Problem 41 - Algebra and Trigonometry

Question

In Exercises 41–64, a. Use the Leading Coefficient Test to determine the graph’s end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. 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. $$f(x)=x^{3}+2 x^{2}-x-2$$

Step-by-Step Solution

Answer

The graph of function \(f(x) = x^3 + 2x^2 - x - 2\) falls to the left and rises to the right, crosses the x-axis at \(x=1\) and \(x=-2\), and has the y-intercept at \(y = -2\). The graph has neither y-axis symmetry nor origin symmetry. When graphed accurately, it will have 2 turning points.

Step 1: Determine the Graph's End Behavior

Using the Leading Coefficient Test, a polynomial with an odd degree and positive leading coefficient will rise to the right and fall to the left.

Step 2: Find the x-intercepts

To find the x-intercepts, set \(f(x)\) to be 0 and solve for \(x\). \[x^{3}+2 x^{2}-x-2 = 0\] This gives the solutions \(x = 1,-2,1\). Therefore, the graph crosses the x-axis at -2 and 1.

Step 3: Find the Y-intercept

To find the y-intercept, set \(x = 0\) in \(f(x)\). \[f(0) = (0)^3 + 2*(0)^2 - 0 - 2 = -2\] Therefore, the y-intercept is -2.

Step 4: Determine the Symmetry

Since the equation doesn't satisfy the conditions \(f(x) = f(-x)\) (for y-axis symmetry) or \(f(x) = -f(-x)\) (for origin symmetry), it has neither y-axis symmetry nor origin symmetry.

Step 5: Graphing the Function

Now that we have the x-intercepts, y-intercepts, and end behavior, we can graph the function. The maximum number of turning points in a polynomial function is one less than the degree, which is 2 in this case. The graph should include the 3 points found earlier and have two turning points.

Key Concepts

Leading Coefficient Test

Understanding the leading coefficient test is crucial for determining the end behavior of a polynomial function. The leading coefficient is the coefficient of the term with the highest degree in the polynomial. In the given function \( f(x) = x^{3}+2x^{2}-x-2 \), the highest degree is 3, and the leading coefficient is 1, which is positive.

This means our polynomial has:

• Odd degree (3)
• Positive leading coefficient (1)

In general, a polynomial with an odd degree and a positive leading coefficient will fall to the left and rise to the right. Think of it as a down-up motion as you move from left to right along the x-axis.

This information gives us insight into how the graph behaves as it extends towards infinity in both directions.

X-intercepts

Finding x-intercepts provides us crucial points where the graph crosses the x-axis. To find these points, we set the equation equal to zero and solve for x. For our polynomial \( x^{3}+2x^{2}-x-2 = 0 \), solving gives the x-intercepts at \( x = 1, -2, 1 \).

At each intercept:

• The graph crosses the x-axis at \( x = 1 \) and \( x = -2 \)

These points are not only zeros of the polynomial but also indicate where the function changes sign, either crossing or touching the axis momentarily before reversing direction.

Y-intercepts

The y-intercept is where the graph crosses the y-axis. For any function, this point is found by evaluating the function at \( x = 0 \).

For our function:

• Set \( x = 0 \) in \( f(x) \) gives \( f(0) = -2 \)

Thus, the y-intercept is at the point \( (0, -2) \). The y-intercept helps us anchor the graph vertically, giving us a point to begin sketching the curve.

Graph Symmetry

Graph symmetry helps us identify unique properties of the function's graph. If a function is symmetrical about the y-axis, it satisfies \( f(x) = f(-x) \), and if it's symmetrical about the origin, it satisfies \( f(x) = -f(-x) \).

For \( f(x) = x^{3}+2x^{2}-x-2 \):

• \( f(x) eq f(-x) \)
• \( f(x) eq -f(-x) \)

This means the graph of the function has neither y-axis symmetry nor origin symmetry, making it distinctly shaped without these reflective properties.

Turning Points

Turning points are where the graph changes direction from increasing to decreasing or vice versa. These points provide critical insight into the shape and flow of the graph.

The maximum number of turning points for a polynomial is one less than its degree. In our case:

• Degree is 3, so maximum turning points is 2

The graph will have two turning points, creating the rising and falling sections in the graph's overall behavior. Identifying these points helps ensure that the drawn curve reflects the function's true behavior and supports accuracy in freehand sketches of the polynomial.