Problem 19 - Precalculus with Calculus Previews

Question

In Problems \(19-22\), construct a polynomial function \(f\) that has the given properties. There is no unique answer. \(f\) is of degree 4 , its graph is symmetric with respect to the \(y\) -axis, \(y\) -intercept is (0,-6)

Step-by-Step Solution

Answer

The polynomial could be \( f(x) = x^4 - 6 \), among other possibilities.

Step 1: Understand Symmetry

Since the graph is symmetric with respect to the y-axis, the polynomial must only contain even powers of x. This means that the polynomial will have the form \( f(x) = ax^4 + bx^2 + c \).

Step 2: Determine Degree of the Polynomial

The problem states that the polynomial should be of degree 4, confirming that the highest power of x should be \( x^4 \), not higher or lower.

Step 3: Determine the Y-Intercept

We are given that the y-intercept of the graph is (0, -6). This means \( f(0) = -6 \). Substituting into the general form, \( f(x) = ax^4 + bx^2 + c \), we can see that \( c = -6 \). Thus, \( f(x) = ax^4 + bx^2 - 6 \).

Step 4: Construct the Polynomial

Using the determined information, we know the polynomial is \( f(x) = ax^4 + bx^2 - 6 \). To fully determine the coefficients, we can choose any suitable values for \( a \) and \( b \) provided \( a eq 0 \) to ensure the degree is 4. An example of such a polynomial could be \( f(x) = 1x^4 + 0x^2 - 6 \).

Key Concepts

Degree of a Polynomial

In mathematics, the degree of a polynomial is the highest power of the variable in the equation. This concept is crucial when determining the nature and shape of the polynomial graph. For instance, if we have a polynomial function, it can be written in the general form:

• \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \)

Here, \( n \) represents the degree of the polynomial, which means it's the highest power of \( x \) in the polynomial. In the problem you're working on, the polynomial is of degree 4, meaning its highest power is \( x^4 \). This directly impacts how the polynomial behaves and appears on a graph. For instance, a polynomial of degree 4 will have 4 roots (real or complex), and its end behaviors will mimic those of the largest degree term, resembling a quadratic equation's end actions only with up to two turning points because it has even degree. Additionally, the degree can help predict the number of turning points and the general curvature of the graph.

Even Functions

An even function is characterized by symmetry about the y-axis. This means for every point on the graph of the function, there is a mirrored point on the opposite side of the y-axis. The mathematical representation of an even function conveys this symmetry, given by the equation:

• \( f(x) = f(-x) \)

This equation implies that substituting \( x \) with \( -x \) in the function does not change its output value.In the exercise you are working on, the polynomial being symmetric with respect to the y-axis means it is an even function. As a result, all the powers of \( x \) in the polynomial must be even. Therefore, the polynomial is structured as \( f(x) = ax^4 + bx^2 + c \), following the even nature rules. Understanding this property helps predict and explain the graph's shape, as the lack of odd-powered terms ensures it will not be skewed to one side.

Y-Intercept

In any graph, the y-intercept is the point where the graph crosses the y-axis. This occurs when the value of \( x \) is zero. Thus, the y-intercept represents the constant term in a polynomial function. For example, in a polynomial described by

• \( f(x) = ax^4 + bx^2 + c \)

When \( x = 0 \), the function evaluates to \( f(0) = c \), identifying that \( c \) is the y-intercept.In the problem's context, the y-intercept is given as (0, -6). This means that when \( x \) is zero, the polynomial function \( f(x) \) equals -6. Therefore, the constant term \( c \) is -6 in this particular polynomial. The y-intercept is vital for graphing as it provides the exact starting point on the y-axis and influences how the polynomial fits onto a coordinate plane, offering a concrete basis for further curve graphing tasks.