Questions: Graph the polynomial function. h(x)=x^3+3x^2-x-3

Graph the polynomial function. h(x)=x^3+3x^2-x-3
Transcript text: Graph the polynomial function. \[ h(x)=x^{3}+3 x^{2}-x-3 \]
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Solution

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Solution Steps

Step 1: Find the x-intercepts

To find the x-intercepts, set h(x) = 0 and solve for x.

$x^3 + 3x^2 - x - 3 = 0$ $x^2(x + 3) - (x + 3) = 0$ $(x^2 - 1)(x + 3) = 0$ $(x - 1)(x + 1)(x + 3) = 0$

The x-intercepts are x = -3, -1, and 1.

Step 2: Find the y-intercept

To find the y-intercept, set x = 0 and solve for h(x).

$h(0) = (0)^3 + 3(0)^2 - (0) - 3 = -3$

The y-intercept is y = -3.

Step 3: Determine the end behavior

Since the leading term is $x^3$ and the coefficient is positive, the end behavior is as follows:

  • As x → -∞, h(x) → -∞
  • As x → ∞, h(x) → ∞

Final Answer:

The x-intercepts are -3, -1, and 1. The y-intercept is -3. As x approaches negative infinity, h(x) approaches negative infinity. As x approaches infinity, h(x) approaches infinity. This information, along with additional plotted points as needed, can be used with the graphing tool.

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