Answer :
To determine the average rates of change of the function [tex]\( y = |x| + 2 \)[/tex] over the intervals [tex]\([-2,0]\)[/tex], [tex]\([0,3]\)[/tex], and [tex]\([-2,3]\)[/tex], we need to follow these steps:
1. Identify the function and intervals:
- The given function is [tex]\( y = |x| + 2 \)[/tex].
- The intervals are [tex]\([-2,0]\)[/tex], [tex]\([0,3]\)[/tex], and [tex]\([-2,3]\)[/tex].
2. Calculate the function values at the endpoints of each interval:
- For the interval [tex]\([-2,0]\)[/tex]:
- [tex]\( x_1 = -2 \)[/tex]
- [tex]\( x_2 = 0 \)[/tex]
- [tex]\( y_1 = |x_1| + 2 = |-2| + 2 = 2 + 2 = 4 \)[/tex]
- [tex]\( y_2 = |x_2| + 2 = |0| + 2 = 0 + 2 = 2 \)[/tex]
- For the interval [tex]\([0,3]\)[/tex]:
- [tex]\( x_1 = 0 \)[/tex]
- [tex]\( x_2 = 3 \)[/tex]
- [tex]\( y_1 = |x_1| + 2 = |0| + 2 = 0 + 2 = 2 \)[/tex]
- [tex]\( y_2 = |x_2| + 2 = |3| + 2 = 3 + 2 = 5 \)[/tex]
- For the interval [tex]\([-2,3]\)[/tex]:
- [tex]\( x_1 = -2 \)[/tex]
- [tex]\( x_2 = 3 \)[/tex]
- [tex]\( y_1 = |x_1| + 2 = |-2| + 2 = 2 + 2 = 4 \)[/tex]
- [tex]\( y_2 = |x_2| + 2 = |3| + 2 = 3 + 2 = 5 \)[/tex]
3. Calculate the average rate of change for each interval:
- For the interval [tex]\([-2,0]\)[/tex]:
[tex]\[ \text{Average rate of change} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 4}{0 - (-2)} = \frac{2 - 4}{0 + 2} = \frac{-2}{2} = -1 \][/tex]
- For the interval [tex]\([0,3]\)[/tex]:
[tex]\[ \text{Average rate of change} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 2}{3 - 0} = \frac{5 - 2}{3} = \frac{3}{3} = 1 \][/tex]
- For the interval [tex]\([-2,3]\)[/tex]:
[tex]\[ \text{Average rate of change} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 4}{3 - (-2)} = \frac{5 - 4}{3 + 2} = \frac{1}{5} = 0.2 \][/tex]
4. Interpret the results:
- The average rate of change over the interval [tex]\([-2,0]\)[/tex] is [tex]\(-1.0\)[/tex]. This indicates that the function [tex]\( y = |x| + 2 \)[/tex] is decreasing at a rate of 1 unit per unit increase in [tex]\( x \)[/tex] over this interval.
- The average rate of change over the interval [tex]\([0,3]\)[/tex] is [tex]\(1.0\)[/tex]. This indicates that the function [tex]\( y = |x| + 2 \)[/tex] is increasing at a rate of 1 unit per unit increase in [tex]\( x \)[/tex] over this interval.
- The average rate of change over the interval [tex]\([-2,3]\)[/tex] is [tex]\(0.2\)[/tex]. This indicates that, when considering the entire interval from [tex]\(-2\)[/tex] to [tex]\(3\)[/tex], the function has a small net increase of 0.2 units per unit increase in [tex]\( x \)[/tex]. This takes into account the mixed behavior of decreasing and then increasing over the different parts of the interval.
These rates of change reflect the piecewise linear nature of the function [tex]\( y = |x| + 2 \)[/tex], which has a V-shape with a vertex at the origin, causing different behaviors in different segments of the domain.
1. Identify the function and intervals:
- The given function is [tex]\( y = |x| + 2 \)[/tex].
- The intervals are [tex]\([-2,0]\)[/tex], [tex]\([0,3]\)[/tex], and [tex]\([-2,3]\)[/tex].
2. Calculate the function values at the endpoints of each interval:
- For the interval [tex]\([-2,0]\)[/tex]:
- [tex]\( x_1 = -2 \)[/tex]
- [tex]\( x_2 = 0 \)[/tex]
- [tex]\( y_1 = |x_1| + 2 = |-2| + 2 = 2 + 2 = 4 \)[/tex]
- [tex]\( y_2 = |x_2| + 2 = |0| + 2 = 0 + 2 = 2 \)[/tex]
- For the interval [tex]\([0,3]\)[/tex]:
- [tex]\( x_1 = 0 \)[/tex]
- [tex]\( x_2 = 3 \)[/tex]
- [tex]\( y_1 = |x_1| + 2 = |0| + 2 = 0 + 2 = 2 \)[/tex]
- [tex]\( y_2 = |x_2| + 2 = |3| + 2 = 3 + 2 = 5 \)[/tex]
- For the interval [tex]\([-2,3]\)[/tex]:
- [tex]\( x_1 = -2 \)[/tex]
- [tex]\( x_2 = 3 \)[/tex]
- [tex]\( y_1 = |x_1| + 2 = |-2| + 2 = 2 + 2 = 4 \)[/tex]
- [tex]\( y_2 = |x_2| + 2 = |3| + 2 = 3 + 2 = 5 \)[/tex]
3. Calculate the average rate of change for each interval:
- For the interval [tex]\([-2,0]\)[/tex]:
[tex]\[ \text{Average rate of change} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 4}{0 - (-2)} = \frac{2 - 4}{0 + 2} = \frac{-2}{2} = -1 \][/tex]
- For the interval [tex]\([0,3]\)[/tex]:
[tex]\[ \text{Average rate of change} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 2}{3 - 0} = \frac{5 - 2}{3} = \frac{3}{3} = 1 \][/tex]
- For the interval [tex]\([-2,3]\)[/tex]:
[tex]\[ \text{Average rate of change} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 4}{3 - (-2)} = \frac{5 - 4}{3 + 2} = \frac{1}{5} = 0.2 \][/tex]
4. Interpret the results:
- The average rate of change over the interval [tex]\([-2,0]\)[/tex] is [tex]\(-1.0\)[/tex]. This indicates that the function [tex]\( y = |x| + 2 \)[/tex] is decreasing at a rate of 1 unit per unit increase in [tex]\( x \)[/tex] over this interval.
- The average rate of change over the interval [tex]\([0,3]\)[/tex] is [tex]\(1.0\)[/tex]. This indicates that the function [tex]\( y = |x| + 2 \)[/tex] is increasing at a rate of 1 unit per unit increase in [tex]\( x \)[/tex] over this interval.
- The average rate of change over the interval [tex]\([-2,3]\)[/tex] is [tex]\(0.2\)[/tex]. This indicates that, when considering the entire interval from [tex]\(-2\)[/tex] to [tex]\(3\)[/tex], the function has a small net increase of 0.2 units per unit increase in [tex]\( x \)[/tex]. This takes into account the mixed behavior of decreasing and then increasing over the different parts of the interval.
These rates of change reflect the piecewise linear nature of the function [tex]\( y = |x| + 2 \)[/tex], which has a V-shape with a vertex at the origin, causing different behaviors in different segments of the domain.
