f(x)=x^2-1 Graph the function. Estimate the intervals on which the function is increasing or decreasing, and estimate any relative maxima or minima.
Having had algebra I, you already know all about parabolas. The vertex is where it changes direction. Use that knowledge.
To graph the function f(x) = x^2 - 1, we can start by plotting a few points:
When x = -2, f(-2) = (-2)^2 - 1 = 3.
When x = -1, f(-1) = (-1)^2 - 1 = 0.
When x = 0, f(0) = (0)^2 - 1 = -1.
When x = 1, f(1) = (1)^2 - 1 = 0.
When x = 2, f(2) = (2)^2 - 1 = 3.
Plotting these points on a graph, we have:
-
|
- |
| |
| |
| |
| |
----|-----|---- x-axis
| |
| |
- |
|
This graph is a parabola that opens upwards, which means it has a minimum point.
To estimate the intervals where the function is increasing or decreasing, we observe that the function is increasing to the left of x = 0 and decreasing to the right of x = 0. Therefore, we can say that the function is increasing on the interval (-∞, 0) and decreasing on the interval (0, +∞).
To estimate the relative maxima or minima, we can see that the minimum point occurs at the vertex of the parabola, which is at x = 0. Therefore, the relative minimum is at f(0) = -1.
So, the function f(x) = x^2 - 1 has a relative minimum at (-1, 0) and is increasing on the interval (-∞, 0) and decreasing on the interval (0, +∞).
To graph the function f(x) = x^2 - 1, we can start by plotting some points and then connecting them to form a smooth curve.
Let's choose a few values for x and calculate the corresponding y-values:
When x = -2, f(x) = (-2)^2 - 1 = 4 - 1 = 3. So, one point is (-2, 3).
When x = -1, f(x) = (-1)^2 - 1 = 1 - 1 = 0. So, another point is (-1, 0).
When x = 0, f(x) = (0)^2 - 1 = 0 - 1 = -1. So, we have another point (0, -1).
When x = 1, f(x) = (1)^2 - 1 = 1 - 1 = 0. Another point is (1, 0).
When x = 2, f(x) = (2)^2 - 1 = 4 - 1 = 3. We have one more point (2, 3).
Now, let's plot these points on a graph:
|
5 |
|
|
3 | o o
|
| o
1 | o
|
|
-1 | o
|________________________
-2 -1 0 1 2
Next, we connect the points to form a smooth curve. Since this is a quadratic function, the graph is a parabola that opens upwards.
By observing the graph, we can estimate the intervals on which the function is increasing or decreasing. From left to right, the function is decreasing from negative infinity to x = -1, and then increasing from x = -1 onwards. Therefore, the function is decreasing on the interval (-∞, -1) and increasing on the interval (-1, ∞).
To find the relative maxima or minima, we can look for the highest point (maximum) or lowest point (minimum) of the graph. In this case, we see that the lowest point of the parabola is at the vertex.
The x-coordinate of the vertex can be found using the formula x = -b/2a, where a and b are the coefficients of x^2 and x, respectively, in the quadratic equation. In this case, a = 1 and b = 0, so the x-coordinate of the vertex is x = 0.
Now, substituting x = 0 into the equation f(x) = x^2 - 1, we get f(0) = 0^2 - 1 = -1. Therefore, the vertex is at the point (0, -1).
We can conclude that there is a relative minimum at the point (0, -1).