Questions: Answer parts (a)-(e) for the function shown below. f(x)=x^3-4x^2-x+4 A. The graph falls to the left and to the right. B. The graph rises to the left and falls to the right. C. The graph falls to the left and rises to the right. D. The graph rises to the left and to the right. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. What are the x-intercepts? x=1,4,-1 (Type an integer or a decimal. Use a comma to separate answers as needed.) At which x-intercept(s) does the graph cross the x-axis? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. x=1,4,-1 (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 x-intercept(s) does the graph 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= (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. The y-intercept is y=

Solution Steps

a. To determine the end behavior of the polynomial function \( f(x) = x^3 - 4x^2 - x + 4 \), we look at the leading term, which is \( x^3 \). Since the coefficient of \( x^3 \) is positive, the graph falls to the left and rises to the right.

b. To find the \( x \)-intercepts, we solve the equation \( f(x) = 0 \). The graph crosses the \( x \)-axis at points where the multiplicity of the root is odd and touches and turns around at points where the multiplicity is even.

c. The \( y \)-intercept is found by evaluating the function at \( x = 0 \).

The function given is \( f(x) = x^3 - 4x^2 - x + 4 \). To determine the end behavior, we look at the leading term, which is \( x^3 \).

• Since the leading term is \( x^3 \), which is an odd degree with a positive coefficient, the graph will fall to the left and rise to the right.

Thus, the correct choice for the end behavior is:

• C. The graph falls to the left and rises to the right.

To find the \( x \)-intercepts, we set \( f(x) = 0 \):

\[ x^3 - 4x^2 - x + 4 = 0 \]

Given the \( x \)-intercepts are \( x = 1, 4, -1 \), we can verify by substituting these values into the equation:

1. For \( x = 1 \): \[ 1^3 - 4(1)^2 - 1 + 4 = 1 - 4 - 1 + 4 = 0 \]

2. For \( x = 4 \): \[ 4^3 - 4(4)^2 - 4 + 4 = 64 - 64 - 4 + 4 = 0 \]

3. For \( x = -1 \): \[ (-1)^3 - 4(-1)^2 - (-1) + 4 = -1 - 4 + 1 + 4 = 0 \]

All values satisfy the equation, confirming the \( x \)-intercepts are correct.

Since the polynomial is of degree 3, each intercept is a simple root, meaning the graph crosses the \( x \)-axis at each intercept.

• At which \( x \)-intercept(s) does the graph cross the \( x \)-axis?
  A. \( x = 1, 4, -1 \)

• At which \( x \)-intercept(s) does the graph touch the \( x \)-axis and turn around?
  B. There are no \( x \)-intercepts at which the graph touches the \( x \)-axis and turns around.

The \( y \)-intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into the function:

\[ f(0) = 0^3 - 4(0)^2 - 0 + 4 = 4 \]

Thus, the \( y \)-intercept is \( y = 4 \).

Final Answer