The graph of the polynomial function P(x) is shown.
a. List the real zeros of P(x).
b. List the turning points of P(x).
c. State the left and right behavior of P(x).
We cannot see the graph.
To find the real zeros of the polynomial function P(x) and the turning points, we would need to analyze the graph of the function or consider its equation if given. Without the graph or equation provided, it is impossible to determine the specific zeros or turning points. However, I can explain the process of finding them in general terms:
a. Finding the real zeros of P(x):
To find the real zeros of a polynomial function, we need to set P(x) equal to zero and solve for x. This involves factoring the polynomial or using methods like the Rational Root Theorem or Descartes' Rule of Signs. Real zeros represent the x-values at which the polynomial intersects or crosses the x-axis.
b. Finding the turning points of P(x):
Turning points, also known as the local extrema, occur where the graph of the function changes from increasing to decreasing or vice versa. They can be found by locating the critical points of the function, which are the x-values where the derivative is equal to zero or does not exist. Then, by evaluating the function at these critical points, we can determine the corresponding y-values of the turning points.
c. Determining the left and right behavior of P(x):
The left and right behavior of a polynomial function refers to how the function behaves as x approaches negative infinity (left) or positive infinity (right). To determine this, we observe the highest degree term of the polynomial. If the degree is odd and the leading coefficient is positive, the function has a positive value for large negative x-values (left) and a negative value for large positive x-values (right). If the degree is odd and the leading coefficient is negative, the behavior is reversed. If the degree is even and the leading coefficient is positive, the function has positive values on both sides. And if the degree is even with a negative leading coefficient, the behavior is negative on both sides.
In summary, the specific real zeros, turning points, and left and right behavior of P(x) can only be determined by analyzing the graph or having the equation of the polynomial function.