Answer parts a-d for the function f(x) = x^3 - 3x^2 - x + 3. a) Use the leading coefficient test to determine the graph's end behavior. Which statement describes the behavior at th... Answer parts a-d for the function f(x) = x^3 - 3x^2 - x + 3. a) Use the leading coefficient test to determine the graph's end behavior. Which statement describes the behavior at the ends of f(x)? b) Find the x-intercepts and state their characteristics regarding the x-axis. c) Find the y-intercept. d) Use a graphing tool to graph the function and verify if it is drawn correctly using the maximum number of turning points. Read more

Steps to Solve

1. Determine End Behavior using Leading Coefficient Test

The leading term of the function $f(x) = x^3 - 3x^2 - x + 3$ is $x^3$. Since the degree is odd and the leading coefficient is positive, the end behavior is:

• As $x \to -\infty$, $f(x) \to -\infty$
• As $x \to +\infty$, $f(x) \to +\infty$

2. Find the Y-Intercept

To find the y-intercept, substitute $x = 0$ into the function:

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

Thus, the y-intercept is at the point $(0, 3)$.

3. Find the X-Intercepts

To find x-intercepts, set $f(x) = 0$ and solve:

$$ x^3 - 3x^2 - x + 3 = 0 $$

We can use the Rational Root Theorem to test potential rational roots: factors of $3$ (constant term). Testing $x = 1$:

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

Thus, $x = 1$ is a root.

Now, use synthetic division to factor $f(x)$ by $(x - 1)$:

Performing synthetic division of $f(x)$ by $(x - 1)$ gives:

$$ f(x) = (x - 1)(x^2 - 2x - 3) $$

Next, factor the quadratic:

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

Combining this gives:

$$ f(x) = (x - 1)(x - 3)(x + 1) $$

Thus, the x-intercepts are $x = 1$, $x = 3$, and $x = -1$.

4. Graph the Function

Using the information gathered:

• End behavior: Down to the left, up to the right.
• Y-intercept: (0, 3)
• X-intercepts: (1, 0), (3, 0), (-1, 0)

Plot these points and sketch the graph according to the end behavior and the x-intercepts.