1)2 ⫺ 4 ⫽ ⫺2
Thus, the vertical line x ⫽ 2 is a line of symmetry—if the graph of equation (1) is drawn on a piece of paper and the paper folded along the line x⫽ 2, then the two sides of the parabola will match exactly.
The above results are illustrated by graphing equation (1) or (2) and the line x ⫽ 2 in a suitable viewing window (Fig. 3).
From the analysis of equation (2), illustrated by the graph in Figure 3, we con-clude that f (x) is decreasing on (⫺⬁, 2] and increasing on [2, ⬁). Furthermore, f (x) can assume any value greater than or equal to ⫺4, but no values less than
⫺4. Thus,
Range of f: y ⱖ ⫺4 or [⫺4, ⬁)
In general, the graph of a quadratic function is a parabola with line of sym-metry parallel to the vertical axis. The lowest or highest point on the parabola, whichever exists, is called the vertex. The maximum or minimum value of a quadratic function always occurs at the vertex of the graph. The vertical line of symmetry through the vertex is called the axis of the parabola. Thus, for f (x) ⫽ 2x2⫺ 8x ⫹ 4, the vertical line x ⫽ 2 is the axis of the parabola and
(2, ⫺4) is its vertex. ⫺10 ⫺4 10 6 f (x) ⫽ 2x2 ⫺ 8x ⫹ 4 ⫽ 2(x ⫺ 2)2 ⫺ 4 Axis of symmetry: x ⫽ 2 Minimum: f (2) ⫽ ⫺4 FIGURE 3
E x p l o r e / D i s c u s s
From equation (2), we can see that the graph of f is simply the graph of g(x) ⫽ 2x2translated to the right 2 units and down 4 units, as shown in Figure 4.
Notice the important results we have obtained from the standard form of the quadratic function f:
➞ The vertex of the parabola ➞ The axis of the parabola ➞ The minimum value of f (x) ➞ The range of f
➞ A relationship between the graph of f and the graph of g
Explore the effect of changing the constants a, h, and k on the graph of f (x) ⫽ a(x ⫺ h)2 ⫹ k.
(A) Let a ⫽ 1 and h ⫽ 5. Graph function f for k ⫽ ⫺4, 0, and 3 simultaneously in the same viewing window. Explain the effect of changing k on the graph of f.
(B) Let a ⫽ 1 and k ⫽ 2. Graph function f for h ⫽ ⫺4, 0, and 5 simultaneously in the same viewing window. Explain the effect of changing h on the graph of f.
(C) Let h ⫽ 5 and k ⫽ ⫺2. Graph function f for a ⫽ 0.25, 1, and 3 simultaneously in the same viewing window. Graph function f for a
⫽ 1, ⫺1, and ⫺0.25 simultaneously in the same viewing window.
Explain the effect of changing a on the graph of f.
(D) Can all quadratic functions of the form y ⫽ ax2⫹ bx ⫹ c be
rewritten as a(x⫹ h)2 ⫹ k?
We generalize the above discussion in the following box: ⫺10 ⫺4 10 6 f (x) ⫽ 2x2⫺ 8x ⫹ 4 ⫽ 2(x ⫺ 2)2 ⫺ 4 g (x) ⫽ 2x2 FIGURE 4
Graph of f is the graph of g translated.
PROPERTIES OF A QUADRATIC FUNCTION AND ITS GRAPH
Given a quadratic function and the standard form obtained by completing the square
f (x) ⫽ ax2⫹ bx ⫹ c ⫽ a(x ⫺ h)2 ⫹ k
a ⫽ 0 we summarize general properties as follows:
1. The graph of f is a parabola:
2. Vertex: (h, k) (parabola increases on one side of the vertex and
decreases on the other).
3. Axis (of symmetry): x ⫽ h (parallel to y axis).
4. f (h) ⫽ k is the minimum if a ⬎ 0 and the maximum if a ⬍ 0. 5. Domain: all real numbers; range: (⫺⬁, k] if a ⬍ 0 or [k, ⬁) if
a ⬎ 0.
6. The graph of f is the graph of g(x) ⫽ ax2
translated horizontally h units and vertically k units.
Analyzing a Quadratic Function
Find the standard form for the following quadratic function, analyze the graph, and check your results with a graphing utility:
f (x) ⫽ ⫺0.5x2 ⫺ x ⫹ 5
S o l u t i o n We complete the square to find the standard form: f (x) ⫽ ⫺0.5x2 ⫺ x ⫹ 5 ⫽ ⫺0.5(x2 ⫹ 2x ⫹ ?) ⫹ 5 ⫽ ⫺0.5(x2 ⫹ 2x ⫹ 1) ⫹ 5 ⫹ 0.5 ⫽ ⫺0.5(x ⫹ 1)2⫹ 5.5 E X A M P L E
2
x f (x) k h Axis x ⫽ h Vertex (h, k) Max f(x) a ⬍ 0 Opens downward x f (x) k h Axis x ⫽ h Vertex (h, k) Min f (x) a ⬎ 0 Opens upwardFrom the standard form we see that h ⫽ ⫺1 and k ⫽ 5.5. Thus, the vertex is (⫺1, 5.5), the axis of symmetry is x ⫽ ⫺1, the maximum value is f(⫺1) ⫽ 5.5, and the range is (⫺⬁, 5.5]. The function f is increasing on (⫺⬁, ⫺1] and decreas-ing on [⫺1, ⬁). The graph of f is the graph of g(x) ⫽ ⫺0.5x2 shifted to the left
1 unit and upward 5.5 units. To check these results, we graph f and g simultane-ously in the same viewing window, use the built-in maximum routine to locate the vertex, and add the graph of the axis of symmetry (Fig. 5).
Find the standard form for the following quadratic function, analyze the graph, and check your results with a graphing utility:
f(x) ⫽ ⫺x2 ⫹ 3x ⫺ 1
Finding the Equation of a Parabola
Find an equation for the parabola whose graph is shown in Figure 6.
(a) (b)
S o l u t i o n Figure 6(a) shows that the vertex of the parabola is (h, k) ⫽ (3, ⫺2). Thus, the standard equation must have the form
f (x) ⫽ a(x ⫺ 3)2 ⫺ 2 (3)
Figure 6(b) shows that f(4) ⫽ 0. Substituting in equation (3) and solving for a, we have
f (4) ⫽ a(4 ⫺ 3)2 ⫺ 2 ⫽ 0
a ⫽ 2 Thus, the equation for the parabola is
f (x) ⫽ 2(x ⫺ 3)2 ⫺ 2 ⫽ 2x2 ⫺ 12x ⫹ 16
Find the equation of the parabola with vertex (2, 4) and y intercept (0, 2).
M A T C H E D P R O B L E M
3
⫺5 0 5 6 ⫺5 0 5 6 FIGURE 6 E X A M P L E3
M A T C H E D P R O B L E M2
⫺6 ⫺6 6 6 FIGURE 5Applications
We now look at several applications that can be modeled using quadratic functions.
Maximum Area
A dairy farm has a barn that is 150 feet long and 75 feet wide. The owner has 240 ft of fencing and wishes to use all of it in the construction of two iden-tical adjacent outdoor pens with the long side of the barn as one side of the pens and a common fence between the two (Fig. 7). The owner wants the pens to be as large as possible.
(A) Construct a mathematical model for the combined area of both pens in the form of a function A(x) (see Fig. 7) and state the domain of A. (B) Find the value of x that produces the maximum combined area. (C) Find the dimensions and the area of each pen.
S o l u t i o n s (A) Since y ⫽ 240 ⫺ 3x,
A(x) ⫽ (240 ⫺ 3x)x ⫽ 240x ⫺ 3x2
The distances x and y must be nonnegative. Since y⫽ 240 ⫺ 3x, it follows that x cannot exceed 80. Thus, a model for this problem is
A(x) ⫽ 240x ⫺ 3x2, 0 ⱕ x ⱕ 80
(B) Omitting the details, the standard form for A is A(x) ⫽ ⫺3(x ⫺ 40)2⫹ 4,800
Thus, the maximum combined area of 4,800 ft2occurs at x⫽ 40. This result
is confirmed in Figure 8.
(C) Each pen is x by y/2 or 40 ft by 60 ft. The area of each pen is 40 ft ⫻ 60 ft ⫽ 2,400 ft2 . x x y x 150 feet 75 feet FIGURE 7 E X A M P L E
4
0 0 5,000 80 FIGURE 8 A(x) ⫽ 240x ⫺ 3x2.Repeat Example 4 with the owner constructing three identical adjacent pens instead of two.
Now that we have added quadratic functions to our mathematical toolbox, we can use this new tool in conjunction with another tool discussed previously— regression analysis. In the next example, we use both of these tools to investigate the effect of recycling efforts on solid waste disposal.
Solid Waste Disposal
Franklin Associates Ltd. of Prairie Village, Kansas, reported the data in Table 1 to the U.S. Environmental Protection Agency.
(A) Let x represent time in years with x ⫽ 0 corresponding to 1960, and let y represent the corresponding annual landfill disposal. Use regression analysis on a graphing utility to find a quadratic function of the form y⫽ ax2⫹ bx ⫹ c that models this data. (Round the constants a, b, and
c to three significant digits* when reporting your results.)
(B) If landfill disposal continues to follow the trend exhibited in Table 1, when (to the nearest year) would the annual landfill disposal return to the 1970 level?
(C) Is it reasonable to expect the annual landfill disposal to follow this trend indefinitely? Explain.
S o l u t i o n s (A) Since the values of y increase from 1970 to 1987 and then begin to decrease, a quadratic model seems a better choice than a linear one. Figure 9 shows the details of constructing the model on a graphing utility.
T A B L E 1 Municipal Solid Waste Disposal
Year 1970 1980 1985 1987 1990 1993 1995
Per Person Per Day (pounds)
Annual Landfill Disposal (millions of tons) 88.2 123.3 136.4 140.0 131.6 127.6 118.4 2.37 2.97 3.13 3.15 2.90 2.70 2.50 E X A M P L E
5
M A T C H E D P R O B L E M4
*For those not familiar with the meaning of significant digits, see Appendix C for a brief discussion of this concept.
Rounding the constants to three significant digits, a quadratic regression equation for this data is
y1 ⫽ ⫺0.187x
2 ⫹ 9.77x ⫹ 7.99
The graph in Figure 9(d) indicates that this is a reasonable model for this data. It is, in fact, the “best” quadratic equation for this data.
(B) To determine when the annual landfill disposal returns to the 1970 level, we add the graph of y2 ⫽ 88.2 to the graph [Fig. 10(a)]. The graphs of y1 and
y2intersect twice, once at x ⫽ 10 (1970), and again at a later date. Using a
built-in intersection routine [Fig. 10(b)] shows that the x coordinate of the second intersection point (to the nearest integer) is 42. Thus, the annual land-fill disposal returns to the 1970 level of 88.2 million tons in 2002. [Note: You will obtain slightly different results if you round the constants a, b, and c before finding the intersection point. As we stated before, we will always use the unrounded constants in calculations and only round the final answer.]
(a) (b)
(C) The graph of y1 continues to decrease and reaches 0 somewhere between
2110 and 2115. It is highly unlikely that the annual landfill disposal will ever reach 0. As time goes by and more data becomes available, new models will have to be constructed to better predict future trends.
Refer to Table 1.
(A) Let x represent time in years with x ⫽ 0 corresponding to 1960, and let y represent the corresponding landfill disposal per person per day. Use regres-sion analysis on a graphing utility to find a quadratic function of the form y ⫽ ax2 ⫹ bx ⫹ c that models this data. (Round the constants a, b, and c
to three significant digits when reporting your results.)
M A T C H E D P R O B L E M
5
0 0 150 60 0 0 150 60 y2⫽ 88.2 FIGURE 10(d) Graph of data and regression equation (c) Regression equation transferred to equation editor (b) Regression equation (a) Data FIGURE 9 0 0 150 60
(B) If landfill disposal per person per day continues to follow the trend exhib-ited in Table 1, when (to the nearest year) would it fall below 1.5 pounds per person per day?
(C) Is it reasonable to expect the landfill disposal per person per day to follow this trend indefinitely? Explain.
A n s w e r s t o M a t c h e d P r o b l e m s
1. (A) (B) x2⫹ 4mx ⫹ 4m2⫽ (x ⫹ 2m)2
2. Standard form: f (x) ⫽ ⫺(x ⫺ 1.5)2⫹ 1.25. The vertex is (1.5, 1.25), the axis of symmetry is x ⫽ 1.5, the maximum value of f (x) is 1.25, and the range of f is (⫺⬁, 1.25]. The function f is increasing on (⫺⬁, 1.5] and decreasing on [1.5, ⬁). The graph of f is the graph of g(x) ⫽ ⫺x2
shifted 1.5 units to the right and 1.25 units upward. 3. f (x) ⫽ ⫺0.5(x ⫺ 2)2⫹ 4 ⫽ ⫺0.5x2⫹ 2x ⫹ 2
4. (A) A(x) ⫽ (240 ⫺ 4x)x, 0 ⱕ x ⱕ 60 (B) The maximum combined area of 3,600 ft2
occurs at x⫽ 30 ft. (C) Each pen is 30 ft by 40 ft with area 1,200 ft2.
5. (A) y⫽ ⫺0.00434x2⫹ 0.202x ⫹ 0.759 (B) 2003 x2⫺ 5x ⫹25 4 ⫽
冢
x⫺ 5 2冣
2E X E R C I S E 2 - 3
A
In Problems 1–6, complete the square and find the standard form of each quadratic function.
1. f (x) ⫽ x2⫺ 4x ⫹ 5 2. g(x) ⫽ ⫺x2⫺ 2x ⫺ 3 3. h(x) ⫽ ⫺x2⫺ 2x ⫺ 1
4. k(x) ⫽ x2⫺ 4x ⫹ 4 5. m(x) ⫽ x2⫺ 4x ⫹ 1 6. n(x) ⫽ ⫺x2⫺ 2x ⫹ 3 In Problems 7–12, write a brief verbal description of the relationship between the graph of the indicated function (from Problems 1–6) and the graph of y ⫽ x2.
7. f (x) ⫽ x2⫺ 4x ⫹ 5 8. g(x) ⫽ ⫺x2⫺ 2x ⫺ 3 9. h(x) ⫽ ⫺x2⫺ 2x ⫺ 1
10. k(x) ⫽ x2⫺ 4x ⫹ 4 11. m(x) ⫽ x2⫺ 4x ⫹ 1 12. n(x) ⫽ ⫺x2⫺ 2x ⫹ 3 In Problems 13–18, match each graph with one of the functions in Problems 1–6. 13. ⫺5 ⫺5 5 5 14. 15. 16. 17. ⫺5 ⫺5 5 5 ⫺5 ⫺5 5 5 ⫺5 ⫺5 5 5 ⫺5 ⫺5 5 5
18.
B
For each quadratic function in Problems 19–24, sketch a graph of the function and label the axis and the vertex. 19. f (x) ⫽ 2x2⫺ 24x ⫹ 90 20. f (x) ⫽ 3x2⫹ 24x ⫹ 30 21. f (x) ⫽ ⫺x2⫺ 6x ⫺ 4
22. f (x) ⫽ ⫺x2⫹ 10x ⫺ 30 23. f (x) ⫽ 0.5x2⫺ 2x ⫺ 7 24. f (x) ⫽ 0.4x2⫹ 4x ⫹ 4 In Problems 25–28, find the intervals where f is increasing, the intervals where f is decreasing, and the range. Express answers in interval notation.
25. f (x) ⫽ 4x2⫺ 18x ⫹ 25 26. f (x) ⫽ 5x2⫹ 29x ⫺ 17 27. f (x) ⫽ ⫺10x2⫹ 44x ⫹ 12 28. f (x) ⫽ ⫺8x2⫺ 20x ⫹ 16
In Problems 29–32, use the graph of the parabola to find the equation of the corresponding quadratic function.
29. 30. 31. ⫺5 ⫺5 5 5 32.
In Problems 33–38, find the equation of a quadratic function whose graph satisfies the given conditions.
33. Vertex: (4, 8); x intercept: 6 34. Vertex: (⫺2, ⫺12); x intercept: ⫺4 35. Vertex: (⫺4, 12); y intercept: 4 36. Vertex: (5, 8); y intercept: ⫺2
37. Vertex: (⫺5, ⫺25); additional point on graph: (⫺2, 20) 38. Vertex: (6, ⫺40); additional point on graph: (3, 50)
39. Graph the line y⫽ 0.5x ⫹ 3. Choose any two distinct points on this line and find the linear regression model for the data set consisting of the two points you chose. Exper-iment with other lines of your choosing. Discuss the rela-tionship between a linear regression model for two points and the line that goes through the two points.
40. Graph the parabola y⫽ x2⫺ 5x. Choose any three distinct points on this parabola and find the quadratic regression model for the data set consisting of the three points you chose. Experiment with other parabolas of your choice. Discuss the relationship between a quadratic regression model for three noncollinear points and the parabola that goes through the three points.
41. Let f (x) ⫽ (x ⫺ 1)2⫹ k. Discuss the relationship between the values of k and the number of x intercepts for the graph of f. Generalize your comments to any function of the form
f (x) ⫽ a(x ⫺ h)2⫹ k, a ⬎ 0
42. Let f (x) ⫽ ⫺(x ⫺ 2)2⫹ k. Discuss the relationship be-tween the values of k and the number of x intercepts for the graph of f. Generalize your comments to any function of the form
f (x) ⫽ a(x ⫺ h)2⫹ k, a ⬍ 0
C
Recall that the standard equation of a circle with radius r and center (h, k) is
(x⫺ h)2⫹ (y ⫺ k)2⫽ r2
In Problems 43–46, use completing the square twice to find the center and radius of the circle with the given equation.
43. x2⫹ y2⫺ 6x ⫺ 4y ⫽ 36 44. x2⫹ y2⫺ 2x ⫺ 10y ⫽ 55 45. x2⫹ y2⫹ 8x ⫺ 2y ⫽ 8 46. x2⫹ y2⫺ 4x ⫹ 12y ⫽ 24
47. Let f (x) ⫽ a(x ⫺ h)2⫹ k. Compare the values of f(h ⫹ r) and f (h⫺ r) for any real number r. Interpret the results in terms of the graph of f.
48. Let f (x) ⫽ ax2⫹ bx ⫹ c, a ⫽ 0. Express each of the fol-lowing in terms of a, b, and c:
(A) The axis of symmetry (B) The vertex
(C) The maximum or minimum value of f, whichever exists.
Problems 49–52 are calculus-related. In geometry, a line that intersects a circle in two distinct points is called a secant line, as shown in figure (a). In calculus, the line through the points (x1, f (x1)) and (x2, f (x2)) is called a
secant line for the graph of the function f, as shown in
figure (b).
In Problems 49 and 50, find the equation of the secant line through the indicated points on the graph of f. Graph f and the secant line on the same coordinate system.
49. f (x) ⫽ x2⫺ 4; (⫺1, ⫺3), (3, 5) 50. f (x) ⫽ 9 ⫺ x2; (⫺2, 5), (4, ⫺7)
51. Let f (x) ⫽ x2⫺ 3x ⫹ 5. If h is a nonzero real number, then (2, f (2)) and (2 ⫹ h, f(2 ⫹ h)) are two distinct points on the graph of f.
(A) Find the slope of the secant line through these two points.
(B) Evaluate the slope of the secant line for h⫽ 1, h⫽ 0.1, h ⫽ 0.01, and h ⫽ 0.001. What value does the slope seem to be approaching?
Secant line for the graph of a function
(b) Secant line for
a circle (a) x f (x) (x1, f (x1)) (x2, f (x2)) P Q
52. Repeat Problem 51 for f (x) ⫽ x2⫹ 2x ⫺ 6.
53. Find the minimum product of two numbers whose differ-ence is 30. Is there a maximum product? Explain.
54. Find the maximum product of two numbers whose sum is 60. Is there a minimum product? Explain.
A P P L I C AT I O N S
55. Construction. A horse breeder wants to construct a corral next to a horse barn 50 feet long, using all of the barn as one side of the corral (see the figure). He has 250 feet of fencing available and wants to use all of it.
(A) Express the area A(x) of the corral as a function of x and indicate its domain.
(B) Find the value of x that produces the maximum area. (C) What are the dimensions of the corral with the
maximum area?
56. Construction. Repeat Problem 55 if the horse breeder has only 140 feet of fencing available for the corral. Does the maximum value of the area function still occur at the ver-tex? Explain.
57. Projectile Flight. An arrow shot vertically into the air from a cross bow reaches a maximum height of 484 feet after 5.5 seconds of flight. Let the quadratic function d(t) represent the distance above ground (in feet) t seconds af-ter the arrow is released. (If air resistance is neglected, a quadratic model provides a good approximation for the flight of a projectile.)
(A) Find d (t) and state its domain.
(B) At what times (to two decimal places) will the arrow be 250 feet above the ground?
58. Projectile Flight. Repeat Problem 57 if the arrow reaches a maximum height of 324 feet after 4.5 seconds of flight. 59. Engineering. The arch of a bridge is in the shape of a
parabola 14 feet high at the center and 20 feet wide at the base (see the figure).
x
y Corral
Horse barn
(A) Express the height of the arch h(x) in terms of x and state its domain.
(B) Can a truck that is 8 feet wide and 12 feet high pass through the arch?
(C) What is the tallest 8-foot-wide truck that can pass through the arch?
(D) What (to two decimal places) is the widest 12-foot-high truck that can pass through the arch?
60. Engineering. The roadbed of one section of a suspension bridge is hanging from a large cable suspended between two towers that are 200 feet apart (see the figure). The ca-ble forms a parabola that is 60 feet above the roadbed at the towers and 10 feet above the roadbed at the lowest point.
(A) Express the vertical distance d(x) (in feet) from the roadbed to the suspension cable in terms of x and state the domain of d.
(B) The roadbed is supported by seven equally spaced vertical cables (see the figure). Find the combined total length of these supporting cables.
61. Break-Even Analysis. Table 1 contains revenue and cost data for the production of lawn mowers where R is the to-tal revenue (in dollars) from the sale of x lawn mowers and C is the total cost (in dollars) of producing x lawn mowers. T A B L E 1 x 200 650 1,000 1,350 1,700 R ($) 95,000 275,000 290,000 260,000 140,000 C ($) 145,000 160,000 210,000 230,000 270,000 200 ft 60 ft x ft d(x) x 20 ft 14 ft h(x)
(A) Find a quadratic regression model for the revenue data using x as the independent variable.
(B) Find a linear regression model for the cost data using x as the independent variable.
(C) Use the regression models from parts A and B to estimate the x coordinates (to the nearest integer) of the break-even points.
62. Profit Analysis. Use the regression models computed in Problem 61 to estimate the indicated quantities.
(A) How many lawn mowers (to the nearest integer) must be produced and sold to realize a profit of $50,000? (B) How many lawn mowers (to the nearest integer) must
be produced and sold to realize the maximum profit? What is the maximum profit (to the nearest dollar)?
63. Water Consumption. Table 2 contains data related to the water consumption in the United States for selected years from 1960 to 1990. This data is based on U.S. Geological Survey, Estimated Use of Water in the United States in 1990, circular 1081, and previous quinquennial issues.
(A) Let the independent variable x represent years since 1960. Find a quadratic regression model for the total daily water consumption.
(B) If daily water consumption continues to follow the trend exhibited in Table 2, when (to the nearest year) would the total consumption return to the 1960 level?
64. Water Consumption. Refer to Problem 63.
(A) Let the independent variable x represent years since 1960. Find a quadratic regression model for the daily water consumption for irrigation.
(B) If daily water consumption continues to follow the trend exhibited in Table 2, when (to the nearest year) would the consumption for irrigation return to the 1960 level?
T A B L E 2 Daily Water Consumption
Year 1960 1965 1970 1975 1980 1985 1990 Total (billion gallons) 61 77 87 96 100 92 94 Irrigation (billion gallons) 52 66 73 80 83 74 76
