Problem 43 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

End behavior is on both sides towards \(\infty\). \(x\)-intercepts are at \(x=0, 3, -3\). \(y\)-intercept is at \(y=0\). Function has y-axis symmetry. For a more precise graph, additional points need to be plotted considering maximum 3 turning points.

Step 1: End Behavior Determination

As per the Leading Coefficient Test, if the degree of the polynomial is even and the leading coefficient is positive, then the end behavior of the function's graph will be: up on the left (\(-\infty\)) and up on the right (\(\infty\)). Here, the leading term is \(x^4\), so the leading coefficient is 1 (which is positive) and the degree is 4 (which is even). Hence, the graph will go up to \(\infty\) as \(x\) goes to \(\infty\) and \(-\infty\).

Step 2: Finding the \(x\)-intercepts

An \(x\)-intercept of a function is a point where the function crosses or touches the \(x\)-axis. To find such points for \(f(x)=x^{4}-9x^{2}\), we set \(f(x)=0\):-- \(x^{4}-9x^{2}=0\) \-- \(x^{2}(x^{2}-9)=0\) \-- Here, the solutions are \(x=0, 3, -3\). Hence, the \(x\)-intercepts are \(x=0, 3, -3\). The function crosses the \(x\)-axis at these points.

Step 3: Finding the \(y\)-intercept

The \(y\)-intercept of a function is a point where the function crosses or touches the \(y\)-axis. To find it for \(f(x)=x^{4}-9x^{2}\), we set \(x=0\): -- \(f(x)=\((0)\)^{4}-9*((0)\)^{2}\) \-- \(f(0)=0\). Hence, the \(y\)-intercept is \(y=0\).

Step 4: Symmetry Determination

If replacing \(x\) with \(-x\) doesn't change the form of the function, then it has y-axis symmetry or is an even function. For this function,-- \(f(-x)=(-x)^{4}-9(-x)^{2}=x^{4}-9x^{2}=f(x)\) \ Since the function remains same after replacing \(x\) with \(-x\), it's an even function. Hence, the graph has y-axis symmetry.

Step 5: Additional Points and Graph

We can choose additional points and find corresponding values of \(f(x)\) to get a more precise graph. Also, considering the nature of this 4th degree polynomial, it can have up to three turning points which should also be confirmed while graphing.

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test is vital for predicting the end behavior of a polynomial's graph. This test inspects the leading term of the polynomial, specifically its degree and coefficient, to determine how the graph behaves as the input values stretch towards positive or negative infinity.

For instance, with the polynomial function given, f(x) = x^4 - 9x^2, we notice that the highest power of x is 4 (an even number) and its leading coefficient is 1 (a positive number). According to the test, since we have an even degree and a positive leading coefficient, the graph will rise on both ends. Therefore, the graph will ascend towards infinity as x approaches both positive and negative infinity.

x-intercepts

The x-intercepts of a graph are points where it intersects the x-axis, which can indicate roots or zeroes of the polynomial function. To find these intercepts, we solve for when f(x) = 0.

In our example with f(x) = x^4 - 9x^2, setting f(x) to zero and factoring out x^2 gives us x^2(x^2 - 9) = 0. Consequently, this yields x-intercepts at 0, 3, and -3. It's crucial to determine whether the graph touches the x-axis at these points and turns around or crosses it. In our case, the graph crosses the x-axis at each x-intercept.

y-intercept

The y-intercept is found where the graph meets the y-axis, which is exceptionally straightforward: we simply evaluate f(x) at x = 0.

For the function f(x) = x^4 - 9x^2, substituting x with 0 yields f(0) = 0^4 - 9(0)^2 = 0. This tells us that the y-intercept is at the point (0, 0), which also happens to be an x-intercept in this scenario.

Graph Symmetry

Analyzing graph symmetry helps us understand the function's behavior more comprehensively and simplifies graphing. A function can be symmetric with respect to the y-axis or the origin. For y-axis symmetry, replacing x with -x will yield the same function (even function). Alternatively, if f(-x) equals -f(x), the function is symmetric around the origin (odd function).

Let's apply this to our polynomial f(x) = x^4 - 9x^2. We find that f(-x) = (-x)^4 - 9(-x)^2 = x^4 - 9x^2, which is identical to the original function f(x). Hence, the graph has symmetry about the y-axis.

Turning Points

Turning points are locations on the graph where the function changes direction from increasing to decreasing or vice versa. The maximum number of turning points is always one less than the degree of the polynomial. Thus, with a 4th-degree polynomial like our f(x) = x^4 - 9x^2, there can be up to 3 turning points.

By graphing or analyzing the derivative of the function, we can locate these turning points. It is essential when graphing to check that your sketch includes all possible turning points, since missing one can result in an inaccurate representation of the function's behavior.