Question

The most basic form of a linear function is f(x)=x. To write linear functions, we have been using the general form of g(x)=m(x−xᵣₑ) +yᵣₑ.

Explain how the general form is a series of transformations on f to create g. That is.
a. What kind of transformation is m ?How does this affect the graph?
b. What kind of transformation is xᵣₑ? How does this affect the graph?
c. What kind of transformation is yᵣₑ? How does this affect the graph?

Answer 1

The general form of a linear function includes a slope (m), a horizontal shift (xₜ), and a vertical shift (yₜ), each of which transforms the basic function f(x)=x to create various linear relationships depicted on a graph.

Step-by-step explanation:

Transformations of the Linear Function f(x)=x

The equation g(x)=m(x-xₜ) +yₜ represents a transformed version of the basic linear function f(x)=x. Each component of g(x) represents a different kind of transformation:

• The factor m represents the slope of the line. It is a scaling transformation that affects the steepness of the graph. A larger absolute value of m results in a steeper line, while a negative value results in the line sloping downwards.
• The value xₜ corresponds to a horizontal shift. If xₜ is positive, the graph of the line is shifted to the right by xₜ units, and if it is negative, to the left.
• The value yₜ is a vertical shift known as the y-intercept. It moves the entire line up if yₜ is positive or down if negative. The graph crosses the y-axis at (0, yₜ).

These transformations allow for the graph of any linear function to be easily plotted and interpreted, representing a variety of relationships between independent and dependent variables.