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Functions and models 1.3: New Functions from Old Functions

In this section, we will learn: How to obtain new functions from old functions and how to combine pairs of functions. Start with the basic functions we discussed in Section 1.2 and obtain new functions by shifting, stretching, and reflecting their graphs. | 1.3 New Functions from Old Functions In this section, we will learn: How to obtain new functions from old functions and how to combine pairs of functions. FUNCTIONS AND MODELS In this section, we: Start with the basic functions we discussed in Section 1.2 and obtain new functions by shifting, stretching, and reflecting their graphs. Show how to combine pairs of functions by the standard arithmetic operations and by composition. NEW FUNCTIONS FROM OLD FUNCTIONS By applying certain transformations to the graph of a given function, we can obtain the graphs of certain related functions. This will give us the ability to sketch the graphs of many functions quickly by hand. It will also enable us to write equations for given graphs. TRANSFORMATIONS OF FUNCTIONS Let’s first consider translations. If c is a positive number, then the graph of y = f(x) + c is just the graph of y = f(x) shifted upward a distance of c units. This is because each y-coordinate is increased by the same number c. Similarly, if g(x) = f(x - c) ,where c > 0, then the value of g at x is the same as the value of f at x - c (c units to the left of x). TRANSLATIONS Therefore, the graph of y = f(x - c) is just the graph of y = f(x) shifted c units to the right. TRANSLATIONS Suppose c > 0. To obtain the graph of y = f(x) + c, shift the graph of y = f(x) a distance c units upward. To obtain the graph of y = f(x) - c, shift the graph of y = f(x) a distance c units downward. SHIFTING To obtain the graph of y = f(x - c), shift the graph of y = f(x) a distance c units to the right. To obtain the graph of y = f(x + c), shift the graph of y = f(x) a distance c units to the left. SHIFTING Now, let’s consider the stretching and reflecting transformations. If c > 1, then the graph of y = cf(x) is the graph of y = f(x) stretched by a factor of c in the vertical direction. This is because each y-coordinate is multiplied by the same number c. STRETCHING AND REFLECTING The graph of y = | 1.3 New Functions from Old Functions In this section, we will learn: How to obtain new functions from old functions and how to combine pairs of functions. FUNCTIONS AND MODELS In this section, we: Start with the basic functions we discussed in Section 1.2 and obtain new functions by shifting, stretching, and reflecting their graphs. Show how to combine pairs of functions by the standard arithmetic operations and by composition. NEW FUNCTIONS FROM OLD FUNCTIONS By applying certain transformations to the graph of a given function, we can obtain the graphs of certain related functions. This will give us the ability to sketch the graphs of many functions quickly by hand. It will also enable us to write equations for given graphs. TRANSFORMATIONS OF FUNCTIONS Let’s first consider translations. If c is a positive number, then the graph of y = f(x) + c is just the graph of y = f(x) shifted upward a distance of c units. This is because each y-coordinate is increased by the same .