x^2 - 4
What are the:
Intervals on which the graph is above the x axis
Intervals on which the graph is below the x axis
Intervals on which the function is increasing
Intervals on which the function is increasing
Im not sure how to write these intervals.. so then i can do the other examples..
let y = x^2 - 4 and sketch it
here is a picture, using that sketch you should be able to answer all questions
http://www.wolframalpha.com/input/?i=plot+y+%3D+x%5E2+-+4
e.g.
"Intervals on which the graph is below the x axis "
----> -2 < x < 2
I will do some of these.
graph is above x axis when x^2-4>0 or
x^2>4 or
-2>x>2
when below
x^2-4<0
x^2<4
-2<x<2
when increasing:
(x+e)^2-4>x^2-4 where e is a very small number Now expanding
x^2+2xe+e^2-4-x^2+4>0
2xe+e^2>0
because e is always a positive number, this is true when x is positive. Answer
increasing when x>0
To find the intervals on which the graph of the function f(x) = x^2 - 4 is above or below the x-axis, we need to determine the sign of the function for different values of x.
1. Intervals above the x-axis:
To find the intervals where the function is above the x-axis, we need to determine where f(x) > 0.
x^2 - 4 > 0
First, let's find the critical points by setting the equation equal to zero:
x^2 - 4 = 0
Factoring, we get:
(x - 2)(x + 2) = 0
Setting each factor equal to zero, we get:
x - 2 = 0 --> x = 2
x + 2 = 0 --> x = -2
These are the critical points.
Next, we can choose test points within each of the intervals formed by the critical points. For example, let's choose x = -3, x = 0, and x = 3.
For x = -3:
f(-3) = (-3)^2 - 4 = 9 - 4 = 5
For x = 0:
f(0) = 0^2 - 4 = -4
For x = 3:
f(3) = 3^2 - 4 = 9 - 4 = 5
Based on these test points, we see that for x < -2 and -2 < x < 2, the function is negative (below the x-axis). And for -2 < x < 2, the function is positive (above the x-axis).
Therefore, the intervals on which the graph is above the x-axis are: (-∞, -2) and (2, ∞).
2. Intervals below the x-axis:
The intervals on which the graph is below the x-axis are between the critical points -2 and 2.
Therefore, the interval on which the graph is below the x-axis is: (-2, 2).
3. Intervals of increasing and decreasing:
To find the intervals of increasing and decreasing, we need to analyze the first derivative of the function f(x).
Taking the derivative of f(x) = x^2 - 4 with respect to x, we get:
f'(x) = 2x
Now, we need to determine the sign of the derivative for different values of x.
For x < 0, f'(x) < 0, which means the function is decreasing.
For x > 0, f'(x) > 0, which means the function is increasing.
Therefore, the interval on which the function is increasing is (0, ∞), and the interval on which the function is decreasing is (-∞, 0).
I hope this helps! Let me know if you have any further questions.
To determine the intervals on which the graph of the function f(x) = x^2 - 4 is above or below the x-axis, as well as the intervals where the function is increasing or decreasing, you need to analyze the signs of the function within specific intervals. Here's a step-by-step explanation:
1. Find the critical points:
To begin, set the function equal to zero and find the x-values where f(x) = 0.
x^2 - 4 = 0
(x - 2)(x + 2) = 0
From this, you can deduce that the critical points are x = -2 and x = 2.
2. Create a number line:
Draw a horizontal number line and mark the critical points on it.
-∞ -2 2 +∞
3. Choose test points:
Pick a test point from each of the intervals created by the critical points.
4. Evaluate the function:
Plug the chosen test points into the original function, f(x) = x^2 - 4, and determine the signs of the evaluations.
5. Analyze the intervals:
Based on the signs obtained, you can draw conclusions for each interval:
- Intervals above the x-axis:
If the function evaluation is positive (+), then the graph is above the x-axis.
If the function evaluation is negative (-), then the graph is below the x-axis.
- Intervals below the x-axis:
If the function evaluation is positive (+), then the graph is below the x-axis.
If the function evaluation is negative (-), then the graph is above the x-axis.
- Intervals of increasing and decreasing:
If the function is increasing, the graph is going "uphill."
If the function is decreasing, the graph is going "downhill."
Now let's proceed with the specific intervals:
- Consider the interval (-∞, -2):
Choose a test point in this interval, such as x = -3.
Evaluate the function: f(-3) = (-3)^2 - 4 = 9 - 4 = 5.
Since the function evaluation is positive (+), the graph is above the x-axis.
- Consider the interval (-2, 2):
Choose a test point in this interval, such as x = 0.
Evaluate the function: f(0) = (0)^2 - 4 = 0 - 4 = -4.
Since the function evaluation is negative (-), the graph is below the x-axis.
- Consider the interval (2, +∞):
Choose a test point in this interval, such as x = 3.
Evaluate the function: f(3) = (3)^2 - 4 = 9 - 4 = 5.
Since the function evaluation is positive (+), the graph is above the x-axis.
To write these intervals, you can use parentheses for open intervals and brackets for closed intervals:
Intervals above the x-axis: (-∞, -2) ∪ (2, +∞)
Intervals below the x-axis: (-2, 2)
Intervals of increasing: (-∞, -2) ∪ (2, +∞)
Intervals of decreasing: None
Remember, interval notation uses parentheses for open intervals (excludes the endpoints) and brackets for closed intervals (includes the endpoints).