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Chapter 11: Problem 14

Graph each function. \(f(x)=-x^{4}+2\)

Short Answer

Expert verified
The graph of the function is a downward-opening quartic curve with a maximum point at (0, 2).

Step by step solution

01

- Identify the Function Type

The given function is a polynomial function. Specifically, it is a quartic function because the highest power of the variable x is 4.
02

- Determine the General Shape

Quartic functions generally have a 'W' or 'M' shape. Since the leading coefficient (the coefficient of the highest power term) is negative, the graph will have a downward opening relative to the standard quartic graph.
03

- Find Key Points

To graph the function, we need to find some key points. Start by evaluating the function at various values of x. For example: - When x = 0, f(0) = -0^4 + 2 = 2. - When x = 1, f(1) = -1^4 + 2 = 1. - When x = -1, f(-1) = -(-1)^4 + 2 = 1. Calculate more points if needed.
04

- Plot the Key Points

Now plot the points found in step 3: - (0, 2)- (1, 1)- (-1, 1). Connect these points with a smooth curve. Ensure the shape of the graph opens downwards, resembling an upside down 'W'
05

- Sketch the Graph

Using the points and the determined shape, draw the complete graph of the function. Make sure the endpoints of the curve go down towards negative infinity as x moves away from the origin in both positive and negative directions.

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Key Concepts

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

Polynomial Functions
Graphing functions involves plotting points and understanding the general behavior of the function. For \(f(x) = -x^4 + 2\), we start by identifying key points, such as where the function crosses the y-axis, and other significant points derived by plugging in different values of x. In this function:
  • When \(x = 0, f(0) = -0^4 + 2 = 2\)
  • When \(x = 1, f(1) = -1^4 + 2 = 1\)
  • When \(x = -1, f(-1) = -(-1)^4 + 2 = 1\)
Plot these points on a coordinate system and connect them with a smooth curve.

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