He decided to draw a graph of his distance from home throughout the day. Table 1 tĮveryday, Jim drives to work and then straight back home. Since the entire function = y, then adding 20 to the function increases the y-values by 20. If the old function was y = f( x), then the new function would be y = f( x) + 20. Notice in figure 5, for each input x-value, the output y-value increased by 20. This will cause the graph to translate up 20 units. The graph can be sketched by adding 20 to each of the y-values of the original function. So, the manager starts charging a $20 cleaning fee in addition to the rent. The manager would like to charge more to rent the pool, but people really like the policy of only charging for the first 10 people. Figure 4 is the graph of the total price for groups. If both h and k are present then the graph translates both horizontally and vertically.Ī small swimming pool lets groups rent the pool for $5 a person, but they only charge for first 10 people. The number k is added outside to the entire function for a vertical shift because the function is y ( y = f( x)) to change the y-value. Notice that the number h is put inside the function with the x for a horizontal translation so that the x-value changes. If h is negative, then it translates left, and if k is negative, then it translated down. In figure 3, the function y = | x| is translated up 2 units. Where k is the distance the graph is translated up. In figure 2, the function y = | x| is translated right 2 units. Where h is the distance the graph is translated to the right. Then horizontal translations are in the form A translation moves a graph horizontally, vertically, or both. The first type of transformation is a translation. Likewise, vertical transformations result from changing the y values. Because the x is the horizontal axis, to transform a graph horizontally, change the x values by addition or multiplication. This lesson looks at transformations that change a graph horizontally or vertically. These changes are transformations which change a graph's position, orientation, or size. This lesson looks at how to change a parent function into a similar function. Mathematicians can transform a parent function to model a problem scenario given as words, tables, graphs, or equations. This lets the functions describe real world situations better. Mathematics can cause the parent functions to transform in ways similar to the mirrors. If the mirror is bent like a fun house mirror, then the image can be stretched or shrunk. If the mirror is tilted, then the image can be shifted horizontally or vertically. credit (wikimedia/Conrad Poirier)Ī flat mirror produces an image called a reflection where everything is inverted left to right. Perform a sequences of transformations.įigure 1: The reflection of two people in a distorting mirror of Cartierville Belmont Park.Graph functions with stretches and shrinks.Most functions do not possess the property of oddness or evenness. For even functions, reflection across the y-axis is the same as the pre-image. Note: For odd functions, reflection across the y-axis gives the same image as reflection across the x-axis. The function f is an odd function if and only if for all x in the domain. Eliminates the part of f for negative values of x.Įven Functions and Odd Functions (all odd exponents) (all even exponents)ĭefinition: Even Function and Odd Function The function f is an even function if and only if for all x in the domain. Reflects the part of the graph for positive values of x to the corresponding negative values of x. Leaves f unchanged if is negative The transformation Leaves f unchanged for nonnegative values of x. Property: Absolute Value Transformations The transformation Reflects f across the x-axis if is nonnegative. Piecewise function: A function follows different rules for different domains. is a horizontal reflection of function f across the y-axis. Property: Reflections Across the Coordinate Axes is a vertical reflection of function f across the x-axis. Plot the pre-image and the two reflections on the same screen. Write an equation for the reflection of this pre-image function across the x-axis. Reflections across the x-axis and y-axis Example 1 Write an equation for the reflection of this pre-image function across the y-axis. Section 1.6 Reflections, Absolute Values, and Other Transformations
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