Final answer:
To graph the function f(x) = -x^3 - 4x^2 + 9x + 36 accurately, the student should consider identifying the x-intercepts, y-intercept, vertex, and end behavior of the graph.
Step-by-step explanation:
In order to graph the function f(x) = -x^3 - 4x^2 + 9x + 36 accurately, there are several characteristics that can be helpful:
- Identifying the x-intercepts: The x-intercepts are the points where the graph crosses the x-axis. To find them, set f(x) equal to zero and solve the resulting equation.
- Identifying the y-intercept: The y-intercept is the point where the graph crosses the y-axis. To find it, substitute x = 0 into the equation and evaluate f(0).
- Finding the vertex: The vertex is the minimum or maximum point of the graph. To find it, first find the x-coordinate of the vertex using the formula x = -b/2a, where the equation is in the form ax^2 + bx + c. Then substitute the x-coordinate back into the equation to find the corresponding y-coordinate.
- Determining the end behavior: Look at the signs of the leading coefficient and the highest power of x to determine the behavior of the graph as x approaches positive and negative infinity.