Problem 18 - Algebra and Trigonometry

Question

Sketch the graph of the function by first making a table of values. \(h(x)=4 x^{2}-x^{4}\)

Step-by-Step Solution

Answer

Plot points from the table (-2,0), (-1,3), (0,0), (1,3), (2,0) and connect them to sketch the graph.

Step 1: Understand the Function

The function is given as \( h(x) = 4x^2 - x^4 \). It represents a polynomial of degree 4, which is an even-degree polynomial and likely has a parabolic shape.

Step 2: Choose Values for x

Select a range of x-values to compute the corresponding y-values for the function. A good range would be from -2 to 2 to capture the general shape of the polynomial.

Step 3: Calculate Corresponding y-values

Calculate \(h(x)\) for the chosen x-values:- \(x = -2\), \(h(-2) = 4(-2)^2 - (-2)^4 = 16 - 16 = 0\)- \(x = -1\), \(h(-1) = 4(-1)^2 - (-1)^4 = 4 - 1 = 3\)- \(x = 0\), \(h(0) = 4(0)^2 - 0^4 = 0\)- \(x = 1\), \(h(1) = 4(1)^2 - 1^4 = 4 - 1 = 3\)- \(x = 2\), \(h(2) = 4(2)^2 - 2^4 = 16 - 16 = 0\)

Step 4: Create the Table of Values

Organize the calculated points into a table:\[\begin{array}{c|c}x & h(x) \\hline-2 & 0 \-1 & 3 \0 & 0 \1 & 3 \2 & 0 \\end{array}\]

Step 5: Plot the Points

Using the table of values, plot the points \((-2, 0)\), \((-1, 3)\), \((0, 0)\), \((1, 3)\), \((2, 0)\) on the Cartesian plane.

Step 6: Sketch the Graph

Connect the plotted points smoothly. The graph should have a symmetrical parabolic shape, opening downwards with the highest points at \(x = -1\) and \(x = 1\), and intercepting the x-axis at \(x = -2\), \(x = 0\), and \(x = 2\).

Key Concepts

Table of Values

Creating a table of values is an essential step when sketching graphs of functions, such as polynomials. It helps visualize the function's behavior by listing specific values of \(x\) along with their corresponding function outputs, \(h(x)\). Follow these simple steps:
- Select a range of \(x\) values that will provide a good sense of the function's shape. For the function \(h(x)=4x^{2}-x^{4}\), using values from \(x = -2\) to \(x = 2\) captures the essence of the curve.- Substitute each \(x\) value into the function to calculate \(h(x)\).- Record the \(x\) and \(h(x)\) pairs in a table.For example:

• \(x = -2\), \(h(-2) = 0\)
• \(x = -1\), \(h(-1) = 3\)
• \(x = 0\), \(h(0) = 0\)
• \(x = 1\), \(h(1) = 3\)
• \(x = 2\), \(h(2) = 0\)

By organizing the values in a table, it's easier to identify any symmetry or patterns in the graph, which aids in sketching.

Even-Degree Polynomial

Even-degree polynomials, like \(h(x)=4x^{2}-x^{4}\), often exhibit symmetrical behavior about the y-axis. A polynomial's degree is determined by the highest power of \(x\) within its expression. Since this function includes the term \(x^4\), it is of degree 4, which is even.
The characteristics of even-degree polynomials include:

• They often have shapes resembling a U or an upside-down U (parabolic)
• Tend to be symmetric about the y-axis
• They may have multiple turning points depending on the specific polynomial

In our case, \(h(x)\) shows an upside-down U shape, indicating the function takes negative values as \(x\) grows very large or very small. Recognizing these traits helps in connecting and extending plotted points in the graph.

Plotting Points

Plotting points to sketch a graph is an intuitive way of visualizing how a function behaves. This involves taking the ordered pairs from the table of values and drawing them on a graph. Here’s how you can effectively plot points:
- Use a grid or graph paper for accuracy.- Mark each pair \((x, h(x))\), such as \((-2, 0)\), \((-1, 3)\), \((0, 0)\), \((1, 3)\), and \((2, 0)\), on the Cartesian plane.- Ensure your graph reflects symmetry, especially with even-degree polynomials where applicable.Once points are plotted, it's time to connect them with smooth lines that represent the overall continuous nature of polynomial graphs. The curve for the function \(h(x)=4x^{2}-x^{4}\) should display a downward-opening parabolic shape, with its peak points at \(x = -1\) and \(x = 1\), showing symmetry about the y-axis. Observing and connecting points as visualized can enhance your understanding of polynomial behavior.