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Chapter 2: Problem 43

19鈥�44 ? Make a table of values and sketch the graph of the equation. Find the x- and y-intercepts and test for symmetry. $$ y=x^{4} $$

Short Answer

Expert verified
The graph of \(y = x^4\) is U-shaped upwards, with both intercepts at (0,0); it is symmetric about the y-axis.

Step by step solution

01

Make a Table of Values

Create a table with two columns: one for \(x\) values and the other for corresponding \(y\) values using the equation \(y = x^4\). Choose a set of \(x\) values, such as \(-2, -1, 0, 1, 2\), and calculate the \(y\) values:\\[\begin{array}{c|c} x & y = x^4 \\hline-2 & 16 \-1 & 1 \ 0 & 0 \ 1 & 1 \ 2 & 16 \\end{array}\]
02

Sketch the Graph

Using the table of values, plot the points \((-2, 16), (-1, 1), (0, 0), (1, 1), (2, 16)\) on a graph. Connect these points smoothly to form the curve of the equation \(y = x^4\), which is a U-shaped curve that opens upwards.
03

Find the x- and y-intercepts

To find the y-intercept, set \(x = 0\) in the equation \(y = x^4\). Solving gives \(y = 0^4 = 0\). Thus, the y-intercept is \((0, 0)\). For the x-intercepts, set \(y = 0\): \(x^4 = 0\). Solving this gives \(x = 0\). The only x-intercept is also \((0, 0)\).
04

Test for Symmetry

Test for symmetry with respect to the y-axis, x-axis, and origin. The equation \(y = x^4\) remains unchanged if we replace \(x\) with \(-x\) (\(y = (-x)^4 = x^4\)), indicating y-axis symmetry. The graph does not change when \(y\) and \(x\) are both negated, indicating no symmetry with the x-axis or origin.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Table of Values
A table of values is an essential tool to help create a visual representation of a polynomial function. By selecting various values for \(x\), you can determine the corresponding \(y\) values by plugging them into the equation. For \(y = x^4\), you can choose \(x\) values such as \(-2, -1, 0, 1, 2\). When these are substituted into the equation, the resulting \(y\) values are \(16, 1, 0, 1, 16\) respectively.
This creates a clear mapping of points:
  • \((-2, 16)\)
  • \((-1, 1)\)
  • \((0, 0)\)
  • \((1, 1)\)
  • \((2, 16)\)
These coordinates are essential for sketching the graph accurately. Once plotted, these points can guide you to connect them smoothly, which in this case forms a distinct U-shaped curve opening upwards.
Exploring X-intercept
The x-intercept of a graph is the point where the graph crosses the x-axis, which occurs when \(y = 0\). For the equation \(y = x^4\), setting \(y = 0\) simplifies to \(x^4 = 0\). The solution to this equation is \(x = 0\).
Therefore, the only x-intercept for this graph is at the origin, which is the point \((0, 0)\). This means that the graph touches the x-axis at just this single point. Identifying the x-intercept is crucial as it provides insights into the behavior of the polynomial at crucial intersections like the origin.
Understanding Y-intercept
The y-intercept is the point where a graph crosses the y-axis, occurring when \(x = 0\). In the case of our equation \(y = x^4\), substituting \(x = 0\) gives \(y = 0^4 = 0\).
Therefore, the y-intercept is also the origin point \((0, 0)\). This point is significant because it acts as a foundational reference for graph symmetry and is often a starting point when drawing any graph manually or digitally.
Symmetry Testing with Polynomial Graphs
Symmetry testing helps us understand the visual balance of a graph. For the polynomial \(y = x^4\), it is beneficial to test symmetry with respect to the y-axis, x-axis, and origin.
To test for y-axis symmetry, substitute \(-x\) for \(x\). If you get the original equation back, the graph is symmetric about the y-axis. Here, substituting \(-x\) yields \(y = (-x)^4 = x^4\); hence, symmetry about the y-axis is confirmed.
  • No symmetry with the x-axis exists, as reversing both \(x\) and \(y\) does not yield the same function form.
  • There is also no origin symmetry, as substituting \((-x, -y)\) doesn't maintain the equation.
Understanding these symmetry properties aids in predicting the overall graph shape and behavior, especially when plotting complex polynomials.

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