Problem 45 - 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. \(f(x)=-x^{4}+16 x^{2}\)

Step-by-Step Solution

Answer

The graph's end behavior is, it falls to the left and right. The \(x\)-intercepts are at \(x = 0, -4, 4\). At \(x=0\) the graph touches the \(x\)-axis and turns around, at \(x=-4\) and \(4\) the graph crosses the \(x\)-axis. The \(y\)-intercept is at \(0\). The graph has y-axis symmetry.

Step 1: Determine the graph's end behavior

To determine the end behavior of the graph, we will use the Leading Coefficient Test. As \(x\rightarrow \pm \infty\), the term with the highest power of \(x\) dominates. In this case, \(f(x)=-x^{4}+16 x^{2}\), the highest power of \(x\) is \(4\). As it's an even power, both ends of the graph go in the same direction. Since the leading coefficient is negative \(-1\), as \(x\rightarrow \pm \infty\), \(f(x)\rightarrow -\infty\). Hence the graph falls to the left and right.

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

To find the \(x\)-intercepts, set \(f(x)\) to zero and solve for \(x\). So, \(-x^{4}+16x^{2}=0 \rightarrow x^{4}=16x^{2} \rightarrow x^{4}-16x^{2}=0 \rightarrow x^{2}(x^{2}-16)=0\). Thus, \(x = 0,-4,4\). To determine whether the graph crosses the \(x\)-axis, we substitute in values slightly more and slightly less than each \(x\)-intercept into the function. Doing that for these \(x\) -intercepts, we find that at \(x=0\) the graph touches the \(x\)-axis and turns around, at \(x=-4\) and \(4\) the graph crosses the \(x\)-axis.

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

The \(y\)-intercept of a function is the value of \(f(0)\). Substituting \(x=0\) into \(f(x)\), we get \(f(0)=-0^{4}+16*0^{2}=0\). Therefore, the \(y\)-intercept is \(0\).

Step 4: Determine the graph's symmetry

To find whether the graph has y-axis symmetry, origin symmetry, or neither we have to check the equation of the function. For y-axis symmetry, the function must be unchanged when \(x\) is replaced with \(-x\). For origin symmetry, \(f(x)\) must be equal to \(-f(-x)\). Testing these conditions, it's found that, \(f(x)\) is unchanged when \(x\) is replaced with \(-x\), hence, the graph has y-axis symmetry. It does not have origin symmetry.

Step 5: Graph the function

Given that the end behavior indicates it is a valley facing downwards, this graph touches the \(x\)-axis at \(x = 0\) and crosses the \(x\)-axis at \(x = -4,4\), plus the \(y\)-intercept is at \(0\), we can graph the function knowing these elements. Then, verify the function with additional points like \(x = -5,5\). The function should have a maximum of \(3\) turning points as per the formula \((n-1)\) where \(n\) is the highest degree of the function.

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test is a useful tool in understanding the end behavior of polynomial functions. In the polynomial function given, \[-x^4 + 16x^2\], we identify the leading term as \(-x^4\). The degree of this term is 4, which is an even number, indicating both ends of the graph will move in the same direction as \(x\) approaches positive and negative infinity. Because the leading coefficient, which is \(-1\), is negative, the graph will fall on both the left and right side, meaning as \(x \rightarrow \pm \infty, f(x) \rightarrow -\infty\). This information is crucial for sketching the overall shape of the graph and places the groundwork for understanding how the function behaves at extreme values.

x-intercepts

Finding the \(x\)-intercepts involves setting the polynomial equal to zero: \[-x^4 + 16x^2 = 0\]. By factoring out \(x^2\), the equation can be solved as follows: \[x^2(x^2 - 16) = 0\]. This yields potential solutions \(x = 0,\ x = -4,\ x = 4\). These \(x\)-intercepts are points where the graph meets or crosses the x-axis. To determine the behavior at each intercept:- At \(x = 0\), the graph only touches the axis and turns around.- At \(x = -4\) and \(x = 4\), the graph crosses the \(x\)-axis. These interactions give hints about the function's roots and are important for plotting where the graph navigates on the coordinate plane.

y-intercepts

To find the \(y\)-intercept of a polynomial, evaluate the function at \(x = 0\). For \[f(x) = -x^4 + 16x^2\],plugging in \(x = 0\) results in:\[f(0) = -0^4 + 16 \times 0^2 = 0\].This indicates that the \(y\)-intercept is at \(0\). The point \((0,0)\) is where the graph intersects the y-axis. This intercept is a critical starting point for sketching the graph and confirms whether the graph passes through the origin, which it clearly does in this case.

Symmetry of Graphs

Symmetry in graphs can simplify understanding the function's behavior and structure. For a graph to have y-axis symmetry, it should satisfy the condition that replacing \(x\) with \(-x\) results in the same function:\[f(-x) = f(x)\].Checking with our function, \[-(-x)^4 + 16(-x)^2\] simplifies to \[-x^4 + 16x^2\],showing it is unchanged and confirming y-axis symmetry.Conversely, for origin symmetry, the function should satisfy \[f(-x) = -f(x)\]. Since this does not hold true, the graph does not possess origin symmetry. Recognizing symmetry helps predict the graph form without plotting many points.

Turning Points in Graphs

Turning points help in understanding how a graph peaks or dips. For the polynomial \[-x^4 + 16x^2\],its degree, 4, suggests it can have up to \(3\) turning points, calculated by \[n - 1\],where \(n\) is the polynomial's highest degree. Turning points are where the graph changes direction, creating peaks or valleys. In our graph, these turning points help shape the overall curve.To accurately graph the function, it's useful to check additional points such as \(x = -5, 5\) to see where these direction changes occur. This will ensure our graph reflects the function's behavior correctly, emphasizing how the polynomial "turns" as it progresses through its defined range.