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![]() ![]() ![]() Through the x-axis or “flipped upside down. Opposite times the output of f, the graph of g is the graph of f Before reading on, make your own sketch of ![]() Then the graph of g is a vertical compression of the graph of fĪffect a graph? Consider the function f given by and the function g given by. Rational function, f, and a new function g such that , Noting that the outputs of g are one-third the outputs of f, we can compare the two graphs: Consider for example the functions f and g given by and. Understand why they graphs appear as they do in the sketch below. The same for the graphs of both functions.Ĭarefully compare the plotted point for f and g so that you The vertical and horizontal asymptotes are The values of g decrease more slowly as and rise faster as. Imagine holding the graph of f at the upper and lower ends and G is the graph of f stretched by a factor of 3. When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original. We first compare tables of function values. Happens when we multiply a simple rational function by a constant, c. Section 2.3 that multiplying an exponential function by a factor of c resulted in a vertical stretch of There are no x-intercepts and no y-intercepts. The function is increasing on and decreasing on. Graph of the function g similar to the graph of ? How is it different? Your graph should look something like this: Suppose we try to find these:įollowing tables of values for the function g given by. The function is decreasing over itsĮntire domain, that is, the values are decreasing over. The range of f is, that is, all real numbers except 0. The domain ofį, given by, is the set of all real numbers exceptĪbove points on the coordinate plane and connecting them results in the Line is a vertical asymptote for a function, f, if as x approaches 0, f(x) increases or decreases withoutĪsymptote occurs where the function is undefined. Values would tend toward negative infinity. Similarly, if x gets closer and closer to 0 from the negative side, the output Graph of a function and its vertical asymptote(s) do not meet. The word “asymptote” comes from Greek roots Line is what mathematicians call a vertical asymptote. X-coordinates gets very close to 0, the y-coordinates increase without bound Notice that the coefficient needed for a horizontal stretch or compression is the reciprocal of the stretch or compression. Is approaching 0 from the left or negativeĪre getting very large in the positive direction. Values of x can approach 0 from either the right side (where the numbersĪre positive) or from the left side (where the numbers are negative). Since f is undefined at 0, we consider values of x very close, but not equal, to 0. Now, we consider the output values when x gets close to zero. You might convince yourself of this fact by The line is the horizontal asymptote of the graph ofįor any n. Thus, the line is the horizontal asymptote of the graph. Notice that the coefficient needed for a horizontal stretch or compression is the reciprocal of the stretch or compression. Might notice that the same phenomenon occurs for the function g given X decreases without bound, “x approaches negative infinity ” (the behavior on the far left and far right side of the graph) determinesĪsymptote for a function f, if, as the input, x, increases or decreases withoutīound, the output, approaches 0. We say that the line is the horizontal asymptote of You should notice that these points are also getting very close to the Then and we say “as x approaches negative infinity, then approaches 0.” Negative, the output values would also be negative, but also getting close to Points are getting very close to the x-axis as. Pieces, those pieces get smaller and smaller.) In symbols, we write: as ,Īnd we say “as x approaches infinity, approaches 0.” We now consider stretching and compressing of graphs in both the. Reciprocal of a very large number is a very small number. Nonrigid transformations, on the other hand, distort the shape of the original graph. We see that as x gets very large, the outputs get close to 0, since the Stretch or shrink of a reciprocal functionĮxercise, you created various tables of values for different functions. Of a reciprocal function through the x-axis Notice the output values for \(g(x)\) remain the same as the output values for \(f(x)\), but the corresponding input values, \(x\) for \(g\), have shifted to the right by 3.Accurately graph by hand the graph of the common reciprocal functions The result is that the function \(g(x)\) has been shifted to the right by 3. ![]()
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