It is difficult to tell from the diagram, but the â¡-coordinate â¡ = 2, and we can see that the roots of the new function have been multiplied by the scale factor and are found at The roots of the original function were at â¡ = â 1 and â¡-coordinate of the minimum point is also unaffected. If we were to analyze this function, then we would find that the â¡-intercept is unchanged and that the ![]() Transformation â¡ ( â¡ ) â â¡ ï¼ 1 2 â¡ ï, and our definition implies that we should then plot ![]() We will choose an arbitrary scale factor of 2 by using the We will first demonstrate the effects of dilation in the horizontal direction. That the new function is a reflection of the function â¡ = 2 â¡ ( â¡ ) in the horizontal axis. The new function is plotted below in green and is overlaid over the previous plot. Local maximum as opposed to a local minimum. The new turning point is ï¼ 1 2, 9 2 ï, but this is now a The â¡-coordinate of the minimum is unchanged, but the â¡-coordinate Once again, the roots of this function are unchanged, but the â¡-intercept has been multiplied by a scale factor of The vertical direction by using the transformation â¡ ( â¡ ) â â 2 â¡ ( â¡ ). Suppose that we had decided to stretch the given function â¡ ( â¡ ) by a scale factor of â 2 in 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 â 2 can be thought of as first stretching the function with the transformation â¡ ( â¡ ) â 2 â¡ ( â¡ ), and then reflecting it by further letting â¡ ( â¡ ) â â â¡ ( â¡ ). For example, stretching the function in the vertical direction by a scale factor of This makes sense, as it is well-known thatĪ function â¡ ( â¡ ) can be reflected in the horizontal axis by applying the transformation Had we chosenĪ negative scale factor, we also would have reflected the function in the horizontal axis. Accordingly, we will begin by studying dilations in the vertical direction before building to this slightly trickier form of dilation.Īt this point it is worth noting that we have only dilated a function in the vertical direction by a positive scale factor. Intuitive to visualize, although their algebraic interpretation can seem a little counterintuitive, especially when stretching in the horizontalÄirection. Geometrically, such transformations can sometimes be fairly ![]() In this explainer, we will investigate the concept of a dilation, which is an umbrella term for stretching or compressing a function (in thisĬase, in either the horizontal or vertical direction) by a fixed scale factor. Subjected to geometric transformations such as rotations, reflections, translations, and dilations. Once anÄ®xpression for a function has been given or obtained, we will often be interested in how this function can be written algebraically when it is Graph of the function can provide a complete picture of the exact, known behavior as well as a more general, qualitative understanding. Coupled with the knowledge of specific information such as the roots, the â¡-intercept, and any maxima or minima, plotting a When working with functions, we are often interested in obtaining the graph as a means of visualizing and understanding the general behavior. Below you can see a variety of vertical stretches, compressions, and/or reflections on the function f\left(x\right)=x.In this explainer, we will learn how to identify function transformations involving horizontal and vertical stretches or compressions. This means the larger the absolute value of m, the steeper the slope. Notice in the figure below that multiplying the equation of f\left(x\right)=x by m vertically stretches the graph of f by a factor of m units if m>1 and vertically compresses the graph of f by a factor of m units if 0 ![]() A function may be transformed by a shift up, down, left, or right. Graphing a Linear Function Using TransformationsĪnother option for graphing linear functions is to use transformations of the identity function f\left(x\right)=x. Use simple transformations to graph linear functions.
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