![]() ![]() STRETCHES AND COMPRESSIONS OF THE PARENT FUNCTION (b) compresses the graph of vertically by a factor of. For example, if we begin by graphing the parent function, we can then graph the stretch, using, to get as shown on the left in Figure 8, and the compression, using, to get as shown on the right in Figure 8.įigure 8 (a) stretches the graph of vertically by a factor of. While horizontal and vertical shifts involve adding constants to the input or to the function itself, a stretch or compression occurs when we multiply the parent function by a constant. (Your answer may be different if you use a different window or use a different value for Guess?) To the nearest thousandth. The -coordinate of the point of intersection is displayed as 2.1661943. The graphs should intersect somewhere near. For a window, use the values –3 to 3 for and –5 to 55 for. The point of intersection gives the value of for the indicated value of the function.ĮXAMPLE 3 Approximating the Solution of an Exponential Equation Select "intersect" and press three times. To find the value of, we compute the point of intersection.Press to observe the graph of the exponential function along with the line for the specified value of.Adjust the -axis so that it includes the value entered for " Y 2=". Enter the given value for in the line headed " Y 2=".Enter the given exponential equation in the line headed " Y 1=". Given an equation of the form for, use a graphing calculator to approximate the solution. The domain is the range is the horizontal asymptote is. Shift the graph of left 1 units and down 3 units. We have an exponential equation of the form, with, , and. ĮXAMPLE 2 Graphing a Shift of an Exponential Function State the domain,, the range,, and the horizontal asymptote.Shift the graph of up units if is positive, and down units if is negative.Shift the graph of left units if is positive, and right units if is negative. HOW TO Given an exponential function with the form, graph the translation. ![]() horizontally units, in the opposite direction of the sign of.vertically units, in the same direction of the sign of.įor any constants and, the function shifts the parent function Again, see that, so the initial value of the function is. When the function is shifted right units to, the -intercept becomes.This is because, so the initial value of the function is. When the function is shifted left units to, the -intercept becomes.Observe the results of shifting horizontally: Both horizontal shifts are shown in Figure 6. For example, if we begin by graphing the parent function, we can then graph two horizontal shifts alongside it, using : the shift left,, and the shift right. The next transformation occurs when we add a constant to the input of the parent function, giving us a horizontal shift units in the opposite direction of the sign. The asymptote also shifts down units to.When the function is shifted down units to :.When the function is shifted up units to :.Observe the results of shifting vertically: Both vertical shifts are shown in Figure 5. For example, if we begin by graphing a parent function,, we can then graph two vertical shifts alongside it, using : the upward shift, and the downward shift. The first transformation occurs when we add a constant to the parent function, giving us a vertical shift units in the same direction as the sign. For instance, just as the quadratic function maintains its parabolic shape when shifted, reflected, stretched, or compressed, the exponential function also maintains its general shape regardless of the transformations applied. Just as with other parent functions, we can apply the four types of transformations – shifts, reflections, stretches, and compressions – to the parent function without loss of shape. ![]() Transformations of exponential graphs behave similarly to those of other functions. ![]()
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