Situation #1: Translating Functions Prepared at University of Georgia William Plummer EMAT 6500 Date last revised: July 28, 2013

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1 Situation #1: Translating Functions Prepared at University of Georgia William Plummer EMAT 6500 Date last revised: July 28, 2013 Prompt An Algebra class is discussing the graphing of quadratic functions using the vertex form f(x) = (x - h) 2 + k where the vertex of the resultant parabola is located at (h,k). This concept is then described as a translation of the quadratic function vertically and horizontally. The following function is presented to the class: f(x) = (x - 3) and many students shifted the parabola down by 3 and to the right by 5. Commentary The graphs of many functions can be regarded as arising from more basic graphs, called parent functions, denoted here as p(x). These graphs come about as a result of one or more transformations including shifting, stretching, compression, and reflection. The following foci describe translations like the one above and their properties. Mathematical Focus 1 The graphs of y = f(x) + h and y = f(x + h) represent vertical and horizontal shifts of the original function f(x) respectively. For any positive number h, the graph of y = f(x) + h is displaced up by h units from the graph of y = f(x) and the graph of y = f(x + h) is displaced to the left h units. To understand why these translations result, consider the point ( a, f(a) ) on the graph of y = f(x). The point ( a, f(a) + h ) can be regarded as the corresponding point on the graph of y = f(x) + h. This point has a y-coordinate h units more than that of the original point ( a, f(a) ). So it is easy to see that it has been shifted up by h units. Likewise, if we let g(x) = f(x g + h), and we compare g(x) to f(x), then in order for g(x) and f(x) to have the same value, f(x g + h) = f(x) x g + h = x x g = x - h So the graph for every point located on g(x) would be located at ( x g, f(x) ) which is h units to the left of every corresponding point on f(x) when h > 0 and h units to the right of f(x) when h < 0. In the case of the quadratic equation above g(x) = x would be a vertical shift up 5 units from p(x) = x 2 and h(x) = (x 3) 2 would be a horizontal shift to the right by 3 units.

2 The function f(x) = (x - 3) incorporates both translations from g(x) and h(x) an is both a vertical shift up by 5 units and a horizontal shift to the right by 3 units.

3 Mathematical Focus 2 The graphs of y = a f(x) and y = f(bx) represent vertical and horizontal stretches of the original function f(x) respectively. For any number a > 1, the graph of y = a f(x) is the graph of y = f(x) vertically stretched with respect to the y-axis by a factor of a When 0 < a < 1, the graph of y = a f(x) is the graph of y = f(x) vertically compressed with respect to the y-axis by a factor of. When a < 0, the graph of y = a f(x) is the reflection of y = a f(x) about the x-axis. For any number b > 1 the graph of y = f(bx) is the graph of y = f(x) horizontally compressed with respect to the x-axis by a factor of b When 0 < b < 1, the graph of y = f(bx) is the graph of y = f(x) horizontally stretched by a factor of. Using the sine function as an example, the graph of g(x) = 5 sin(x) is the graph of p(x) = sin(x) vertically stretched by a factor of 5 and the graph of sin(x) is the graph of p(x) = sin(x) Just as with vertical and horizontal shifting, vertical and horizontal stretching / compressing as well as reflection can all occur in conjunction. Mathematical Focus 3 Since a translation is an isometry, the graphs of many other mathematical functions can experience the same movements as the quadratic equation above. Here we display horizontal and vertical translation of various functions encountered in the Common Core Curriculum. A translation is a bijection of the plane onto itself. For each point A and B on the plane the translation assigns the points A and B such that the length of segment AB is equal to the length of segment A B, that is the translation is distance preserving and is, therefore, an isometry (rigid motion). Cubic function The function f(x) = (x - 3) is a cubic function that assigns every point P on the graph of p(x) = x 3 a point P on the plane vertically shifted up by 5 units and horizontally shifted to the right by 3 units.

4 Exponential function The function f(x) = 2 x is an exponential function that assigns every point P on the graph of p(x) = 2 x a point P vertically shifted up by 5 units and horizontally shifted to the right by 3 units.

5 Rational Function The function f(x) = + 5 is a rational function that assigns every point P on the graph of p(x) = a point P vertically shifted up by 5 units and horizontally shifted to the right by 3 units.