Question

Graph each polynomial function. $$ f(x)=x^{4}-x^{3}-6 x^{2}+4 x+8 $$

Short answer

Degree 4 polynomial with positive leading coefficient means both ends go to infinity. Find and plot roots and the y-intercept at (0, 8). Sketch accordingly.

Step-by-step solution

- Identify the Degree and Leading Coefficient

The given polynomial is \(f(x) = x^4 - x^3 - 6x^2 + 4x + 8\). Identify the degree of the polynomial and the leading coefficient. Here the degree is 4, and the leading coefficient is 1.

- Determine the End Behavior

The degree is even and the leading coefficient is positive. Therefore, as \( x \to \pm \infty \), \( f(x) \to \infty \). The ends of the graph will rise to infinity.

- Calculate the Polynomial's Roots

Solve \(x^4 - x^3 - 6x^2 + 4x + 8 = 0\). This can be complex, so use numerical or graphical methods if necessary. Factor as much as possible or use software/calculators for approximations.

- Plot the Roots on the x-axis

Plot the roots calculated in the previous step on the x-axis. These are the points where the graph intersects the x-axis.

- Determine the Y-intercept

Find the y-intercept by evaluating \(f(0) \). Substituting in 0 for x, we get \( f(0) = 8 \). So, the y-intercept is (0,8). Plot this point.

- Sketch the Graph

Using the information from steps 2-5, sketch the graph. Remember to reflect the appropriate end behavior (ends going up to infinity) and connecting the root points smoothly.

Key concepts

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. In our polynomial function \({\ f(x) = x^4 - x^3 - 6x^2 + 4x + 8 }\), the term with the highest degree is \( x^4 \). Therefore, the leading coefficient is 1. This number plays a crucial role in determining the end behavior of the polynomial graph and overall shape of the function.

Degree of a Polynomial

The degree of a polynomial indicates the highest power of x present in the polynomial. For our polynomial \({\ f(x) = x^4 - x^3 - 6x^2 + 4x + 8 }\), the highest power of x is 4, so it is a fourth-degree polynomial. The degree helps us understand the number of possible roots and the general shape of the graph. In this case, being an even-degree polynomial, the graph will typically start and end in the same direction.

End Behavior

End behavior describes how the function behaves as x approaches positive or negative infinity. \({\ f(x) = x^4 - x^3 -6x^2+4x+8 }\)\ has a degree of 4 (even) and a positive leading coefficient (1). For such polynomials, as \( x \to \pm \infty \), the function \( f(x) \to \infty \). This means both ends of the polynomial will point upward, giving a 'W' shaped curve.

Polynomial Roots

Roots of a polynomial are the values of x where the polynomial equals zero. For the function \({\ f(x) = x^4 - x^3 - 6x^2 + 4x + 8 }\), finding the exact roots involves solving the equation \({\ x^4 - x^3 - 6x^2 + 4x + 8 = 0 }\). In many cases, it can be complex, so we may use numerical methods or graphing tools to find approximate values of the roots. The roots are important because these are the points where the function crosses the x-axis.

Y-intercept

The y-intercept of a polynomial is the point where the graph crosses the y-axis. It is found by substituting x = 0 into the polynomial function. For \({\ f(x) = x^4 - x^3 - 6x^2 + 4x + 8 }\), the y-intercept is \({\ f(0) = 8 }\). This gives us the point (0, 8) which is easily plotted on the graph. The y-intercept is often a starting point when sketching the graph of the polynomial.