Problem 43 - 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. $$f(x)=-2 x^{4}+4 x^{3}$$

Step-by-Step Solution

Answer

The function \(f(x) = -2x^{4}+4x^{3}\) end behavior is: falls to the left and falls to the right. The function crosses the \(x\)-axis at \(x = 0\) and \(x = 2\). The \(y\)-intercept is at coordinate (0,0) and there is no symmetry. The graph can be drawn knowing these facts along with a few additional random points for a more accurate representation.

Step 1: Determine the end behavior using Leading Coefficient Test

For a polynomial function, if the degree of the polynomial is even and the leading coefficient is negative, then the graph falls to the left and falls to the right. So, the end behavior of given polynomial will be: As \(x\) approaches \(-\infty\), \(f(x)\) approaches \(\infty\), and as \(x\) approaches \(\infty\), \(f(x)\) approaches \(\infty\).

Step 2: Finding \(x\)-intercepts

To obtain the \(x\)-intercepts of the curve, we set \(f(x) = 0\). This gives us \(-2x^{4}+4x^{3} = 0\). Now, factoring \(2x^3\) out, we get \(2x^{3}(-x+2) = 0\). So, \(x = 0\) or \(x = 2\). The behavior at the \(x\)-intercepts is to look at the corresponding factor of \(f(x)\). When the power of a factor is even, the graph touches the \(x\)-axis and turns around. When it is odd, the graph crosses the \(x\)-axis. Here, both factors \(-x^{3}\) and \(-x+2\) have odd powers. Thus, the graph crosses the \(x\)-axis at both 0 and 2.

Step 3: Finding \(y\)-intercept

Setting \(x=0\) and computing \(f(0)\), we get \(y = -2(0)^{4} + 4(0)^{3}=0\). Hence, \(y\)-intercept is at coordinate (0,0).

Step 4: Determining any symmetry

A graph of a function \(f\) is symmetric with respect to the \(y\)-axis if and only if \(f(-x) = f(x)\) for every number \(x\) in its domain. The given function when replaced with \(-x\) does not produce the same function and also not a function that is a reflection in the origin. Therefore, it does not have origin symmetry nor \(y\)-axis symmetry.

Step 5: Additional points and graph it.

From the end behavior, \(x\)-intercepts, \(y\)-intercept, and symmetry, we can sketch a rough plot of the function with turning points. We can estimate additional points by picking random values for \(x\) and solving for \(f(x)\). Remember, the maximum number of turning points is \(n-1\), where \(n\) is the degree of the function, so the maximum turning points here would be \(4-1=3\).

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test helps predict the end behavior of a polynomial function. It's like a crystal ball for graphing! For example, consider the polynomial function \[f(x) = -2x^4 + 4x^3.\] ​The degree of the polynomial is 4, which is even, and the leading coefficient is -2, which is negative. This combination means the graph will fall to the left and right. Essentially, as \(x\) approaches \(-\infty\) or \(+\infty\), \(f(x)\) will also fall towards \(-\infty\).
So, the end behavior gives you insight into how the arms of the graph behave in opposite directions.

x-intercepts

Finding the \(x\)-intercepts involves setting the function to zero and solving the equation. For instance, to find where the graph of \(f(x)=-2x^4+4x^3\) crosses the \(x\)-axis, set \(f(x) = 0\):

• \(-2x^4 + 4x^3 = 0.\)
• This can be factored as \(-2x^3(x-2) = 0.\)
• So, the solutions are \(x = 0\) and \(x = 2\).

The graph's behavior at these intercepts depends on the factors' powers at these points:

• If the power is odd, the graph crosses the \(x\)-axis.
• In this case, both factors \(-x^3\) and \(-x+2\) have odd exponents, meaning the graph crosses the \(x\)-axis both at 0 and 2.

y-intercepts

The \(y\)-intercept of a polynomial is found by evaluating the function at \(x = 0\). Think of it as figuring out where the graph touches or crosses the \(y\)-axis.For \(f(x) = -2x^4 + 4x^3\), substitute \(x=0\) and calculate \(f(0)\):

• \(f(0) = -2(0)^4 + 4(0)^3 = 0.\)

Therefore, the \(y\)-intercept is at (0, 0), meaning the graph passes through the origin. It's a crucial starting point to confirm your graph’s accuracy!

Symmetry of Graphs

Symmetry in polynomial graphs can guide you on how the curve is mirrored about certain axes or points. For example, a graph might be symmetric about the \(y\)-axis if \(f(-x)=f(x)\), or symmetric to the origin if \(f(-x) = -f(x)\). For \(f(x) = -2x^4 + 4x^3\), replace \(x\) with \(-x\):

• You get \(-2(-x)^4 + 4(-x)^3.\)
• After simplifying, the equation doesn’t match the original, nor is it the negative. This means there’s neither \(y\)-axis symmetry nor origin symmetry.

Symmetry is an elegant guide in graphing, making predictions easier about how parts of a graph can reflect each other.