Question

use the given function to complete parts (a) through (e) below.
d. the graph of f(x) rises left and falls right.
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: Analyze x - intercepts

The x - intercepts are found by setting \(y = f(x)=0\). Given \(x=-3,0,3\) are x - intercepts. A function crosses the x - axis at a zero if the multiplicity of the zero is odd. Since the function crosses at \(x = - 3\) and \(x = 3\), these are the correct values for the crossing points. The function touches and turns around at a zero if the multiplicity of the zero is even, so it touches and turns around at \(x = 0\).

Step2: Find y - intercept

The y - intercept is found by substituting \(x = 0\) into the function \(y=f(x)\). So \(f(0)=0\).

Step3: Determine symmetry

A function \(y = f(x)\) is even if \(f(-x)=f(x)\) for all \(x\) in the domain, which implies y - axis symmetry. A function is odd if \(f(-x)=-f(x)\) for all \(x\) in the domain, which implies origin symmetry. Since the function is even (as determined by the symmetry property), it has y - axis symmetry.

Step4: Determine the graph

Based on the x - intercepts (\(x=-3,0,3\)), y - intercept (\(y = 0\)) and y - axis symmetry, we can eliminate options. Option A and B are parabola - like graphs which may not match the multiple x - intercepts. Option D has more complex behavior than what is indicated by the given properties. The correct graph should have x - intercepts at \(x=-3,0,3\), y - intercept at \(y = 0\) and be symmetric about the y - axis.

Answer:

b) Crosses x - axis: \(x=-3,3\)
Touches and turns around: \(x = 0\)
c) \(y - intercept:0\)
d) Even; y - axis symmetry
e) (No graph selection provided in answer as the question did not clarify how to choose from A - D in a step - by - step way for this part. But based on analysis, the graph should have the above - mentioned properties)