Section 4.1 Linear Functions and Their Properties
Section 4.1 Linear Functions and Their Properties
Linear Function A linear function is a function of the form , where is the slop and is y-intercept. ∆ Slope Average Rate of Change ∆ Only linear functions have a constant average rate of change. If a function has a constant average rate of change, the function is linear. If a function does not have a constant average rate of change, the function is nonlinear.
Increasing, Decreasing and Constant A linear function is increasing, if . A linear function is decreasing, if . A linear function is constant, if .
Exercises 1. (Solution 1) (a) g The slope y-intercept
(b) Adjust “Window” 10, 10, 2 by 10, 10, 2
(c) average rate of change = slope
(d) g 3 4 is increasing because the slope of the function is positive; 3 0
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(Solution 2) 2.
y-intercept 0 3 3 1 0 3 0 3 2 1 6 3 3 3 2 6 3 3 3 2
Because the rate of change, 3, is constant, the function is linear. ← 3, 3 3 3 Thus, the equation of the line is 3 3
(Solution 3) 3.
10 30 20 2 1 2 10 8 3 2 6 2 4 4 3 4 6 10 5 4
Because the rate of change is not constant, the function is nonlinear.
4.
(Solution 4.a) 0 0 7 8 ← 0 8 8 8 8 7 ⇒ 7 (Solution 4.b) 0 7 8 0 8 8 8 7 8 ⇒ 7 8 Thus, the interval is ,∞ . 7
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4. Continued… (Solution 4.c) g 7 8 5 4 5 5 12 8 4 8 8 12 12 1 (Solution 4.d) g 7 8 5 4 5 5 12 12 1 Thus, the interval is ∞, 1 . (Solution 4.e) The Solution to the system of linear equations is the point of intercept. In (c) we have 1 for g . Because 1 7 1 8 1 and g 1 5 1 4 1, the intercept is 1, 1 .
5.
(Solution 5.a) is the cost for the given value of x in miles. 0.25 40 ← 120 120 0.25 120 40 70 Therefore, the cost is $70 at 120 miles.
(Solution 5.b) The given coast is $90 0.25 40 ← 90 90 0.25 40 40 40 50 0.25 50 0.25 (Solution 5.c) 0.25 0.25 80 200 0.25 40 80 Therefore, the person drove 200 miles with $90. 40 40 0.25 40 (Solution 5.d) 0.25 40 The cost is a function of x and the x is the number of 0.25 0.25 miles driven. Thus, the domain is the number of miles 160 driven. The number of miles driven cannot be negative. Thus, the person can drive 160 miles maximum. Therefore, the domain of is 0, ∞ .
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5. Continued…
(Solution 5.e.f) The cost 0.25 40 is a linear function of x. Because 0.25 is the positive slope, the cost increases by 0.25. Because 40 is the y-intercept and is the cost for 0 mile driven, it is the minimum payment for renting the moving truck.
6.
(Solution 6.a) Because a new computer is $3200, the value 0 3200. So, the first point is 0, 3200 that is the y-intercept. Because the value of the computer depreciate for 4 years, the value 4 0. So the second point is 4, 0 . 0 3200 800 4 0 ← 800, 3200 800 3200 (Solution 6.b) Because the company chooses to depreciate for 4 years, the domain is 0, 4 .
(Solution 6.c) Plot two points obtained in (a) 0, 3200 , 4, 0
(Solution 6.d)
(Solution 6.d) 2, 1600 (Solution 6.e) 800 3200 ← 2 3, 800 2 800 2 3200 1600
(Solution 6.e) 800 3200 ← 800 800 800 3200 3200 3200 2400 800 ⇒ 3
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7.
(Solution 7) Let be the cost for renting a truck and x be the number of miles driven. $26 is a fixed charge and the average rate of change is $0.10 for each mile driven. Then a linear equation for the cost is 0.10 26
If the truck is driven 151 miles, the cost is If the truck is driven 500 miles, the cost is 0.10 26 ← 151 0.10 26 ← 151 151 0.10 151 26 500 0.10 500 26 41.1 76
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