Problem 2 - Precalculus Mathematics for Calculus

Question

Every polynomial has one of the following behaviors: (i) \(y \rightarrow \infty\) as \(x \rightarrow \infty\) and \(y \rightarrow \infty\) as \(x \rightarrow-\infty\) (ii) \(y \rightarrow \infty\) as \(x \rightarrow \infty\) and \(y \rightarrow-\infty\) as \(x \rightarrow-\infty\) (iii) \(y \rightarrow-\infty\) as \(x \rightarrow \infty\) and \(y \rightarrow \infty\) as \(x \rightarrow-\infty\) (iv) \(y \rightarrow-\infty\) as \(x \rightarrow \infty\) and \(y \rightarrow-\infty\) as \(x \rightarrow-\infty\) For each polynomial, choose the appropriate description of its end behavior from the list above. (a) \(y=x^{3}-8 x^{2}+2 x-15\); end behavior _____. (b) \(y=-2 x^{4}+12 x+100 ;\) end behavior ______.

Step-by-Step Solution

Answer

(a) (ii), (b) (iv).

Step 1: Identify the degree and leading coefficient for Polynomial (a)

For the polynomial \(y = x^{3} - 8x^{2} + 2x - 15\), the degree is 3 because the highest power of \(x\) is 3. The leading coefficient, which is the coefficient of \(x^3\), is 1.

Step 2: Determine the end behavior for an odd degree and positive leading coefficient

Polynomials with an odd degree and a positive leading coefficient, like polynomial (a), have end behaviors where \(y \rightarrow \infty\) as \(x \rightarrow \infty\) and \(y \rightarrow -\infty\) as \(x \rightarrow -\infty\). Therefore, the end behavior for polynomial (a) is (ii).

Step 3: Identify the degree and leading coefficient for Polynomial (b)

For the polynomial \(y = -2x^4 + 12x + 100\), the degree is 4 because the highest power of \(x\) is 4. The leading coefficient, which is the coefficient of \(x^4\), is -2.

Step 4: Determine the end behavior for an even degree and negative leading coefficient

Polynomials with an even degree and a negative leading coefficient, like polynomial (b), have end behaviors where \(y \rightarrow -\infty\) as \(x \rightarrow \infty\) and \(y \rightarrow -\infty\) as \(x \rightarrow -\infty\). Therefore, the end behavior for polynomial (b) is (iv).

Key Concepts

End Behavior

When discussing polynomials, understanding their 'end behavior' is crucial. End behavior describes how the function behaves as the input value \(x\) moves towards infinity or negative infinity. This characteristic of polynomials helps in predicting their graph's direction at the far ends of the horizontal axis.

There are four main types of end behavior for polynomials based on their degree and leading coefficient:

• If both ends of the polynomial graph go upwards, this usually indicates \(y \rightarrow \infty\) as \(x \rightarrow \infty\) and \(x \rightarrow -\infty\), which corresponds to an even degree with a positive leading coefficient.
• Conversely, if both ends go downwards, both \(y \rightarrow -\infty\) as \(x \rightarrow \infty\) and \(x \rightarrow -\infty\), suggesting an even degree with a negative leading coefficient.
• When the right side of the graph goes up (\(y \rightarrow \infty\) as \(x \rightarrow \infty\)) and the left side goes down (\(y \rightarrow -\infty\) as \(x \rightarrow -\infty\)), one can infer an odd degree with a positive leading coefficient.
• Whereas, if the right side goes down and the left side goes up, it signifies an odd degree with a negative leading coefficient.

Recognizing these patterns will give you an immediate understanding of how different polynomials behave at extreme values of \(x\).

Leading Coefficient

The leading coefficient in a polynomial is a key player in defining its characteristics. Located in the term with the highest power, this coefficient dictates the stretch and direction of the polynomial's graph.

Here is a breakdown of its role:

• Positive Leading Coefficient: Generally causes the right side of the graph to rise. This means as \(x \rightarrow \infty\), the function \(y\) will also increase for polynomials with an odd degree.
• Negative Leading Coefficient: The opposite happens, where the right side of the graph typically falls. Hence, as \(x \rightarrow \infty\), \(y\) will decrease for polynomials of odd degrees.

The leading coefficient also influences the graph's steepness – larger absolute values imply steeper rises or falls. Mastering the role of the leading coefficient helps shape the general perception of how the polynomial may appear on a graph.

Polynomial Degree

The degree of a polynomial is determined by the highest power of \(x\) present in its expression. This integer number greatly influences the polynomial's shape and end behavior.

Here are key points about polynomial degree:

• Odd Degree Polynomials: These typically have graph behaviors where the ends go in opposite directions. For example, rising to infinity in one direction while dipping to negative infinity in the other.
• Even Degree Polynomials: Tend to have ends that move in the same direction, either both going up or both going down.

Moreover, the degree suggests the potential number of roots or x-intercepts a polynomial can have. For instance, a third-degree polynomial has up to three real roots. Understanding the degree aids in forming a fundamental grasp of the polynomial’s graph and possible intersections on the axis.