A good approximation of the normal weight of a person 60...

A good approximation of the normal weight of a person 60 inches or taller but not taller than 80 inches is given by $w(x)=5.5 x-220,$ where $x$ is height in inches and $w(x)$ is weight in pounds. (A) Describe how the graph of function $w$ can be obtained from the graph of one of the basic functions in Figure 1 , page $19 .$ (B) Sketch a graph of function $w$ using part (A) as an aid.

A good approximation of the normal weight of a person 60 inches or taller but not taller than 80 inches is given by $w(x)=5.5 x-220,$ where $x$ is height in inches and $w(x)$ is weight in pounds.
(A) Describe how the graph of function $w$ can be obtained from the graph of one of the basic functions in Figure 1 , page $19 .$
(B) Sketch a graph of function $w$ using part (A) as an aid.

Key Concepts

Linear Functions

A linear function is one whose graph is a straight line. It is characterized by a constant rate of change (slope) and can be expressed in the form f(x) = mx + b, where m represents the slope and b represents the y-intercept. This concept is fundamental for understanding how changes in x correspond to constant changes in f(x).

Basic Function Transformation

Graph transformations involve altering a simple, well?understood base function to create a more complex function. In this case, the base function is the identity function f(x) = x. Transformations such as scaling (multiplying the independent variable or the function by a constant) and shifting (adding or subtracting a constant) are applied to this base function to obtain the desired function.

Vertical Scaling

Vertical scaling is the process of stretching or compressing the graph of a function by multiplying its output by a constant factor. When a function f(x) is multiplied by a constant, the steepness or slope of its graph changes accordingly. This transformation changes the rate at which the function’s values grow or decline as x increases.

Vertical Translation

Vertical translation is the process of shifting the graph of a function up or down by adding or subtracting a constant to the function’s output. This shifts the y-intercept of the graph without altering its slope, resulting in the entire graph moving uniformly in the vertical direction.

Domain Restrictions

Domain restrictions limit the set of input values (x-values) for which a function is defined. When graphing a function with a restricted domain, it is important to clearly indicate the portion of the graph that applies, even if the function's formula is defined in a broader context. This ensures that the graph accurately represents the situation being modeled.

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