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Chapter 2: Problem 49

a. Use the Leading Coefficient Test to determine the graph's end behavior. b. Find the \(x\) -intercepts. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each intercept. c. Find the \(y\) -intercept. 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 maximum number of turning points to check whether it is drawn correctly. $$f(x)=-2 x^{4}+4 x^{3}$$

Short Answer

Expert verified
End behavior: \(f(x) \to -\infty\) as \(x \to \infty\) or \(x \to -\infty\). \(x\)-intercepts: at \(x=0\) and \(x=1\), crosses \(x\)-axis. \(y\)-intercept: at \(y=0\). Symmetry: Neither \(y\)-axis symmetry nor origin symmetry. The graph crosses the \(x\)-axis at \(x=0\) and \(x=1\) and has at most 3 turning points.

Step by step solution

01

Determine the Graph's End Behavior

Using the Leading Coefficient Test, the end behavior of a polynomial function's graph is determined by the degree and the sign of the leading coefficient. Here, the degree of the function \(f(x)=-2 x^{4}+4 x^{3}\) is 4, an even number, and the leading coefficient is -2, a negative number. Hence, as \(x\) approaches positive or negative infinity, \(f(x)\) approaches negative infinity. The end behavior can be written as \(f(x) \to -\infty\) as \(x \to \infty\) or \(x \to -\infty\).
02

Find the \(x\)-intercepts

To find the \(x\)-intercepts, set the function equal to zero and solve for \(x\). So, set \(-2 x^{4}+4 x^{3}=0\). Factoring out \( -2x^{3}\) gives \(-2x^{3}(2x-2)=0\). Thus, the \(x\)-intercepts are at \(x=0\) and \(x=1\). As the multiplicity of \(x\) at these points is odd, the graph crosses the \(x\)-axis at each intercept.
03

Find the \(y\)-intercept

To find the \(y\)-intercept, replace \(x\) with zero. So, substituting \(x=0\) into \(f(x)=-2 x^{4}+4 x^{3}\), results in \(f(0)=0\). Hence, the \(y\)-intercept is at \(y=0\).
04

Determine the Graph Symmetry

Next, check whether the graph has \(y\)-axis symmetry, origin symmetry, or neither. Replace \(x\) with \(-x\) in the equation \(f(x)=-2 x^{4}+4 x^{3}\) to get \(f(-x)=-2 (-x)^{4}+4 (-x)^{3} = -2 x^{4}-4 x^{3}\). Since \(f(-x) \neq f(x)\) and \(f(-x) \neq -f(x)\), the graph has neither \(y\)-axis symmetry nor origin symmetry.
05

Graph the Function

To sketch the graph, place the \(x\)-intercept points at \(x=0\) and \(x=1\), and the \(y\)-intercept at \(y=0\). From the end behavior, the graph starts and ends in the 3rd quadrant, crossing the \(x\)-axis at \(x=0\) and \(x=1\). Since the highest degree of the polynomial is 4, the graph has at most 3 turning points.

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