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 $$ f(x)=m x+b $$ Graph both $y_{1}$ and $y_{2}$ in the standard viewing window of your calculator, and describe how the graph of $y_{2}$ can be obtained by vertically translating the graph of $y_{1}$ What is the value of the constant in this vertical translation? Where do you think it comes from?

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
$$
f(x)=m x+b
$$
Graph both $y_{1}$ and $y_{2}$ in the standard viewing window of your calculator, and describe how the graph of $y_{2}$ can be obtained by vertically translating the graph of $y_{1}$ What is the value of the constant in this vertical translation? Where do you think it comes from?

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Key Concepts

Linear Functions in Slope?Intercept Form

This concept involves representing a line using the formula f(x) = mx + b, where m is the slope (indicating the steepness and direction of the line) and b is the y-intercept (the point where the line crosses the y-axis). This form quickly communicates key properties of a linear function and facilitates graphing and analyzing lines.

Vertical Translation

A vertical translation is a transformation that shifts a graph up or down without altering its shape. In the context of linear functions, adding or subtracting a constant to the function's output results in the entire line moving vertically by that constant value. The constant in the translation represents the difference in the y-intercepts of the two functions, and it originates from how the function is adjusted to achieve the shift.

Function Transformations

This broader concept encompasses various changes applied to the basic graph of a function — including vertical and horizontal translations, reflections, stretches, and compressions. Vertical translations, in particular, are important because they show how adding a constant value to a function's output directly affects the position of the graph in the coordinate plane without altering its slope or general shape.

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