Question

b) find the x - intercepts.
x = - 3,0,3
(type an integer or a decimal. use a comma to separate answers as needed.)
at which zeros does the graph of the function cross the x - axis? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. x = - 3,3 (type an integer or a decimal. use a comma to separate answers as needed.)
b. there are no x - intercepts at which the graph crosses the x - axis.
at which zeros does the graph of the function touch the x - axis and turn around? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. x = 0 (type an integer or a decimal. use a comma to separate answers as needed.)
b. there are no x - intercepts at which the graph touches the x - axis and turns around.
c) find the y - intercept by computing f(0).
f(0)=0
d) determine the symmetry of the graph.
odd; origin symmetry
even; y - axis symmetry
neither
e) determine the graph of the function.

Explanation:

Step1: Recall x - intercept concept

The x - intercepts are the values of x for which y = f(x)=0. Given x - intercepts are x=-3,0,3. The graph crosses the x - axis at the zeros where the function changes sign. Since the function changes sign at x = - 3 and x = 3, these are the points where the graph crosses the x - axis.

Step2: Recall touch - turn concept

The graph of the function touches the x - axis and turns around at the zero where the multiplicity of the zero is even. Here, x = 0 has an even - like behavior (implied by the problem context), so the graph touches and turns around at x = 0.

Step3: Find y - intercept

To find the y - intercept, we substitute x = 0 into the function. Given f(0)=0, so the y - intercept is 0.

Step4: Determine symmetry

A function is even if f(x)=f(-x). Since the graph has y - axis symmetry (even function), for every point (x,y) on the graph, the point (-x,y) is also on the graph.

Step5: Determine graph

Based on x - intercepts (-3,0,3), y - intercept (0,0), and y - axis symmetry, we can analyze the shape of the graph.

Answer:

b) At which zeros does the graph of the function cross the x - axis? A. x=-3,3
At which zeros does the graph of the function touch the x - axis and turn around? A. x = 0
c) f(0)=0
d) Even; y - axis symmetry
e) (No graph selection is provided in the answer as no information about the characteristics of graphs A, B, C, D is given other than the visual appearance. If more details about the graphs were provided, we could further analyze and select the correct one.)