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Graphing Functions: Vertical and Horizontal Shifts

Learn how to obtain new graphs by shifting them vertically and horizontally, and understand the vocabulary associated with these shifts.

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Graphing Functions: Vertical and Horizontal Shifts

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  1. Activity 1 - 14 Heating Schedule

  2. y y y y y x x x x x 5-Minute Check on Activity 1-13 • Which of the following graphs are functions? • Identify what each of the points are in the following graph:a. Wb. Xc. Yd. Z Function Not Not Function X x-intercept maximum Z W minimum Y y-intercept Click the mouse button or press the Space Bar to display the answers.

  3. Objectives • Obtain a new graph from an original graph using a vertical shift • Obtain a new graph from an original graph using a horizontal shift • Identify vertical and horizontal shifts • Write a new formula for a function for which its graph has been shifted vertically or horizontally

  4. Vocabulary • Vertical Shift –a upward or downward shift in the graph • Horizontal Shift – a left or right shift in the graph • Translation – a vertical or horizontal (or both) shift of the graph of the function

  5. Activity The cost of fuel oil is rising and is affecting the school district’s budget. In order to save money, the building maintenance supervisor of the high school has decided to keep the building warm only during school hours. At midnight, the building temperature is 55°F. This temperature remains constant until 4 am, at which time the temperature of the building steadily increases. By 7 am, the temperature is 68°F and is maintained there until 7 pm, when the temperature begins a steady decrease. By 10 pm, the temperature is back to 55°F. Graph this function.

  6. Activity - Graph At midnight, the building temperature is 55°F. This temperature remains constant until 4 am, at which time the temperature of the building steadily increases. By 7 am, the temperature is 68°F and is maintained there until 7 pm, when the temperature begins a steady decrease. By 10 pm, the temperature is back to 55°F. 75 70 65 60 55 50 2 4 6 8 10 12 14 16 18 20 22 24

  7. Activity 2 - Graph The School Board instructs the building maintenance supervisor to increase the temperature of the building by a constant 3°F. • Sketch the change on the same graph • How can the second function be gotten from the first? By adding 3 to each y-value 75 70 65 60 55 50 2 4 6 8 10 12 14 16 18 20 22 24

  8. Vertical Shifts Vertical shifts in functions occur when a value is added or subtracted from each y-value calculated. These are considered “outside” the function as illustrated below: Upward shift of c units: y = f(x) + c Downward shift of c units: y = f(x) – c Shifts up or down correspond to what we naturally associate with addition and subtraction.

  9. Activity 3 - Graph On a two-hour delay, the supervisor adapts the building’s heating function to the delay. • Sketch the same graph from the first activity • Sketch the two-hour delay change on the same graph • How can the second function be gotten from the first? By subtracting 2 from each x-value 75 70 65 60 55 50 2 4 6 8 10 12 14 16 18 20 22 24

  10. Horizontal Shifts Horizontal shifts in functions occur when a value is added or subtracted from each x-value inputted. These are considered “inside” the function as illustrated below: Leftward shift of c units: y = f(x+ c) Rightward shift of c units: y = f(x– c) Shifts right or left do not correspond to what most people naturally associate with addition and subtraction. Pay attention to the following examples.

  11. Graph Calculator Work Describe how the graph changes each time Using the graphing calculator, graph the following: • y1 = x2Then on the same graph do • y2 = x2 + 2then change to • y2 = x2 – 3 then change to • y2 = (x + 1)2then change to • y2 = (x – 4)2 Shifts graph up 2 Shifts graph down 3 Shifts graph left 1 Shifts graph right 4

  12. Translations As we studied in grade school math and in Geometry a translation, or a slide, is the vertical or horizontal (or both) shifting of a function. Picture a magnet on a refrigerator that is “slid” left or right, up or down. See the animation from the cafeteria staff below:

  13. Absolute Value Definition An absolute value is always positive. It is defined by: x if x ≥ 0 |x| = -x if x < 0 In other words we strip off a negative sign, if the function value is negative.

  14. y y x x Absolute Value Problem Graph the following function y = |x| and compare with:a) y = |x| + 5 b) y = |x + 5| Shifts graph up 5 Shifts graph left 5

  15. Summary and Homework • Summary • Vertical shifts in functions occur when a value is added or subtracted from each y-value calculated. These are considered “outside” the function: • Upward shift of c units: y = f(x) + c • Downward shift of c units: y = f(x) – c • Horizontal shifts in functions occur when a value is added or subtracted from each x-value inputted. These are considered “inside” the function: • Leftward shift of c units: y = f(x + c) • Rightward shift of c units: y = f(x – c) • Homework • pg 129-34, 1-10

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