Provide step-by-step explanations. This allows us to think about reflecting a function in the horizontal axis as stretching it in the vertical direction by a scale factor of. However, both the -intercept and the minimum point have moved. This means that the function should be "squashed" by a factor of 3 parallel to the -axis.
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. The function is stretched in the horizontal direction by a scale factor of 2. E. If one star is three times as luminous as another, yet they have the same surface temperature, then the brighter star must have three times the surface area of the dimmer star. Complete the table to investigate dilations of Whi - Gauthmath. Check the full answer on App Gauthmath. Suppose that we take any coordinate on the graph of this the new function, which we will label. 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. The -coordinate of the turning point has also been multiplied by the scale factor and the new location of the turning point is at. The function represents a dilation in the vertical direction by a scale factor of, meaning that this is a compression. For example, the points, and. If this information is known precisely, then it will usually be enough to infer the specific dilation without further investigation.
As with dilation in the vertical direction, we anticipate that there will be a reflection involved, although this time in the vertical axis instead of the horizontal axis. Complete the table to investigate dilations of exponential functions in different. Example 5: Finding the Coordinates of a Point on a Curve After the Original Function Is Dilated. Still have questions? This new function has the same roots as but the value of the -intercept is now. We will use this approach throughout the remainder of the examples in this explainer, where we will only ever be dilating in either the vertical or the horizontal direction.
For example, suppose that we chose to stretch it in the vertical direction by a scale factor of by applying the transformation. We will use the same function as before to understand dilations in the horizontal direction. Express as a transformation of. Are white dwarfs more or less luminous than main sequence stars of the same surface temperature? In these situations, it is not quite proper to use terminology such as "intercept" or "root, " since these terms are normally reserved for use with continuous functions. The roots of the original function were at and, and we can see that the roots of the new function have been multiplied by the scale factor and are found at and respectively. This explainer has so far worked with functions that were continuous when defined over the real axis, with all behaviors being "smooth, " even if they are complicated. We can confirm visually that this function does seem to have been squished in the vertical direction by a factor of 3. Complete the table to investigate dilations of exponential functions in real life. Had we chosen a negative scale factor, we also would have reflected the function in the horizontal axis. Note that the roots of this graph are unaffected by the given dilation, which gives an indication that we have made the correct choice. As we have previously mentioned, it can be helpful to understand dilations in terms of the effects that they have on key points of a function, such as the -intercept, the roots, and the locations of any turning points.
Find the surface temperature of the main sequence star that is times as luminous as the sun? Once again, the roots of this function are unchanged, but the -intercept has been multiplied by a scale factor of and now has the value 4. We would then plot the function. Complete the table to investigate dilations of exponential functions in two. Example 2: Expressing Horizontal Dilations Using Function Notation. By clicking Sign up you accept Numerade's Terms of Service and Privacy Policy. Regarding the local maximum at the point, the -coordinate will be halved and the -coordinate will be unaffected, meaning that the local maximum of will be at the point. Stretching a function in the horizontal direction by a scale factor of will give the transformation.
The figure shows the graph of and the point. Crop a question and search for answer. 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 could investigate this new function and we would find that the location of the roots is unchanged. Then, the point lays on the graph of. We will first demonstrate the effects of dilation in the horizontal direction. We should double check that the changes in any turning points are consistent with this understanding. The roots of the function are multiplied by the scale factor, as are the -coordinates of any turning points.
Suppose that we had decided to stretch the given function by a scale factor of in the vertical direction by using the transformation. We note that the function intersects the -axis at the point and that the function appears to cross the -axis at the points and. Thus a star of relative luminosity is five times as luminous as the sun.