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Chapter 4: Problem 28

Determine whether each function is linear or nonlinear. If it is linear, determine the slope. $$ \begin{array}{|rc|} {\boldsymbol{x}} & \boldsymbol{y}=\boldsymbol{f}(\boldsymbol{x}) \\ \hline-2 & 0 \\ -1 & 1 \\ 0 & 4 \\ 1 & 9 \\ 2 & 16 \\ \hline \end{array} $$

Short Answer

Expert verified
The function is nonlinear.

Step by step solution

01

- Understanding the Table

The table gives pairs \(x, f(x)\). A function is linear if the rate of change (slope) between any two points is constant.
02

- Calculate the Slopes

Calculate the slope between each pair of consecutive points:1. \((x_1, y_1) = (-2, 0)\) and \((x_2, y_2) = (-1, 1)\)\[ \text{Slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 0}{-1 - (-2)} = \frac{1}{1} = 1 \]2. \((x_1, y_1) = (-1, 1)\) and \((x_2, y_2) = (0, 4)\)\[ \text{Slope} = \frac{4 - 1}{0 - (-1)} = \frac{3}{1} = 3 \]3. \((x_1, y_1) = (0, 4)\) and \((x_2, y_2) = (1, 9)\)\[ \text{Slope} = \frac{9 - 4}{1 - 0} = \frac{5}{1} = 5 \]4. \((x_1, y_1) = (1, 9)\) and \((x_2, y_2) = (2, 16)\)\[ \text{Slope} = \frac{16 - 9}{2 - 1} = \frac{7}{1} = 7 \]
03

- Determine Linearity

The slopes calculated are 1, 3, 5, and 7, which are not constant. Therefore, the function is nonlinear.

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Key Concepts

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

Calculating Slope
The slope in a function describes how steep a line is. To find the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the formula: \[ \text{Slope} = \frac{y_2 - y_1}{x_2 - x_1} \] This formula indicates the change in y divided by the change in x. Slope is a crucial concept in understanding linear functions. If the slope between every pair of points in a dataset remains the same, the function is linear. In the given exercise, we calculated slopes between consecutive points: \[ \frac{1 - 0}{-1 + 2} = 1, \ \frac{4 - 1}{0 + 1} = 3, \ \frac{9 - 4}{1 - 0} = 5, \ \frac{16 - 9}{2 - 1} = 7 \] These calculations clearly show the slope is not constant.
Understanding Rate of Change
Rate of change is another term for slope when talking about functions. It represents how one quantity changes in relation to another. In linear functions, the rate of change between any two points is always the same. This means that in a graph, the line is straight. If the rate of change varies between points, this indicates a nonlinear function. From the exercise, we observed slopes of 1, 3, 5, and 7. This varying rate of change tells us the function is nonlinear. Knowing if a function's rate of change is constant or not helps to quickly determine if it's linear or nonlinear.
Determining Function Type
To determine whether a function is linear or nonlinear, follow these steps:
  • Calculate the slope between several pairs of points.
  • Check if the slopes are constant.
If every calculated slope equals, the function is linear. Linear functions create straight lines on a graph. Conversely, if the slopes differ, the function is nonlinear. Nonlinear functions result in curved lines on a graph. In the provided exercise, varying slopes of 1, 3, 5, and 7 confirmed the function was nonlinear. Understanding these methods enables quick verification of a function's type and aids in deeper comprehension of mathematical relationships.

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