Problem 35 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The x- and y-intercepts are both at (0, 0).

Step 1: Create a Table of Values

Begin by selecting several values for the variable \(x\). Typically, we choose values such as \(-2, -1, 0, 1, 2\). For each \(x\) value, calculate the corresponding \(y\) value using the equation \(y = x^4\). This will help us in plotting the graph later. The table will look like this:- \(x = -2\), \(y = (-2)^4 = 16\)- \(x = -1\), \(y = (-1)^4 = 1\)- \(x = 0\), \(y = 0^4 = 0\)- \(x = 1\), \(y = 1^4 = 1\)- \(x = 2\), \(y = 2^4 = 16\)

Step 2: Plot the Points

Using the table of values, plot the points on a coordinate plane. For example, plot the points \((-2, 16), (-1, 1), (0, 0), (1, 1), (2, 16)\). Connect these points smoothly to show the shape of the graph, which resembles a parabola but opens like a bowl as it gets wider.

Step 3: Identify the Intercepts

The intercepts are found where the graph crosses the axes. The \(y\)-intercept is where the graph crosses the \(y\)-axis, which happens when \(x = 0\). For \(y = x^4\), when \(x = 0\), \(y = 0\). Thus, the \(y\)-intercept is at the point \((0, 0)\). The \(x\)-intercept is where the graph crosses the \(x\)-axis, which also occurs at \(x = 0\), hence the \(x\)-intercept is also at \((0, 0)\).

Step 4: Describe the Graph

Upon sketching, you'll notice that the graph of \(y = x^4\) is symmetric with respect to the \(y\)-axis, forming a U-shape. It touches the origin and rises steeply on both sides as \(|x|\) increases.

Key Concepts

Intercepts

In the context of polynomial functions, intercepts are crucial points that show where the graph of an equation crosses the axes. For the function \(y = x^4\), both the \(x\)-intercept and \(y\)-intercept occur at the origin (0, 0).

• \(x\)-intercept: This happens where the graph crosses the \(x\)-axis. In our equation, when \(y = 0\), \(x = 0\) as well, indicating the graph crosses at (0, 0).
• \(y\)-intercept: Located where the graph meets the \(y\)-axis, or where \(x = 0\). Substituting \(x = 0\) into \(y = x^4\) gives \(y = 0\), confirming the intercept at (0, 0).

Intercepts are significant in graphing as they provide insight into the shape and position of the graph on the coordinate plane.

Table of Values

Creating a table of values is a fundamental step when graphing functions, as it offers a clear view of how \(x\) values affect \(y\) values. Here's how to construct this for \(y = x^4\):

• Choose several \(x\) values such as -2, -1, 0, 1, and 2 to ensure a broad view of the function.
• Substitute each \(x\) into the equation to get corresponding \(y\) values:
• - When \(x = -2\), \(y = 16\)
• - When \(x = -1\), \(y = 1\)
• - When \(x = 0\), \(y = 0\)
• - When \(x = 1\), \(y = 1\)
• - When \(x = 2\), \(y = 16\)

This table helps visualize how the function behaves at different points, revealing its overall structure and pattern. By plotting these points, you see how \(y\) changes as \(x\) increases or decreases.

Symmetric Graph

A graph is symmetric when it mirrors itself across a specific axis, providing a beautiful balance in its layout. For \(y = x^4\), the graph displays symmetry around the \(y\)-axis.

• Such symmetry means for each point \((x, y)\) on one side, there's a matching point \((-x, y)\) on the opposite side.
• This property makes understanding and predicting the graph's shape much easier, as you can focus on just one half and then reflect it.

Symmetry in graphs is beneficial for sketching them quickly and accurately since it implies predictable behavior and patterns.

U-shaped Curve

The graph of the function \(y = x^4\) forms a U-shaped curve, which is distinct yet similar to a parabola. Here's what makes it unique:

• The graph starts at the origin (0, 0) and opens upward.
• Unlike the typical parabolic shape, which might be narrow, the curve for \(x^4\) is more bowl-like as it widens with increasing \(|x|\).
• This U-shape signifies how quickly \(y\) rises as \(|x|\) grows, creating steep arches on either side of the origin.

Understanding this unique shape is essential in recognizing how higher power polynomial functions, like \(x^4\), behave differently from quadratic ones like \(x^2\).