Question

Jake claims that all graphs of polynomial functions of the form \(y=a x^{n}+x+b\) where \(a, b,\) and \(n\) are even integers, extend from quadrant II to quadrant I. Do you agree? Use examples to explain your answer.

Short answer

Jake's claim is not always true; negative 'a' results in graphs extending from quadrant III to quadrant IV.

Step-by-step solution

- Understand polynomial functions

A polynomial function of the form y = ax^n + x + b has three components: a coefficient 'a' for the term with the highest degree, a coefficient of 1 for the linear term 'x', and a constant term 'b'. Here, 'a', 'b', and 'n' are even integers.

- Consider the even power of x

When n is an even integer, the term 'ax^n' will determine the end behavior of the function. For example, if n = 2, the term would be 'ax^2'. Since 'a' is also even, it could be either positive or negative.

- Analyze case when 'a' is positive

If 'a' is positive (let's assume a = 2 and n = 2), the function would be y = 2x^2 + x + b. As x increases in the positive direction (to the right), the function increases rapidly due to the '2x^2' term. As x becomes more negative (to the left), the function still increases rapidly due to the '2x^2' term. This means the graph opens upwards. But because of the x term, there is asymmetry about the y-axis. However, the overall behavior will remain such that the function extends from quadrant II to quadrant I.

- Analyze case when 'a' is negative

If 'a' is negative (let's assume a = -2 and n = 2), the function would be y = -2x^2 + x + b. As x increases in the positive direction, the function decreases (forming a downwards-opening parabola), and as x becomes more negative, the function also decreases. This means the graph opens downwards, making it extend from quadrant III to quadrant IV.

- Draw a conclusion based on cases

By analyzing both scenarios when 'a' is positive and negative, it is clear that the initial claim is not always true. In the case where 'a' is negative, the function does not extend from quadrant II to quadrant I.

- Provide an example

For a concrete example with even integers, consider the function f(x) = 2x^2 + x + 4 (a=2, b=4, n=2). The graph extends from quadrant II to quadrant I. However, for the function f(x) = -2x^2 + x + 4 (a=-2, b=4, n=2), it extends from quadrant III to quadrant IV, demonstrating that Jake's claim isn't universally valid.

Key concepts

End Behavior of Polynomial Functions

The end behavior of a polynomial function describes how the function behaves as x approaches positive or negative infinity. This is mainly determined by the term with the highest degree.
For a polynomial function of the form \(y = ax^n + x + b\), where both 'a' and 'n' are even integers, the highest degree term \(ax^n\) dominates the end behavior:

• If 'a' is positive, the function extends upwards on both ends - thus extending from quadrant II (top-left) to quadrant I (top-right).
• If 'a' is negative, the function extends downwards on both ends - extending from quadrant III (bottom-left) to quadrant IV (bottom-right).

This analysis helps us understand the general shape and direction of the polynomial function's graph.

Graphing Quadratic Equations

Quadratic equations are polynomial functions where the highest degree is 2. These functions generally have the form \y = ax^2 + bx + c\. The graph of a quadratic equation is a parabola.
Depending on the sign of 'a':

• If 'a' is positive, the parabola opens upwards.
• If 'a' is negative, the parabola opens downwards.

To graph these equations, we start by identifying key points like the vertex and intercepts. The vertex can be located using the formula \(x = -\frac{b}{2a}\), substituting this value back into the equation to get the y-coordinate of the vertex. Plotting multiple points around the vertex and using symmetry helps to draw an accurate parabolic graph.

Even and Odd Functions

Functions can be classified as even, odd, or neither based on their symmetry properties. Understanding whether a function is even or odd helps in simplifying and predicting its behavior:

• Even functions satisfy \(f(-x) = f(x)\). These functions are symmetric about the y-axis.
• Odd functions satisfy \(f(-x) = -f(x)\). These functions are symmetric about the origin.

For polynomial functions such as \(y = ax^n + x + b\), if 'n' is even and 'a', 'b' are integers, the function will not exhibit pure even or odd symmetry due to the linear term 'x'. Instead, these functions show more complex behaviors depending on the coefficients.

Quadrants of the Coordinate Plane

The coordinate plane is divided into four quadrants:

• Quadrant I: positive x and y coordinates (top-right).
• Quadrant II: negative x and positive y coordinates (top-left).
• Quadrant III: negative x and y coordinates (bottom-left).
• Quadrant IV: positive x and negative y coordinates (bottom-right).

When analyzing the graphs of polynomial functions, understanding these quadrants helps us predict where the graph will lie based on the function's end behavior. For example, with the form \(y = ax^n + x + b\), a function with a positive 'a' extends upward through quadrants II and I, while a function with a negative 'a' extends downward through quadrants III and IV.