The distance from the roots to the origin has doubled, which means that we have indeed dilated the function in the horizontal direction by a factor of 2. We would then plot the function. When dilating in the vertical direction, the value of the -intercept, as well as the -coordinate of any turning point, will also be multiplied by the scale factor. Therefore, we have the relationship. Complete the table to investigate dilations of Whi - Gauthmath. Now take the original function and dilate it by a scale factor of in the vertical direction and a scale factor of in the horizontal direction to give a new function. Coupled with the knowledge of specific information such as the roots, the -intercept, and any maxima or minima, plotting a graph of the function can provide a complete picture of the exact, known behavior as well as a more general, qualitative understanding.
We will now further explore the definition above by stretching the function by a scale factor that is between 0 and 1, and in this case we will choose the scale factor. This makes sense, as it is well-known that a function can be reflected in the horizontal axis by applying the transformation. The value of the -intercept has been multiplied by the scale factor of 3 and now has the value of. Definition: Dilation in the Horizontal Direction. Such transformations can be hard to picture, even with the assistance of accurate graphing tools, especially if either of the scale factors is negative (meaning that either involves a reflection about the axis). This problem has been solved! Complete the table to investigate dilations of exponential functions based. Referring to the key points in the previous paragraph, these will transform to the following, respectively:,,,, and. To create this dilation effect from the original function, we use the transformation, meaning that we should plot the function.
Get 5 free video unlocks on our app with code GOMOBILE. Much as the question style is slightly more advanced than the previous example, the main approach is largely unchanged. Example 6: Identifying the Graph of a Given Function following a Dilation. Much as this is the case, we will approach the treatment of dilations in the horizontal direction through much the same framework as the one for dilations in the vertical direction, discussing the effects on key points such as the roots, the -intercepts, and the turning points of the function that we are interested in. The new turning point is, but this is now a local maximum as opposed to a local minimum. Solved by verified expert. However, both the -intercept and the minimum point have moved. Complete the table to investigate dilations of exponential functions at a. This result generalizes the earlier results about special points such as intercepts, roots, and turning points. Then, we would obtain the new function by virtue of the transformation. We can dilate in both directions, with a scale factor of in the vertical direction and a scale factor of in the horizontal direction, by using the transformation. Accordingly, we will begin by studying dilations in the vertical direction before building to this slightly trickier form of dilation. Feedback from students. Express as a transformation of. Had we chosen a negative scale factor, we also would have reflected the function in the horizontal axis.
If this information is known precisely, then it will usually be enough to infer the specific dilation without further investigation. Once an expression for a function has been given or obtained, we will often be interested in how this function can be written algebraically when it is subjected to geometric transformations such as rotations, reflections, translations, and dilations. This transformation will turn local minima into local maxima, and vice versa. The plot of the function is given below. The function is stretched in the horizontal direction by a scale factor of 2.