91Ó°ÊÓ

To find the y-intercept of a polynomial function, we need to understand what a y-intercept represents. Simply put, the y-intercept is the point where the graph of the function crosses the y-axis. This point occurs when the input variable, often denoted as \( x \), is zero. It's a critical piece of information because it tells us where the function starts along the y-axis when visualizing the function as a whole.

To find the y-intercept, you substitute \( x = 0 \) into the given polynomial function. For example, in the given function \( f(x)=x^{2}(2-x)(x+3) \), you replace \( x \) with zero. This gives us:

\[ f(0) = 0^2(2-0)(0+3) = 0 \]

Therefore, the y-intercept of the function is at the origin, or the point \( (0, 0) \). This means that when \( x \) is zero, the value of the function itself is zero, hence the function intersects the y-axis at the origin. It's helpful to visualize this on a graph, but even without a sketch, this arithmetic approach gives a precise answer.

X-intercepts are just as important as y-intercepts. They indicate the points where the graph of the polynomial function crosses the x-axis. These are the values of \( x \) for which the function's output, \( f(x) \), equals zero.

To find the x-intercepts, set the polynomial equal to zero and solve for \( x \). In the function \( f(x)=x^{2}(2-x)(x+3) \), solve:

• \( x^2 = 0 \)
• \( 2-x = 0 \)
• \( x+3 = 0 \)

Each equation helps in finding a possible x-intercept. By solving these, we find:

• From \( x^2 = 0 \), \( x = 0 \)
• From \( 2-x = 0 \), \( x = 2 \)
• From \( x+3 = 0 \), \( x = -3 \)

Thus, the x-intercepts are \( (0, 0) \), \( (2, 0) \), and \( (-3, 0) \). At these points, the graph touches or crosses the x-axis.

Identifying intervals where a function is positive or negative is key to understanding its behavior. These intervals give us insights into where the graph of the polynomial is above or below the x-axis.

To determine this, examine the sign of the function in the intervals created by the x-intercepts. In this case, the critical points where the sign might change are at \( x = -3 \), \( x = 0 \), and \( x = 2 \), separating the function into different intervals:

• \( x < -3 \)
• \( -3 < x < 0 \)
• \( 0 < x < 2 \)
• \( x > 2 \)

We check each interval by choosing a test point within it and evaluating the function:

- For \( x < -3 \), choose \( x = -4 \), resulting in a negative value, so \( f(x) < 0 \).
- For \( -3 < x < 0 \), choose \( x = -1 \), resulting in a positive value, so \( f(x) > 0 \).
- For \( 0 < x < 2 \), choose \( x = 1 \), resulting in a positive value, so \( f(x) > 0 \).
- For \( x > 2 \), choose \( x = 3 \), resulting in a negative value, so \( f(x) < 0 \).

In summary, the function is positive on the intervals \( (-3, 0) \) and \( (0, 2) \), and negative on \( (-\infty, -3) \) and \( (2, \infty) \). Recognizing these intervals allows you to understand where the graph lies in relation to the x-axis.