Questions: (a) Choose the end behavior of the graph of f. Choose One Falls to the left and rises to the right Rises to the left and falls to the right Rises to the left and rises to the right Falls to the left and falls to the right Zero(s) where the graph touches, but does not cross the x-axis: 1 (c) Find the y-intercept of the graph of f: 2 (d) Graph f(x)=(x+1)(x+2)(x-1)^2 by doing the following. - Plot all points where the graph of f intersects the x-axis or y-axis. - For each point on the x-axis, select the correct behavior. - Click on the graph icon.

(a) Choose the end behavior of the graph of f. Choose One Falls to the left and rises to the right Rises to the left and falls to the right Rises to the left and rises to the right Falls to the left and falls to the right Zero(s) where the graph touches, but does not cross the x-axis: 1 (c) Find the y-intercept of the graph of f: 2 (d) Graph f(x)=(x+1)(x+2)(x-1)^2 by doing the following. - Plot all points where the graph of f intersects the x-axis or y-axis. - For each point on the x-axis, select the correct behavior. - Click on the graph icon.
Transcript text: (a) Choose the end behavior of the graph of $f$. Choose One Falls to the left and rises to the right Rises to the left and falls to the right Rises to the left and rises to the right Falls to the left and falls to the right Zero(s) where the graph touches, but does not cross the $x$-axis: 1 (c) Find the $y$-intercept of the graph of $f$: 2 (d) Graph $f(x)=(x+1)(x+2)(x-1)^{2}$ by doing the following. - Plot all points where the graph of $f$ intersects the $x$-axis or $y$-axis. - For each point on the $x$-axis, select the correct behavior. - Click on the graph icon.
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Solution

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Solution Steps

Step 1: Determine the end behavior of the graph

The polynomial function \( f(x) = (x+1)(x+2)(x-1)^2 \) is a degree 4 polynomial with a positive leading coefficient. Therefore, the end behavior is:

  • Rises to the left and rises to the right.
Step 2: Find the zeros of the function

The zeros of the function are found by setting \( f(x) = 0 \): \[ (x+1)(x+2)(x-1)^2 = 0 \] The zeros are: \[ x = -1, -2, 1 \]

Step 3: Find the y-intercept of the function

The y-intercept is found by evaluating \( f(0) \): \[ f(0) = (0+1)(0+2)(0-1)^2 = 1 \cdot 2 \cdot 1 = 2 \]

Final Answer

  • End behavior: Rises to the left and rises to the right.
  • Zeros: \( x = -1, -2, 1 \)
  • y-intercept: \( y = 2 \)

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