Recall from Chapter 1 that a unique line is determined by...

Recall from Chapter 1 that a unique line is determined by two distinct points on the line and that the values of $m$ and $b$ can then be determined for the general form of the linear function $$ f(x)=m x+b $$ Keeping the same two $x$ -values as given in Exercise 81 add 6 to each $y$ -value. What are the coordinates of the two new points?

Recall from Chapter 1 that a unique line is determined by two distinct points on the line and that the values of $m$ and $b$ can then be determined for the general form of the linear function
$$
f(x)=m x+b
$$
Keeping the same two $x$ -values as given in Exercise 81 add 6 to each $y$ -value. What are the coordinates of the two new points?

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Key Concepts

Linear Functions

A linear function is expressed as f(x) = mx + b, where m represents the slope and b the y-intercept. It defines a straight-line graph in the coordinate plane, emphasizing the constant rate of change between x and y. This concept is fundamental in algebra and analytic geometry for understanding relationships and predicting outcomes.

Vertical Translation

A vertical translation involves adding or subtracting a constant to each output value of a function, resulting in the graph shifting upward or downward. For linear functions, this means every point on the line moves the same distance vertically, preserving the slope and shape of the line while changing its vertical position.

Coordinate Points in the Context of Graphs

Coordinate points describe specific locations on the graph of a function, representing the relationship between the input (x-value) and the output (y-value). Identifying and manipulating these points, such as by applying vertical translations, is key to understanding and altering the graph's position and behavior.

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