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Which Table Represents a Linear Function

In mathematics, a linear function is a function that can be represented by a straight line. It is a fundamental concept in algebra and is used extensively in various fields of mathematics, science, engineering, and economics. A linear function has a constant rate of change, which means that for every unit change in the input variable, the output variable changes by a constant amount. In this article, we will discuss what a linear function is, how to identify a linear function from a table, and how to graph a linear function.

A linear function is a function of the form f(x) = mx + b, where m and b are constants. The constant m is the slope of the line, which represents the rate of change of the function, and b is the y-intercept, which represents the value of the function when x = 0. The slope of the line is the ratio of the change in the output variable to the change in the input variable.

For example, consider the following function:

f(x) = 2x + 1

The slope of the line is 2, which means that for every unit change in the input variable x, the output variable y changes by 2 units. The y-intercept is 1, which means that when x = 0, the value of the function is 1.

A linear function can be identified from a table by looking for a constant rate of change in the output variable for every unit change in the input variable. To find the slope of the line from a table, we can use the formula:

slope = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are any two points on the line. If the slope is constant for all pairs of points, then the function is linear.

For example, consider the following table:

x | y |

0 | 1 |

1 | 3 |

2 | 5 |

3 | 7 |

To find the slope of the line, we can choose any two points on the line, for example, (0, 1) and (1, 3). Then the slope is:

slope = (3 - 1) / (1 - 0) = 2

If we choose any other pair of points, we will get the same slope, which means that the function is linear. We can also find the y-intercept by choosing any point on the line and solving for b in the equation f(x) = mx + b.

To graph a linear function, we need to plot at least two points on the line and connect them with a straight line. We can use the slope-intercept form of the equation to find the y-intercept and plot the point (0, b). Then we can use the slope to find another point on the line, for example, by starting from the y-intercept and moving up or down by the slope for each unit change in the x-coordinate. We can then connect the two points with a straight line.

For example, consider the function:

f(x) = 2x + 1

To graph this function, we can start by plotting the y-intercept (0, 1). Then we can use the slope 2 to find another point on the line by moving up by 2 units for every unit change in x. For example, when x = 1, y = 3, and when x = 2, y = 5. We can plot these points and connect them with a straight line.

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