Questions: Use the given function to complete parts a) through e) below. f(x) = -3(x-2)^2(x^2-1) (Type an integer or a decimal. Use a comma to separate answers At which zero(s) does the graph of the function touch the x-axis and turn if necessary, fill in the answer box to complete your choice. A. The graph touches the x-axis and turns around at the zero(s) 2. (Type an integer or a decimal. Use a comma to separate answers B. There are no zeros at which the graph touches the x-axis and turns c) Find the y-intercept by computing f(0). The y-intercept is 12. d) Determine the symmetry of the graph. Odd; origin symmetry Even; y-axis symmetry neither f(x) nor -f(x)

Use the given function to complete parts a) through e) below.
f(x) = -3(x-2)^2(x^2-1)
(Type an integer or a decimal. Use a comma to separate answers At which zero(s) does the graph of the function touch the x-axis and turn if necessary, fill in the answer box to complete your choice.
A. The graph touches the x-axis and turns around at the zero(s) 2.
(Type an integer or a decimal. Use a comma to separate answers
B. There are no zeros at which the graph touches the x-axis and turns
c) Find the y-intercept by computing f(0).

The y-intercept is 12.
d) Determine the symmetry of the graph.
Odd; origin symmetry
Even; y-axis symmetry
neither f(x) nor -f(x)

Transcript text: Use the given function to complete parts a) through e) below. \[ f(x)=-3(x-2)^{2}\left(x^{2}-1\right) \] (Type an integer or a decimal. Use a comma to separate answers At which zero(s) does the graph of the function touch the x-axis and turn if necessary, fill in the answer box to complete your choice. A. The graph touches the x-axis and turns around at the zero(s) 2 . (Type an integer or a decimal. Use a comma to separate answers B. There are no zeros at which the graph touches the $x$-axis and turns c) Find the $y$-intercept by computing $f(0)$. The $y$-intercept is 12 . d) Determine the symmetry of the graph. Odd; origin symmetry Even; $y$-axis symmetry neither $f(x)$ nor $-f(x)$

Solution

Solution Steps

Step 1: Finding the x-intercepts (roots) of the polynomial

The roots of the polynomial are found by solving the equation $f(x) = 0$. For the given polynomial, the roots are: x = -0.76, x = 0.76, x = -0.76_I, x = 0.76_I.

Step 2: Determine the multiplicity of each root and analyze behavior

The root $x = -0.76$ has a multiplicity of 1, indicating that the graph crosses the x-axis at this point. The root $x = 0.76$ has a multiplicity of 1, indicating that the graph crosses the x-axis at this point. The root $x = -0.76_I$ has a multiplicity of 1, indicating that the graph crosses the x-axis at this point. The root $x = 0.76_I$ has a multiplicity of 1, indicating that the graph crosses the x-axis at this point.

Step 3: Determine the end behavior of the graph

Given that the degree of the polynomial is 4 and the leading coefficient is 1, the graph rises to the left and right.

Final Answer:

The x-intercepts of the polynomial are determined by its roots and their multiplicities. The behavior at each root is analyzed based on the multiplicity, and the end behavior of the graph is determined by the degree of the polynomial and the sign of the leading coefficient. This analysis provides a comprehensive understanding of how the graph behaves across the entire real number line.

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