Question

Graph each polynomial function. $$ f(x)=x^{4}+x^{3}-3 x^{2}-x+2 $$

Short answer

Plot key points, trace curves respecting end behavior.

Step-by-step solution

- Identify the Degree and Leading Coefficient

The given polynomial is \(f(x) = x^4 + x^3 - 3x^2 - x + 2\). Identify the degree (highest power of x) which is 4, and the leading coefficient (coefficient of the highest power) which is 1.

- Determine End Behavior

Since the degree is 4 (even) and the leading coefficient is positive, as \(x\) approaches infinity, \(f(x)\) approaches infinity. As \(x\) approaches negative infinity, \(f(x)\) also approaches infinity.

- Find the Zeros

Find the zeros of the polynomial by solving \(f(x) = 0\). This may involve factoring or using the Rational Root Theorem. For a rough sketch, approximate real roots if exact values are difficult to find.

- Calculate Intermediate Values

Evaluate the polynomial at various points like \(x = -2, -1, 0, 1, 2\) to get an idea of the polynomial's path. Plug in the x-values into the function to find corresponding y-values.

- Determine Symmetry

Check if the polynomial function is symmetric. Since the polynomial f(x) is not the same when \(f(-x)\) is substituted, it does not have even or odd symmetry.

- Sketch the Graph

Using the identified zeros, end behavior, and intermediate values, plot these points on a coordinate plane. Connect the points smoothly, following the polynomial's end behavior.

Key concepts

Degree of Polynomial

The degree of a polynomial is a key feature that helps us understand the behavior of the function. It's the highest power of the variable (in this case, x) present in the polynomial.
In the exercise given, the degree of our polynomial function \(f(x)=x^{4}+x^{3}-3x^{2}-x+2\) is 4 since the highest exponent of x is 4.
This tells us a couple of important things:

• The function will have at most 4 roots or zeros.
• The graph will have up to 3 turning points.
• Polynomials of even degree have similar end behavior on both ends of the x-axis.

Knowing your polynomial's degree helps you predict the overall shape of its graph.

Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree in the polynomial.
For \(f(x)=x^{4}+x^{3}-3x^{2}-x+2\), the leading coefficient is 1, which is the coefficient of \(x^4\).
This coefficient greatly influences the graph's end behavior:

• If the leading coefficient is positive, as x approaches infinity, f(x) also approaches infinity.
• If the leading coefficient is negative, as x approaches infinity, f(x) approaches negative infinity.

In our example, since the leading coefficient is 1 (positive), the ends of the graph go up as we move left and right along the x-axis.

End Behavior

End behavior describes what happens to the function's values (y-values) as x approaches infinity and negative infinity.
In simpler terms, it tells us which direction the graph is heading towards the ends.
Here’s how to determine end behavior easily:

• For polynomials with an even degree and positive leading coefficient, both ends of the graph rise towards infinity.
• For an even degree and negative leading coefficient, both ends of the graph fall towards negative infinity.
• For an odd degree with a positive leading coefficient, the left end falls, and the right end rises.
• For an odd degree with a negative leading coefficient, the left end rises, and the right end falls.

Since our polynomial has degree 4 (even) and a positive leading coefficient, the end behavior is such that as x approaches both positive and negative infinity, f(x) approaches positive infinity.

Zeros of the Polynomial

Zeros (or roots) of a polynomial are the x-values where the function equals zero, i.e., where the graph crosses the x-axis.
To find the zeros, solve the equation \(f(x) = 0\). This can be done by factoring, using synthetic division, or applying the Rational Root Theorem.
In our example, finding the exact zeros might be tricky without advanced algebraic techniques. But knowing the degree of 4 tells us there can be up to 4 real zeros.
Another approach is to estimate and plot approximate values by evaluating the polynomial at various points like \(x = -2, -1, 0, 1, 2\). Plugging these into the function will give us corresponding y-values.
This, combined with other clues like end behavior, helps in sketching a rough graph of the polynomial.