Transformations: Shifts
Warm Up Find y-intercept, domain, range of Given two points (0, -2) and (-1, -8) of the exponential graph, write the equation. Growth or Decay? b) 4. Describe the Shift: (Up, Down, Left, Right)
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 k ● f(x), where k is positive or negative.
Check for Understanding 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.)
Check for Understanding 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.
Mini Lesson One type of non-rigid transformation is a stretch or compression. A vertical stretching is the stretching of the graph away from the x-axis. A vertical compression is the squeezing of the graph towards the x-axis. A compression is a stretch by a factor less than 1.
Mini Lesson Vertical Stretches and Compressions For the parent function y = f(x), the vertical stretching or compression of the function is a ● f(x). If | a | < 1 (a fraction between 0 and 1), then the graph is compressed vertically by a factor of a units. If | a | > 1, then the graph is stretched vertically by a factor of a units. *For values of a that are negative, then the vertical compression or vertical stretching of the graph is followed by a reflection across the x-axis. To determine the coordinates of a specific point under a vertical stretch or compression, the x-value remains the same and the y-value is multiplied by a.
Mini Lesson
Mini Lesson A horizontal stretching is the stretching of the graph away from the y-axis. A horizontal compression is the squeezing of the graph towards the y-axis.
Mini Lesson Horizontal Stretches and Compressions A compression is a stretch by a factor less than 1. If | b | < 1 (a fraction between 0 and 1), then the graph is stretched horizontally by a factor of b units. If | b | > 1, then the graph is compressed horizontally by a factor of b units. *For values of b that are negative, then the horizontal compression or horizontal stretching of the graph is followed by a reflection across the y- axis. To determine the coordinates of a specific point under a horizontal stretch or compression, the y-value remains the same and the x-value is divided by b.
Mini Lesson
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. Point M is on the graph of f(x) = (1/2)x, as shown. Determine the coordinates of point M under the transformation 5/4f(x).
Work Session Example 1. Point M is on the graph of f(x) = (1/2)x, as shown. Determine the coordinates of point M under the transformation 5/4f(x). Solution: g(x) = af(bx) is a vertical stretch or compression by a factor of a and a horizontal compression or stretch by a factor of b. 5/4f(x) is a vertical stretch by a factor of 5/4. (-3, 8) translates to (-3, 8 • 5/4) = (-3, 10).
Work Session Example 2. Point G is on the graph of f(x) = 3x, as shown. Determine the coordinates of point G under the transformation f(-4x).
Work Session Example 2. Point G is on the graph of f(x) = 3x, as shown. Determine the coordinates of point G under the transformation f(-4x). Solution: g(x) = af(bx) is a vertical stretch or compression by a factor of a and a horizontal compression or stretch by a factor of b. f(-4x) is a horizontal compression by a factor of 4, followed by a reflection across the y-axis since b is negative. (2, 9) translates to (2 ÷ -4, 9) = (-1/2, 9).
Work Session Example 3. The graph of f(x) = 2x is shown below. Determine the coordinates of point Q after the transformation f((2(x + 3)).
Work Session Example 3. The graph of f(x) = 2x is shown below. Determine the coordinates of point Q after the transformation f((2(x + 3)). Solution: g(x) = af(bx – h) + k is a vertical stretch or compression by a factor of a, a horizontal compression or stretch by a factor of b, a horizontal shift by h units and a vertical shift by k units. f ((2(x + 3)) is a horizontal compression by a factor of 2 followed by a horizontal translation 3 units to the left. Perform the horizontal compression first: (2, 4) translates to (2 ÷ 1/2, 4) = (1, 4). Then, shift the point 3 units left: (1,4) translates to (1 - 3, 4) = (-2, 4).
Work Session Example 4. Describe the transformations of g(x) = 1/3 • 2x + 1 – 4 from f(x) = 2x.
Work Session Example 4. Describe the transformations of g(x) = 1/3 • 2x + 1 – 4 from f(x) = 2x. Solution: g(x) = af(bx – h) + k is a vertical stretch or compression by a factor of a, a horizontal compression or stretch by a factor of b, a horizontal shift by h units and a vertical shift by k units. For g(x), a = 1/3, h = -1 and k = -4. The graph is compressed vertically by a factor of 1/3, shifted 1 unit to the left, and 4 units down.
Homework Complete: Summary of the effects of the transformation af(bx – h) + k on f(x)
Work Session