Problem 36 - Algebra and Trigonometry

Question

\(11-36\) Make a table of values and sketch the graph of the equation. Find the \(x\) - and \(y\) -intercepts. $$ y=16-x^{4} $$

Step-by-Step Solution

Answer

\(x\)-intercepts: \((-2, 0)\) and \((2, 0)\); \(y\)-intercept: \((0, 16)\).

Step 1: Create a Table of Values

To construct a table of values for the equation \(y = 16 - x^4\), choose a range of \(x\) values to calculate corresponding \(y\) values. 1. Let \(x = -2\): \(y = 16 - (-2)^4 = 16 - 16 = 0\)2. Let \(x = -1\): \(y = 16 - (-1)^4 = 16 - 1 = 15\)3. Let \(x = 0\): \(y = 16 - 0^4 = 16\)4. Let \(x = 1\): \(y = 16 - 1^4 = 15\)5. Let \(x = 2\): \(y = 16 - 2^4 = 0\)6. Let \(x = 3\): \(y = 16 - 3^4 = -65\)(point goes off visible graph, included to indicate shape). This gives the table of values: \[\begin{array}{c|c}x & y \\hline-2 & 0 \-1 & 15 \0 & 16 \1 & 15 \2 & 0 \3 & -65 \\end{array}\]

Step 2: Plot the Points and Sketch the Graph

Using the table of values from Step 1, plot the points on a coordinate system: \((-2, 0), (-1, 15), (0, 16), (1, 15), (2, 0), (3, -65)\). The points \((-2, 0)\) and \((2, 0)\) show the \(x\)-intercepts, and the graph is symmetric with respect to the \(y\)-axis. The plotted points will outline a curve opening downwards. Draw this graph smoothly passing through the plotted points.

Step 3: Determine the x-intercepts

The \(x\)-intercepts are the points where the graph crosses the \(x\)-axis. These occur where \(y = 0\).From the table, see that the \(x\)-intercepts occur at \(x = -2\) and \(x = 2\). Thus, the \(x\)-intercepts are \((-2, 0)\) and \((2, 0)\).

Step 4: Determine the y-intercept

The \(y\)-intercept is the point where the graph crosses the \(y\)-axis. This occurs where \(x = 0\).From the table, when \(x = 0\), \(y = 16\). Thus, the \(y\)-intercept is \((0, 16)\).

Key Concepts

Polynomial Functions

Polynomial functions are expressions involving variables raised to powers, combined using addition, subtraction, and multiplication. The general form of a polynomial function is:

• \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\)

In this formula:

• \(a_n, a_{n-1}, \ldots, a_0\) are constants
• \(x\) represents the variable
• \(n\) is a non-negative integer, determining the degree of the polynomial

For example, the equation \(y = 16 - x^4\) illustrates a fourth-degree polynomial. This means the graph will feature a shape typical of fourth-degree functions. Understanding polynomials helps visualize how changes in affected terms modify the overall pattern of the graph.

X-intercepts

X-intercepts are the points where a graph crosses the x-axis. These points occur where the output of the function \(y\) is zero. For our example, \(y = 16 - x^4\), the x-intercepts are places where the equation equals zero. To find these:

• Set \(y = 0\).
• Resulting in \(16 - x^4 = 0\).
• Solving gives \(x^4 = 16\), which results in \(x = \pm2\).

Thus, the x-intercepts for this function are \((-2, 0)\) and \((2, 0)\). Observing x-intercepts help us grasp where the graph will touch or cross the x-axis.

Y-intercepts

Y-intercepts occur where a graph crosses the y-axis. For this to happen, the value of \(x\) is 0. So in our function \(y = 16 - x^4\), simply substitute \(x = 0\) into the expression.The equation becomes:

• \(y = 16 - 0^4\)
• This simplifies to \(y = 16\).

Hence, the y-intercept is at point \((0, 16)\). Y-intercepts can solidly indicate where the graph starts when you are sketching it, effectively acting as an anchor point.

Coordinate System

A coordinate system, like the Cartesian plane, serves to visually plot mathematical functions. It consists of two perpendicular axes, the x-axis and y-axis. Each point on this plane is defined by a pair of numerical coordinates \((x, y)\).In the context of our example \(y = 16 - x^4\), these coordinates allow us to precisely mark points derived from the table of values. This helps create a visual representation of the data and the possible curve of the polynomial function.

• The x-coordinate represents horizontal placement.
• The y-coordinate provides vertical positioning.

Once you accurately plot these across the defined axes, joining them smoothly illustrates the function's behavior.

Table of Values

Creating a table of values is a methodical way of understanding functions. It involves selecting various x-values, plugging each into your polynomial equation, and solving for the corresponding y-values. For \(y = 16 - x^4\), we calculated some values:

• \(x = -2\), then \(y = 0\)
• \(x = -1\), then \(y = 15\)
• \(x = 0\), then \(y = 16\)
• \(x = 1\), then \(y = 15\)
• \(x = 2\), then \(y = 0\)

These points aid in predicting the actual curve when plotted, showing how the function behaves over its domain. The curve becomes clearer as more points are added, helping to sketch or visualize the graph of a polynomial function effectively.