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Chapter 5: Problem 31

Use transformations of the graph of \(y=x^{4}\) or \(y=x^{5}\) to graph each function. $$ f(x)=\frac{1}{2} x^{4} $$

Short Answer

Expert verified
Vertically compress the graph of \( y = x^4 \) by a factor of \( \frac{1}{2} \).

Step by step solution

01

- Understand the Parent Function

The parent function is given as \( y = x^4 \). This is a basic polynomial function where the graph has a characteristic U-shape that is steeper than \( y = x^2 \).
02

- Identify the Transformation

The given function is \( f(x) = \frac{1}{2} x^4 \). This indicates that the graph of the parent function \( y = x^4 \) will be vertically compressed by a factor of \( \frac{1}{2} \).
03

- Effect of the Vertical Compression

A vertical compression by a factor of \( \frac{1}{2} \) means that every point on the graph of the parent function \( y = x^4 \) will be half as far from the x-axis. For example, if a point on \( y = x^4 \) had coordinates (1, 1), then on \( f(x) = \frac{1}{2} x^4 \) it will be (1, \( \frac{1}{2} \)).
04

- Plot Key Points

To create the graph, compute key points by taking values of x and applying the transformation. For instance: - When \( x = 0 \), \( f(x) = \frac{1}{2} (0)^4 = 0 \)- When \( x = 1 \), \( f(x) = \frac{1}{2} (1)^4 = \frac{1}{2} \)- When \( x = 2 \), \( f(x) = \frac{1}{2} (2)^4 = 8 \)Do this for both positive and negative values of x.
05

- Draw the Transformed Graph

Using the computed points, plot the transformed function on a coordinate plane. It will look like a broader U-shape compared to the parent function \( y = x^4 \). Connect the points smoothly, ensuring the shape remains continuous and symmetric about the y-axis.

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Key Concepts

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

parent function
The parent function is the most basic form of a function without any transformations applied. In this case, the parent function is defined as \( y = x^4 \). This polynomial function forms a U-shaped curve that is symmetrically centered on the y-axis. The steepness of the curve is greater than that of the simpler quadratic function \( y = x^2 \). Understanding the inherent shape and properties of the parent function is crucial because it serves as the foundation upon which transformations will be applied.
vertical compression
In our given function \( f(x) = \frac{1}{2}x^4 \), a vertical compression has been applied to the parent function \( y = x^4 \). A vertical compression is a transformation that squeezes the graph towards the x-axis. The factor of compression in this case is \( \frac{1}{2} \), meaning each point on the graph of the parent function will be half as far from the x-axis as it originally was. For instance, a point at (1, 1) on \( y = x^4 \) will move to (1, \( \frac{1}{2} \)) on \( f(x) = \frac{1}{2}x^4 \). This results in a graph that still maintains its U-shape but appears broader compared to the original.
plotting points
Plotting points is a critical step in graphing any function, especially after transformations. Start by selecting key values of x. For instance:
  • When \( x = 0 \), \( f(x) = \frac{1}{2}(0)^4 = 0 \). The point is (0, 0).
  • When \( x = 1 \), \( f(x) = \frac{1}{2}(1)^4 = \frac{1}{2} \). The point is (1, \( \frac{1}{2} \)).
  • When \( x = 2 \), \( f(x) = \frac{1}{2}(2)^4 = 8 \). The point is (2, 8).
Repeat this process for both positive and negative values of x to ensure the graph is complete. By calculating and plotting these key points, we create a skeleton of the transformed function.
transformed graph
To visualize the transformed graph, we use the key points obtained from the previous step. Start by marking the plotted points on a coordinate plane. The transformed graph of \( f(x) = \frac{1}{2}x^4 \) will have a shape similar to the parent function but will appear wider or more compressed vertically.
The curve remains continuous and symmetric about the y-axis, just like the parent. Connect the points smoothly to form the final graph of the transformed function. The result will be a broader U-shaped curve, distinct yet related to the parent function. Understanding the impact of each transformation helps in accurately predicting and sketching the graph.

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Most popular questions from this chapter

Solve each equation in the real number system. $$ 2 x^{4}-19 x^{3}+57 x^{2}-64 x+20=0 $$

Determine whether the graph of $$\left(x^{2}+y^{2}-2 x\right)^{2}=9\left(x^{2}+y^{2}\right)$$ is symmetric with respect to the \(x\) -axis, \(y\) -axis, origin, or none of these.

We begin with two consecutive integers, \(a\) and \(a+1,\) for which \(f(a)\) and \(f(a+1)\) are of opposite sign. Evaluate \(f\) at the midpoint \(m_{1}\) of \(a\) and \(a+1 .\) If \(f\left(m_{1}\right)=0,\) then \(m_{1}\) is the zero of \(f,\) and we are finished. Otherwise, \(f\left(m_{1}\right)\) is of opposite sign to either \(f(a)\) or \(f(a+1) .\) Suppose that it is \(f(a)\) and \(f\left(m_{1}\right)\) that are of opposite sign. Now evaluate \(f\) at the midpoint \(m_{2}\) of \(a\) and \(m_{1} .\) Repeat this process until the desired degree of accuracy is obtained. Note that each iteration places the zero in an interval whose length is half that of the previous interval. Use the bisection method to approximate the zero of \(f(x)=8 x^{4}-2 x^{2}+5 x-1\) in the interval [0,1] correct to three decimal places. [Hint: The process ends when both endpoints agree to the desired number of decimal places.

Let \(f(x)\) be a polynomial function whose coefficients are integers. Suppose that \(r\) is a real zero of \(f\) and that the leading coefficient of \(f\) is \(1 .\) Use the Rational Zeros Theorem to show that \(r\) is either an integer or an irrational number.

Find bounds on the real zeros of each polynomial function. $$ f(x)=3 x^{4}-3 x^{3}-5 x^{2}+27 x-36 $$
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