![]() ![]() Authored by: John Jarvis and Hazel McKenna. Interactive transformation of f(x)=x to f(x)=mx+b. Interactive transformation from f(x)=x to f(x)=mx. Interactive transformation from f(x)=x to f(x)=x+b. All graphs created by Desmos graphing calculator.Authored by: James Sousa () for Lumen Learning. Graph a Linear Function as a Transformation of f(x)=x.Authored by: Hazel McKenna and Leo Chang. 2.4: Transformations of Linear Functions.That is, the y value in y=f(x)=x becomes y=f(x)=-x (e.g., positive y values become negative, or negative y values become positive). From an algebraic point of view, vertically flipping the graph of the function f(x)=x is equivalent to multiplying the x-values by –1. To reflect a linear function across the x-axis means all points on the line will see their y-coordinates change sign positive x-coordinates will become negative (e.g, (2, 2) becomes (2, -2)), and negative y-coordinates will become positive (e.g., (1, –5) becomes (1, 5)). This results in a mirror image of the original function with respect to the x-axis. Reflecting a linear function across the x-axis means to flip the line representing the function across the x-axis. ![]() It is important to note that the position of the y-intercept does not change because there is no vertical shift.įigure 4 illustrates the graphs of various functions that are the result of stretching (m>1 or compressing (0≤m<1 the function f(x)=x to f(x)=mx.Ĥ. For example, compressing the function f(x)=x to one-half of its height means multiplying the slope by \dfracx. To decrease the steepness of a line means to multiply the original slope of 1 by a number that is between 0 and 1. The result would be to decrease the steepness (slope) of the line. If instead of being able to stretch the ends of the line in opposite directions we were able to push the ends of the line, causing the slope of the line to decrease, we would witness vertical compression. It is important to note that the position of the y-intercept does not change because there is no vertical shift. Therefore, the stretched function becomes f(x)=2x. For example, stretching the function f(x)=x to be twice as steep means multiplying the original slope of 1 by 2. To increase the steepness of a function the slope must increase from 1 to a number that is greater than 1. This is vertical stretching of a linear function. If we could grab both ends of the line and pull vertically in opposite directions we would stretch the line causing the slope of the line to increase. ![]() Decrease in y-valuesį(x)=x+3 Vertical Stretching or Compressing Vertically stretching Therefore, the function becomes f(x)=x-2. From another perspective, this transformation results in the original y-intercept (0, 0) moving to (0, –2). If the function f(x)=x is shifted down two units, it means all the y-coordinates of the points on the line are decreased by 2 (increased by –2) (Table 2). In other words, the function f(x)=x becomes f(x)=x-2 because all of the y-values are decreased by 2. In other words, the function f(x)=x transforms to f(x)=x+2 because all of the y-values are increased by 2. See how this is applied to solve various problems. We can even reflect it about both axes by graphing y-f(-x). If fact, every point on the graph shifts up by 2 units so for every point on the line y=x, the y-coordinate of each point increases by 2. We can reflect the graph of any function f about the x-axis by graphing y-f(x) and we can reflect it about the y-axis by graphing yf(-x). If we shift the line up two units, the slope does not change, but the y-intercept moves to (0, 2) (Figure 1). For example, the function f(x)=x has a slope of 1 and a y-intercept at (0, 0). When we shift a line up or down, the slope (steepness) of the line does not change. Making a vertical shift means to move the graph of the linear function vertically up or down a certain number of units. We will apply transformations graphically and consider what these transformations mean algebraically. The original function f(x)=x is also known as the parent function and is the linear function used for transformations in this section. A function may also be transformed using a reflection, stretch, or compression. A function may be transformed by a shift up, down, left, or right. Recall the introduction to graphical transformations of a function we saw in section 1.3. Write an algebraic function after completing transformations on the parent function.Explain the transformations performed on f(x)=x given the transformed function.Perform a combination of transformations on a linear function.Perform a reflection of a linear function across the x-axis.Perform a vertical stretch or compression on a linear function.Perform a vertical shift on a linear function.The equation of a parabola is $$$y = \frac = 0.25 $$$ A.ĭomain: $$$\left(-\infty, \infty\right) $$$ A. ![]()
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