Problem 36 - College Algebra

Question

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=x^{3}+x^{2}-4 x-4$$

Step-by-Step Solution

Answer

The end behavior of the function is 'rises to the right, falls to the left'. The x-intercepts would need to be found using numerical methods or software, but after these are determined, would indicate where the graph crosses or touches the x-axis. The y-intercept is at (0, -4). The function has neither y-axis symmetry nor origin symmetry. The sketch of the function should show the correct end behavior and at most 2 turning points.

Step 1: End Behavior Using the Leading Coefficient Test

The given function is \(f(x) = x^{3} + x^{2} - 4x - 4\). The degree of the function is 3 and the leading coefficient is 1. For odd degree functions, if the leading coefficient is positive, the end behavior is 'rises to the right, falls to the left'. So this function rises to the right and falls to the left.

Step 2: Finding x-intercepts

Setting \(f(x) = 0\), we get the polynomial equation \(x^{3} + x^{2} - 4x - 4 = 0\). Solving this equation will yield the x-intercepts. Since the degree of the polynomial is 3, we expect up to 3 real roots. And no simple rational roots exist for this polynomial equation, we need to use numerical methods or software tools to find the roots. So depending on these roots, we determine whether the graph crosses or touches the x-axis at each intercept.

Step 3: Finding y-intercept

Setting \(x = 0\) in the function. We get \(f(0) = -4\). So the y-intercept of the graph of the function is at (0,-4)

Step 4: Checking for Symmetry

This function is neither even nor odd, because if we replace \(x\) with \(-x\), the function does not remain the same (no y-axis symmetry) and the function also does not become the negative of the original function (no origin symmetry). So the graph has neither y-axis symmetry nor origin symmetry.

Step 5: Plot the Function

Now we know the end behavior, roots and the y-intercept, we can sketch the polynomial. Plot the points, draw the curve through the points and the x and y intercept(s), making sure it has the correct end behavior. The number of turning points should be less than or equal to 2 (n-1, where n is the degree of the polynomial), check if it's correct.

Key Concepts

Leading Coefficient Test

Understanding the end behavior of a polynomial function can be a breeze with the Leading Coefficient Test. For any polynomial, the sign of the leading coefficient and the degree of the function jointly predict how the graph behaves as the values approach positive and negative infinity.

Let's apply this to a given polynomial function, say, \(f(x) = x^{3} + x^{2} - 4x - 4\). Recognize that the leading coefficient is 1 (the number in front of the highest degree term, \(x^3\)), and the degree is 3, which is odd. Here's the rule of thumb: if the leading coefficient is positive and the degree is odd, the graph will rise to the right and fall to the left. This means as \(x\) gets very large, the value of \(f(x)\) gets very large, and as \(x\) gets very small, the value of \(f(x)\) goes down without bound.

x-intercepts of a polynomial

Finding where a polynomial function meets the x-axis is like uncovering hidden treasure — those points are the x-intercepts. To find them, set \(f(x)\) to zero and solve the resulting equation. For our polynomial \(f(x) = x^{3} + x^{2} - 4x - 4\), this translates to solving \(x^{3} + x^{2} - 4x - 4 = 0\).

The degree of the polynomial tells us we can have up to 3 real roots, or x-intercepts. However, discovering these roots may require numerical methods or graphing tools since they may not be simple rational numbers. When you pinpoint these intercepts, notice if the graph crosses or merely touches the x-axis at these points.

y-intercept calculation

To catch the y-intercept of a polynomial graph, simply set \(x\) to zero in the function and evaluate. For \(f(x) = x^{3} + x^{2} - 4x - 4\), putting \(x=0\) yields \(f(0) = -4\). So the graph crosses the y-axis at the coordinate (0,-4).

This single point can anchor the entire graph on the coordinate plane and ensures that we have a starting point for sketching our graph accurately.

polynomial symmetry

Symmetry in polynomials is like a magic mirror that reflects the graph over a line or through a point. To determine if a graph has y-axis symmetry, substitute \(-x\) for \(x\) and see if you get the original function back; this would indicate even symmetry. However, for the function \(f(x) = x^{3} + x^{2} - 4x - 4\), replacing \(x\) with \(-x\) does not return the original function; thus, it lacks y-axis symmetry.

To check for origin symmetry, which denotes odd symmetry, replace \(x\) with \(-x\) and see if the output is \(-f(x)\). Once again, our function doesn't fulfill this criterion, indicating the absence of origin symmetry. Recognizing symmetry helps simplify graphing and understanding the behavior of polynomials.

turning points of a polynomial

The twists and turns of a polynomial graph are formally known as its turning points. These are points where the graph changes direction from increasing to decreasing or vice versa. A fundamental rule states that a polynomial of degree \(n\) can have up to \(n - 1\) turning points.

For instance, \(f(x) = x^{3} + x^{2} - 4x - 4\) is a third-degree polynomial, so it may have up to \(2\) turning points. By plotting our polynomial, we must ensure that there are no more than 2 turning points to validate the accuracy of the graph. These inflection points are central to accurately capturing the essence of the polynomial's shape and behavior.