Problem 40 - A Graphical Approach to Precalculus with Limits

Question

Sketch by hand the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned} P(x) &=3 x^{4}-4 x^{3}-22 x^{2}+15 x+18 \\ &=(3 x+2)(x-3)\left(x^{2}+x-3\right) \end{aligned}$$

Step-by-Step Solution

Answer

Graph is a quartic with roots \(-2.3, -\frac{2}{3}, 1.3, 3\) and y-intercept (0, 18).

Step 1: Identify the Degree and Leading Coefficient

The polynomial \( P(x) = 3x^4 - 4x^3 - 22x^2 + 15x + 18 \) is a quartic function, as the highest degree term is \( x^4 \). The leading coefficient is 3, which is positive.

Step 2: Determine End Behavior

Since the leading term of the polynomial is \( 3x^4 \), as \( x \to \pm \infty \), \( P(x) \to +\infty \). This means both ends of the graph will head upwards.

Step 3: Find the Roots of the Polynomial

The polynomial is factored as \( (3x+2)(x-3)(x^2+x-3) \). Setting each factor to zero gives the roots: \( x = -\frac{2}{3}, \) \( x = 3, \) and solving \( x^2 + x - 3 = 0 \) using the quadratic formula gives \( x = \frac{-1 \pm \sqrt{13}}{2} \). The approximate roots are \( x \approx 1.3 \) and \( x \approx -2.3 \).

Step 4: Determine Behavior Near Each Root

The roots \( x = -\frac{2}{3}, x = 1.3, x = -2.3, x = 3 \) all have a multiplicity of 1, meaning the graph crosses the x-axis at these points.

Step 5: Find the Y-intercept

The y-intercept occurs when \( x = 0 \). Plugging it into the polynomial gives \( P(0) = 18 \). Thus, the y-intercept is (0, 18).

Step 6: Sketch the Graph

Using the information about the roots, end behavior, and the y-intercept, sketch the graph. Start from left (-∞) and make sure it passes through the y-intercept (0, 18), crosses at each root, and ends up in the positive direction as \( x \to +\infty \).

Key Concepts

Quartic Functions

Quartic functions are polynomials of degree four. In the expression of a quartic polynomial, the highest power of the variable is four. The general form of a quartic function is \( ax^4 + bx^3 + cx^2 + dx + e \), where \( a eq 0 \). In this specific problem, the quartic polynomial is \( P(x) = 3x^4 - 4x^3 - 22x^2 + 15x + 18 \). Here, the term \( 3x^4 \) indicates that it's a quartic function due to the highest exponent being 4.

One key aspect of quartic functions is their versatility: they can have up to four real roots and can display complex turning points. They might look quite wavy when plotted and can have varying numbers of x-intercepts (points where the function crosses the x-axis). Understanding these functions helps us grasp more about polynomial behavior in general.

Roots of Polynomial

To determine the behavior and shape of a polynomial graph, it's essential to find its roots. Roots, or zeros, are values of \( x \) where the polynomial equals zero. For \( P(x) = 3x^4 - 4x^3 - 22x^2 + 15x + 18 \), the roots are obtained from its factored form: \((3x+2)(x-3)(x^2+x-3)\). Solving these factors gives us the roots: \( x = -\frac{2}{3} \), \( x = 3 \), and solving for \( x^2 + x - 3 = 0 \) using the quadratic formula provides the approximate roots \( x \approx 1.3 \) and \( x \approx -2.3 \).

• Each root indicates where the graph of the polynomial will intersect the x-axis.
• All roots here have a multiplicity of 1, meaning the graph will cross the x-axis at each root rather than just touch it.

Having a clear understanding of these roots helps you accurately sketch the polynomial graph and predict its crossings.

End Behavior of Polynomial

The end behavior of a polynomial describes what happens to the values of the polynomial as \( x \) becomes very large or very small, essentially heading to infinity in either direction. For any polynomial, the term with the highest degree dictates this behavior. In our polynomial \( P(x) = 3x^4 - 4x^3 - 22x^2 + 15x + 18 \), the leading term is \( 3x^4 \), a crucial factor in determining end behavior.

Since \( 3x^4 \) is both positive and even, as \( x \to \pm \infty \), \( P(x) \to +\infty \). This tells us:

• Both ends of the graph will rise upwards.

Understanding end behavior gives insight into the general shape of the graph, even without knowing exact values of the function for every \( x \).

Y-intercept of Polynomial

The y-intercept of a polynomial function is the point where the graph crosses the y-axis. This occurs when \( x = 0 \). To find it, simply substitute \( x = 0 \) into the polynomial and solve for \( y \). For our function \( P(x) = 3x^4 - 4x^3 - 22x^2 + 15x + 18 \), plugging in zero gives:

\[ P(0) = 3(0)^4 - 4(0)^3 - 22(0)^2 + 15(0) + 18 = 18 \]
Thus, the y-intercept is the point \((0, 18)\).

This information is valuable while sketching the graph as it indicates the starting point where the graph will intercept the y-axis. It's a fixed point that allows us to anchor our graph and better understand its layout and orientation.