This means that the function should be "squashed" by a factor of 3 parallel to the -axis. Complete the table to investigate dilations of exponential functions. Enjoy live Q&A or pic answer. Enter your parent or guardian's email address: Already have an account? 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.
In practice, astronomers compare the luminosity of a star with that of the sun and speak of relative luminosity. Furthermore, the location of the minimum point is. We will not give the reasoning here, but this function has two roots, one when and one when, with a -intercept of, as well as a minimum at the point. 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. 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). 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.
To create this dilation effect from the original function, we use the transformation, meaning that we should plot the function. In this new function, the -intercept and the -coordinate of the turning point are not affected. Check Solution in Our App. We have plotted the graph of the dilated function below, where we can see the effect of the reflection in the vertical axis combined with the stretching effect. Please check your spam folder. Solved by verified expert. Then, the point lays on the graph of. We will use the same function as before to understand dilations in the horizontal direction. The new function is plotted below in green and is overlaid over the previous plot. Does the answer help you? This means that we can ignore the roots of the function, and instead we will focus on the -intercept of, which appears to be at the point. 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. We will begin by noting the key points of the function, plotted in red. We will demonstrate this definition by working with the quadratic.
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. In this explainer, we will investigate the concept of a dilation, which is an umbrella term for stretching or compressing a function (in this case, in either the horizontal or vertical direction) by a fixed scale factor. At first, working with dilations in the horizontal direction can feel counterintuitive. Still have questions? However, both the -intercept and the minimum point have moved. Thus a star of relative luminosity is five times as luminous as the sun. When working with functions, we are often interested in obtaining the graph as a means of visualizing and understanding the general behavior. This transformation will turn local minima into local maxima, and vice versa.
Crop a question and search for answer. However, in the new function, plotted in green, we can see that there are roots when and, hence being at the points and. Write, in terms of, the equation of the transformed function. This information is summarized in the diagram below, where the original function is plotted in blue and the dilated function is plotted in purple. Unlimited access to all gallery answers. C. About of all stars, including the sun, lie on or near the main sequence. For example, the points, and. Just by looking at the graph, we can see that the function has been stretched in the horizontal direction, which would indicate that the function has been dilated in the horizontal direction. According to our definition, this means that we will need to apply the transformation and hence sketch the function. We will first demonstrate the effects of dilation in the horizontal direction. This problem has been solved! Therefore, we have the relationship. One of the most important graphical representations in astronomy is the Hertzsprung-Russell diagram, or diagram, which plots relative luminosity versus surface temperature in thousands of kelvins (degrees on the Kelvin scale). Get 5 free video unlocks on our app with code GOMOBILE.
In terms of the effects on known coordinates of the function, any noted points will have their -coordinate unaffected and their -coordinate will be divided by 3. In our final demonstration, we will exhibit the effects of dilation in the horizontal direction by a negative scale factor. We know that this function has two roots when and, also having a -intercept of, and a minimum point with the coordinate. Example 2: Expressing Horizontal Dilations Using Function Notation. Additionally, the -coordinate of the turning point has also been halved, meaning that the new location is. 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. Although we will not give the working here, the -coordinate of the minimum is also unchanged, although the new -coordinate is thrice the previous value, meaning that the location of the new minimum point is. Had we chosen a negative scale factor, we also would have reflected the function in the horizontal axis.
The roots of the function are multiplied by the scale factor, as are the -coordinates of any turning points. Since the given scale factor is, the new function is. Try Numerade free for 7 days. For the sake of clarity, we have only plotted the original function in blue and the new function in purple. The figure shows the graph of and the point. 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. Approximately what is the surface temperature of the sun? Feedback from students. For example, suppose that we chose to stretch it in the vertical direction by a scale factor of by applying the transformation. By paying attention to the behavior of the key points, we will see that we can quickly infer this information with little other investigation. As a reminder, we had the quadratic function, the graph of which is below. Are white dwarfs more or less luminous than main sequence stars of the same surface temperature? The transformation represents a dilation in the horizontal direction by a scale factor of.
The plot of the function is given below. We note that the function intersects the -axis at the point and that the function appears to cross the -axis at the points and. Gauthmath helper for Chrome. There are other points which are easy to identify and write in coordinate form.
Good Question ( 54). Students also viewed. For example, stretching the function in the vertical direction by a scale factor of can be thought of as first stretching the function with the transformation, and then reflecting it by further letting. The new turning point is, but this is now a local maximum as opposed to a local minimum. Identify the corresponding local maximum for the transformation. Firstly, the -intercept is at the origin, hence the point, meaning that it is also a root of. When considering the function, the -coordinates will change and hence give the new roots at and, which will, respectively, have the coordinates and. Now comparing to, we can see that the -coordinate of these turning points appears to have doubled, whereas the -coordinate has not changed. A function can be dilated in the horizontal direction by a scale factor of by creating the new function.
Suppose that we had decided to stretch the given function by a scale factor of in the vertical direction by using the transformation. Stretching a function in the horizontal direction by a scale factor of will give the transformation. In particular, the roots of at and, respectively, have the coordinates and, which also happen to be the two local minimums of the function.
If this information is known precisely, then it will usually be enough to infer the specific dilation without further investigation. Check the full answer on App Gauthmath. Given that we are dilating the function in the vertical direction, the -coordinates of any key points will not be affected, and we will give our attention to the -coordinates instead. The function is stretched in the horizontal direction by a scale factor of 2.
The dilation corresponds to a compression in the vertical direction by a factor of 3. This indicates that we have dilated by a scale factor of 2. Example 4: Expressing a Dilation Using Function Notation Where the Dilation Is Shown Graphically. From the graphs given, the only graph that respects this property is option (e), meaning that this must be the correct choice. The result, however, is actually very simple to state. You have successfully created an account. We solved the question! Gauth Tutor Solution. Provide step-by-step explanations. In this explainer, we will learn how to identify function transformations involving horizontal and vertical stretches or compressions.