Graphing the Function (f(x) x^2 - x - 2)

The function (f(x) x^2 - x - 2) can be graphed by understanding its behavior in different intervals. This guide will walk you through the process of graphing this function, including the key points and the step-by-step analysis of the intervals where the function changes its behavior.

Introduction to the Function

The function (f(x) x^2 - x - 2) is a quadratic function, but it can be more complex when we consider the absolute values involved. However, in this specific case, we don't have absolute values, so we proceed directly with the given function. The goal is to identify the key points and intervals where the function's behavior changes.

Identifying Critical Points

The critical points for (f(x) x^2 - x - 2) are not related to absolute values but can be found by solving the equation (x^2 - x - 2 0).

Solving the quadratic equation:

[x^2 - x - 2 0]

Factoring the equation:

[(x - 2)(x 1) 0]

So, the critical points are:

[x 2 quad text{and} quad x -1]

These points will help us divide the real number line into intervals and analyze the function in each interval.

Breaking Down the Function into Intervals

We will now break down the function into different intervals based on the critical points (x -1) and (x 2).

Interval 1: (x

In this interval, the function (f(x) x^2 - x - 2) behaves as expected from a quadratic function. Let's analyze it:

[f(x) x^2 - x - 2]

This is a standard quadratic function, and its graph is a parabola opening upwards.

Interval 2: (-1 leq x

In this interval, we need to check the behavior of the function. However, for (f(x) x^2 - x - 2), there are no absolute values affecting the function in this interval, so it remains the same quadratic function.

[f(x) x^2 - x - 2]

This confirms that the function behaves the same way as in the interval (x

Interval 3: (x geq 2)

For (x geq 2), the function (f(x) x^2 - x - 2) also remains the same quadratic function.

[f(x) x^2 - x - 2]

So, in summary, the function (f(x) x^2 - x - 2) is the same quadratic function in all the intervals.

Graphing the Function

The graph of (f(x) x^2 - x - 2) can be visualized as follows:

For (x At (x -1), the vertex of the parabola occurs. For (-1 leq x At (x 2), the vertex of the parabola occurs. For (x geq 2), the parabola continues to open upwards.

The graph can be plotted using graphing software, but for manual plotting, you can use the vertex form of the quadratic function, which is (y a(x - h)^2 k), where (h) and (k) are the coordinates of the vertex. For our function:

[f(x) (x - 0.5)^2 - 2.25]

The vertex is at ((0.5, -2.25)), and the parabola opens upwards.

Calculating Key Points

It’s useful to calculate the values of the function at key points, such as the turning points and the intercepts:

At (x -1): (f(-1) (-1)^2 - (-1) - 2 0) At (x 2): (f(2) 2^2 - 2 - 2 0) The y-intercept: (f(0) 0^2 - 0 - 2 -2)

Final Graph

The graph of (f(x) x^2 - x - 2) is a parabola opening upwards with the vertex at ((0.5, -2.25)). The parabola intersects the x-axis at (x -1) and (x 2), and it intersects the y-axis at (y -2).

The final graph will look like:

The graph of (f(x) x^2 - x - 2) showing the parabola opening upwards with the vertex at (0.5, -2.25).

By understanding the key points and the intervals where the function changes, you can effectively graph the function and interpret its behavior.

Conclusion

Graphing functions, especially piecewise and absolute value functions, requires understanding the behavior of the function in different intervals. This guide provides a comprehensive approach to graphing (f(x) x^2 - x - 2), and the key points and intervals help you visualize and plot the function accurately.

If you have any further questions or need more assistance, feel free to reach out.