- Domain:
which is the set of all real numbers
- Range:
which is the set of y values such that
- x intercepts: none
- y intercept: only one y intercept and it is at (0,7) this is ordered pair notation and not interval notation
- Interval positive:
- Interval negative: none
- Interval increasing:
- Interval decreasing:
- Average rate of change over [-2,0] is 2
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Explanation:
The domain is the set of all real numbers. This is because the graph goes on forever to the left and right. We can use any x value we want as an input. There are no restrictions to worry about such as to prevent dividing by zero errors.
To say "all real numbers" in interval notation, we write which is another way of saying
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The range in interval notation is since y = 3 is the smallest y output possible, and we could have larger y values as well. So basically
can be used to describe the range without saying much else.
Note the use of a square bracket to include 3 as part of the interval.
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There are no x intercepts because this graph does not cross the x axis. The lowest point is at (-2,3) so there's no way we could reach y = 0. Put another way, y = 0 is not part of the range so we cannot have any x intercepts.
There is one y intercept and it is at (0,7) where the graph crosses the y axis. For any function, the max number of y intercepts is 1.
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When it says "interval positive", its asking "which part(s) of the graph are above the x axis?". That would be the entire graph meaning we have the interval . Every point on this V shaped curve is of the form (x,y) where y is positive.
So this means that we do not have any points with a negative y value, and therefore the answer to "interval negative" is none.
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Now onto the "interval increasing". This is similar to the previous section, but now we're looking when the graph is going uphill as we read from left to right. This happens on the interval or put another way when x > -2.
The graph goes downhill whenever x < -2. So that's why the answer for the "interval decreasing" is
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Note the points (-2,3) and (0,7) are on the V shaped graph. These points have x coordinates of x = -2 and x = 0, which are the endpoints of the interval we're focusing on.
Compute the slope of the line through those two points
m = (y2-y1)/(x2-x1)
m = (7-3)/(0-(-2))
m = (7-3)/(0+2)
m = 4/2
m = 2
The positive slope means the line goes uphill as we read from left to right
The average rate of change on the interval [-2, 0] is the value 2
In other words, we go up 2 units each time we move to the right 1 unit.