Transformations: Shifts

Warm Up Find y-intercept, domain, range of Given two points (0, 3) and (1, 5) of the exponential graph, write the equation. Growth or Decay? b)

Standard Standard: MCC9-12.F.BF.3 Identify the effects on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs.

Learning Target · I can describe the effect on the graph of f(x) by using the transformation f(x) + k, where k is positive or negative. · I can describe the effect on the graph of f(x) by using the transformation f(x + k), where k is positive or negative.

Mini Lesson A transformation of a function takes whatever f(x) is and transforms it, meaning that the graph gets changed somehow. Transformations are rigid if they keep the same size and shape of the original function, and non-rigid if they do not. The original function is referred to as the parent function. One type of rigid transformation is a translation, in which the graph keeps the same size and shape, but is shifted (moved), either left, right, up, or down. A translation is sometimes referred to as a shift.

Mini Lesson If the parent function is y = f(x), then the translation of the function horizontally to the left or right (in the  x-direction) is given by the function f(x – h). if h > 0, the graph translates to the right. if h < 0, the graph translates to the left. Remember, the value of h is subtracted from x. Thus, f(x + 2) is f(x – (-2)), a = -2, and the graph shifts 2 units to the left.

Mini Lesson If the parent function is y = f(x), then the translation of the function vertically to the left or right (in the x-direction) is given by the function f(x) + k. if k > 0, the graph translates to the up. if k < 0, the graph translates to the down. Remember, the value of h is subtracted from x. Thus, f(x + 2) is f(x – (-2)), a = -2, and the graph shifts 2 units to the left.

Mini Lesson Multiple transformations may be performed on one parent graph, that is if y = f(x), then f(x – h) + k will translate the graph horizontally h units and vertically k units.

Work Session Example 1. The graph of f(x) = 4x is shown below. Determine the coordinates of point A after the transformation g(x) = f(x) + 2.

Work Session Example 1. The graph of f(x) = 4x is shown below. Determine the coordinates of point A after the transformation g(x) = f(x) + 2. Solution:  g(x) = f(x – h) + k is a transformation of f h units in the x-direction and k units in the y-direction. For f(x) + 2, h = 0 and k = 2. The graph shifts 2 units up and Point A ends up at (6, 1).

Work Session Example 2. Describe how the graph of y = 2x is translated to get the graph of y = 2(x − 4) + 3.

Work Session Example 2. Describe how the graph of y = 2x is translated to get the graph of y = 2(x − 4) + 3. Solution:  g(x) = f(x – h) + k is a transformation of f h units in the x-direction and k units in the y-direction. For y = 2(x – 4) + 3, h = 4 and k = 3. The graph is shifted 4 units to the right and 3 units up.

Work Session Example 3. Given the graph and the original function f(x) = 2(x - 1), find k and the new function. Describe the translation. (The original graph is in red and the transformed graph is in blue.)

Work Session Solution: The graph indicates a positive shift up. Example 3. Given the graph and the original function f(x) = 2(x - 1), find k and the new function. Describe the translation. (The original graph is in red and the transformed graph is in blue.) Solution: The graph indicates a positive shift up. The y-intercepts indicate a vertical translation of positive 3. The new function is written as  f(x) + 3 = 2(x − 1) + 3.

Work Session Example 4. Given the graph and the original function f(x) = 2x, find k and the new function. Describe the translation. (The original graph is in blue and the transformed graph is in red.)

Work Session Example 4. Given the graph and the original function f(x) = 2x, find k and the new function. Describe the translation. (The original graph is in blue and the transformed graph is in red.) Solution: The graph indicates a downward vertical translation. In looking at the y-intercepts, there was a downward shift of 3, so k = -3. The new function is written as g(x) − 3 = 2x – 3.

Work Session Example 5. The function g(x) is obtained by translating f(x) = 2x to the left 4 units and up 1 units. Write an equation for g(x).

Work Session Example 5. The function g(x) is obtained by translating f(x) = 2x to the left 4 units and up 1 units. Write an equation for g(x). Solution: If g(x) = f(x – h) + k, then the graph shifts h units in the horizontal direction and k units in the vertical direction. h = -4, k = 1. g(x) = 2 (x +  4) + 1

Work Session The graph of f(x) = 2x is shown to the right. Determine the coordinates of point Q after the transformation g(x) = f(x - 3).  Given the graph and the original function f(x) = 2x, find h and the new function. Describe the translation. (The original graph is in green and the transformed graph is in red.) Describe how the graph of y = 4x is translated to get the graph of y = 4(x − 1) – 2. The function g(x) is obtained by translating f(x) = 2x + 3 up 5 units. Write an equation for g(x). Do NOT simplify.

Closing