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Chapter 3: Problem 43

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. $$f(x)=x^{4}-9 x^{2}$$

Short Answer

Expert verified
The end behavior is that as \(x\) approaches positive or negative infinity, \(f(x)\) approaches positive infinity. The x-intercepts are at \(x=0\), \(x=3\), and \(x=-3\), at each of which the graph crosses the x-axis. The y-intercept is (0,0). The graph has neither y-axis symmetry nor origin symmetry. After graphing the function along with the intercepts and taking the end behavior into account, the graph correctly turns at most 3 times as per the degree of the polynomial.

Step by step solution

01

Determine the graph's end behavior

To find the end behavior, look at the degree and leading coefficient of the polynomial. The degree of the function \(f(x)=x^{4}-9 x^{2}\) is 4, which is even and the leading coefficient is positive. For polynomials with even degree and positive leading coefficient, the end behavior is that as \(x\) approaches positive or negative infinity, \(f(x)\) approaches positive infinity.
02

Find the x-intercepts

The x-intercepts are found by setting the function equal to zero and solving for \(x\), \(0=x^{4}-9x^{2}\). Factoring out \(x^2\) leads to \(0=x^{2}(x^{2}-9)\). This gives us three solutions: \(x=0\), \(x=3\), \(x=-3\). Thus, the graph crosses the x-axis at these points.
03

Find the y-intercept

The y-intercept is found by setting \(x=0\) in the equation, which gives us \(f(0)=0\). Thus, the y-intercept is \((0,0)\).
04

Determine symmetry

There is y-axis symmetry if replacing \(x\) with \(-x\) results in the same function, and origin symmetry if replacing \(x\) with \(-x\) gives the negation of the function. Doing this, one finds that neither condition applies. Thus, the graph has neither y-axis symmetry nor origin symmetry.
05

Graph the function

Now we want to graph the function. The maximum number of turning points for a degree 4 polynomial is \(4-1=3\). The points derived from the previous steps (\(x=0\), \(x=3\), \(x=-3\), y-intercept \((0,0)\)) are used to sketch the graph. First plot the intercepts, then, considering the end behavior and the symmetry, add relevant turning points while making sure the graph doesn't turn more than 3 times.

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