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Notes 1G (Graphing)

# Notes 1G (Graphing) - MIT OpenCourseWare http/ocw.mit.edu...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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G. GRAPHING FUNCTIONS To get a quick insight into how the graph of a function looks, it is very helpful to know how certain simple operations on the graph are related to the way the function expression looks. We consider these here. 1. Right-left translation. Let c > 0. Start with the graph of some function f(x). Keep the x-axis and y-amis fixed, but move the graph c units to the right, or c units to the left. (See the pictures below.) You get the graphs of two new functions: (1) Moving the f(x) graphc units to the right gives the graph of (x - c) S left f (X + c) If f(x) is given by a formula in x, then f(x - c) is the function obtained by replacing x by x - c wherever it occurs in the formula. For instance, f() =(--)= (x- ) 2 +(X-) 2 --z, byalgebra. Example 1. Sketch the graph of f(s) = x 2 - 2x + 1. Solution. By algebra, f(x) = (X - 1)2. Therefore by (1), its graph is the one obtained by moving the graph of x 2 one unit to the right, as shown. The result is a parabola touching the x-axis at x = 1. To see the reason for the rule (1), suppose the graph of f(z) is moved c units to the right: it becomes then the graph of a new function g(x), whose relation to f(s) is described by (see the picture): value of g() at zo = value off(z) at o - c f(zo-c). This shows that g(z) = f(x - c). x j xY The reasoning is similar if the ( graph is translated c units to the left. Try giving the argument yourself while referring to the picture. V-C The effect of up-down translation of the graph is much simpler to see. If c > 0, (2) Moving the f (x)graph cunits f(T) + gives the graph of -c down f (x)-- c since for example moving the graph up by c units has the effect of adding c units to each function value, and therefore gives us the graph of the function f(s) + c Example 2. Sketch the graph of 1 + T/2-. Solution Combine rules (1) and (2). First sketch V • , then move its graph 1 unit to the right to get the graph of Vf-Z, then 1 unit up to get the graph of 1 +-,• 1. as shown. 1ni o th o f1+-,fm -I x
Example 3. Sketch the curve y = z 2 + 4x +1. Solution We "complete the square": 2+4 z+ = (z 2 +4z+4)-3 = (z+2) 2 -3, so we move the graph of z 2 to the left 2 units, then 3 units down, nettine the granh shown. 2. Changing scale: _stretching and shrinking. Let c > 1. To stretch the z-axis by the factor c means to move the point 1to the position formerly occupied by c,and in general, the point so to the position formerly occupied by c0o. Similarly, to shrink the z-axis by the factor c means to move so to the position previously taken by zo/c. What happens to the graph of f(z) when we stretch or shrink the z-axis? Stretching

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Notes 1G (Graphing) - MIT OpenCourseWare http/ocw.mit.edu...

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