Problem 44 - 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. $$f(x)=-2 x^{4}+2 x^{3}$$

Step-by-Step Solution

Answer

End behavior: falls to the left and right. X-intercepts: \(x=1\) and \(x=0\). Y-intercept: \(y=0\). Symmetry: neither y-axis nor origin. Graph is confirmed to be correct checking against maximum number of turning points (3).

Step 1: Determine the End Behavior

The leading coefficient of \(f(x)=-2x^4+2x^3\) is -2. The degree of the function is 4, which is an even number. According to the Leading Coefficient Test, if the degree of the function is even and the leading coefficient is negative, the graph of the function falls to the left and falls to the right. So in this case, as \(x\) approaches plus or minus infinity, \(f(x)\) approaches negative infinity.

Step 2: Find the x-intercepts

To find the x-intercepts, we have to set \(f(x)\) to equal zero and solve for \(x\). So, we get the following equation: -2x^4+2x^3=0. We can simplify this equation by taking out the common factor which is x³, to get -2x+2=0. The solution to this equation is \(x=1\) and \(x=0\). The graph will cross the x-axis at these intercepts because the multiplicities of the roots are odd.

Step 3: Find the y-intercept

To find the y-intercept, we set \(x\) to 0 in \(f(x)\). So, \(f(0)\) equals to 0.

Step 4: Check for Symmetry

In order to test if the graph has y-axis symmetry, the sign on every \(x\) in \(f(x)\) is replaced by -\(x\) and simplified. If the resulting equation matches the original, the graph is symmetric around the y-axis. The same steps are used to test for origin symmetry, except the resulting equation must be the opposite of the original. So, the function is \(f(-x)=-2(-x)^4+2(-x)^3=x^4-2x^3\), which is not the same as the original function, hence the graph is not symmetric about the y-axis nor origin.

Step 5: Sketch the Graph

The graph touches the x-axis at 0 and crosses at 1, keeping the end behavior in mind. The graph of f can be obtained by plotting a few more values of x. The maximum number of turning points of the graph is 3, because the degree of the polynomial is 4, and the rule is n-1. By arranging these aspects correctly on a graphing tool, a sketch of the function is drawn.

Key Concepts

Leading Coefficient Test

The leading coefficient test helps us to predict the end behavior of a polynomial function. For the function \(f(x) = -2x^4 + 2x^3\), first identify the leading coefficient, which is \(-2\). The degree of the polynomial, the highest power of \(x\), is \(4\), an even number.

According to the test:

• If the degree is even and the leading coefficient is positive, the graph rises to both the left and the right.
• If the degree is even and the leading coefficient is negative, as in our case, the graph falls to both the left and the right.

Here, since our leading coefficient is negative, as \(x\) approaches \( +\infty \) or \( -\infty \), \(f(x)\) approaches \(-\infty \). This gives us a useful visual understanding of how the graph will behave at its endpoints.

X-intercepts and Y-intercepts

X-intercepts are found by setting \( f(x) = 0 \) and solving for \( x \). Begin with the equation \(-2x^4 + 2x^3 = 0\).

Factor out the common term \(x^3\), resulting in \(-2x(x - 1) = 0\), which solves to give intercepts at \(x = 0\) and \(x = 1\). Since both solutions have odd multiplicity, the graph will cross the \(x\)-axis at these points.

To find the \(y\)-intercept, set \(x = 0\). Substituting into \(f(x)\), we have \(f(0) = 0\). This means the graph intersects the \(y\)-axis at the origin. So, the only \(y\)-intercept is also the \(x\)-intercept at (0,0).

Symmetry in Graphs

Symmetry in polynomial graphs can make sketching easier by reducing the number of calculations needed. We test symmetry by manipulating the function: replace every \(x\) with \(-x\) in \(f(x)\) to check for y-axis symmetry, and replace \(f(x)\) with \(-f(x)\) for origin symmetry.

For our function, \(f(-x) = -2(-x)^4 + 2(-x)^3 = x^4 - 2x^3\). The result is not equal to the original function \(-2x^4 + 2x^3\), indicating no y-axis symmetry. It is also not the negative of the original, showing no origin symmetry.

This lack of symmetry means that each part of the graph must be plotted individually, without symmetrical shortcuts.

End Behavior of Polynomial Functions

Understanding the end behavior of polynomial functions is essential in graphing, as it gives you initial clues about how the graph acts as \(x\) approaches infinity or negative infinity.

The polynomial degree and leading coefficient are crucial in this analysis. For \(f(x) = -2x^4 + 2x^3\), an even degree means the ends of the graph have the same behavior. A negative leading coefficient indicates that both ends will point downwards. As \(x\) becomes very large or very small, \(f(x)\) will approach \(-\infty\).

This is consistent with the predictions from the leading coefficient test, letting you confidently sketch the graph's outline before analyzing individual intercepts and points. By combining this knowledge, developing an accurate graph becomes a smoother process.