Questions: Answer parts (a)-(e) for the function shown below. f(x)=x^3-2x^2-x+2 The y-intercept is y=2. (Type an integer or a decimal.) d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. Choose the correct answer below. A. origin symmetry B. y-axis symmetry C. neither e. Use the graphing tool to graph the function. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly.

Answer parts (a)-(e) for the function shown below. f(x)=x^3-2x^2-x+2 The y-intercept is y=2. (Type an integer or a decimal.) d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. Choose the correct answer below. A. origin symmetry B. y-axis symmetry C. neither e. Use the graphing tool to graph the function. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly.
Transcript text: Answer parts (a)-(e) for the function shown below. \[ f(x)=x^{3}-2 x^{2}-x+2 \] The $y$-intercept is $y=2$. (Type an integer or a decimal.) d. Determine whether the graph has $y$-axis symmetry, origin symmetry, or neither Choose the correct answer below. A. origin symmetry B. $y$-axis symmetry C. neither e. Use the graphing tool to graph the function. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly.
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Solution

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Solution Steps

Step 1: Identify the y-intercept

The y-intercept of a function is the value of \( f(x) \) when \( x = 0 \). For the function \( f(x) = x^3 - 2x^2 - x + 2 \): \[ f(0) = 0^3 - 2(0)^2 - 0 + 2 = 2 \] So, the y-intercept is \( y = 2 \).

Step 2: Determine the symmetry of the graph

To determine the symmetry of the function \( f(x) = x^3 - 2x^2 - x + 2 \):

  • Y-axis symmetry: \( f(x) = f(-x) \)
  • Origin symmetry: \( f(x) = -f(-x) \)

Calculate \( f(-x) \): \[ f(-x) = (-x)^3 - 2(-x)^2 - (-x) + 2 = -x^3 - 2x^2 + x + 2 \]

Compare \( f(-x) \) with \( f(x) \): \[ f(x) = x^3 - 2x^2 - x + 2 \] \[ f(-x) = -x^3 - 2x^2 + x + 2 \]

Since \( f(x) \neq f(-x) \) and \( f(x) \neq -f(-x) \), the function has neither y-axis symmetry nor origin symmetry.

Step 3: Graph the function

To graph the function \( f(x) = x^3 - 2x^2 - x + 2 \), plot several points and use the maximum number of turning points to ensure accuracy. The function is a cubic polynomial, so it can have up to 2 turning points.

Final Answer

  1. The y-intercept is \( y = 2 \).
  2. The graph has neither y-axis symmetry nor origin symmetry.
  3. Graph the function \( f(x) = x^3 - 2x^2 - x + 2 \) by plotting points and identifying turning points.
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