a Recall that if a≥ 0, then |a|=a. Conversely, if a<0, then |a|=- a.
B
b Make a table of values, plot, and connect the obtained points. Use the graph to determine domain and range.
A
aSlope for the uphill portion of the trip: 14 Slope for the downhill portion of the trip: - 14
B
bGraph:
Domain: 0≤ x ≤ 800 Range: 0≤ y≤ 100
Practice makes perfect
a The absolute value of a number is the distance from that number to 0. For example, the absolute value of 3, expressed as |3|, is 3.
The absolute value applies to negative numbers as well. The absolute value of -3 is the distance from - 3 to 0 on the number line.
Therefore, the absolute value is the positive value of a number. Hence, the definition of the absolute value of a is divided into two cases: the first where a is positive or 0 , and the second when a is negative.
|a|= a, a≥ 0 - a, a<0
We can apply the above definition to our function.
y= - 14(x-400)+100, x-400≥ 0 - 14[- (x-400)]+100, x-400 < 0
⇕
y= - 14x+200, x≥ 400 14x, x < 400
Since the second piece of the function has a slope of 14, which is a positive number, it represents the uphill portion of the trip. Conversely, the first piece of the function has a negative slope, which is - 14. This piece represents the downhill portion of the trip.
ccc
Slope for the uphill & & Slope for the downhill
portion of the trip & & portion of the trip [0.8em]
1/4 & & - 1/4
b We will make a table of values to graph this function. To do so, we will assign some values to the x-variable and calculate the corresponding values for y. For simplicity, we will choose multiples of 10 to substitute for x. Recall that from Part A we know that x is between 0 and 800, inclusive.
x
- 1/4|x-400|+100
y=- 1/4|x-400|+100
0
- 1/4| 0-400|+100
0
100
- 1/4| 100-400|+100
25
200
- 1/4| 200-400|+100
50
300
- 1/4| 300-400|+100
75
400
- 1/4| 400-400|+100
100
500
- 1/4| 500-400|+100
75
600
- 1/4| 600-400|+100
50
700
- 1/4| 700-400|+100
25
800
- 1/4| 800-400|+100
0
Now, we will plot and connect the obtained points. Recall that the graph of an absolute value function has a V shape.
We see above, and we know from Part A, that x takes values that are greater than or equal to 0 andless than or equal to 800. Moreover, we see that y takes values that are greater than or equal to 0 andless than or equal to 100. With this in mind, we can write the domain and range of the function.
Domain:& 0≤ x ≤ 800
Range:& 0≤ y≤ 100
