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Activity 1 - 14 Heating Schedule

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

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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

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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

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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.

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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. 2 4 6 8 10 12 14 16 18 20 22 24 75 70 65 60 55 50

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

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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.

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

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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.

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Graph Calculator Work Using the graphing calculator, graph the following: a)y 1 = x 2 Then on the same graph do b)y 2 = x 2 + 2 then change to c)y 2 = x 2 – 3 then change to d)y 2 = (x + 2) 2 then change to e)y 2 = (x – 4) 2 Describe how the graph changes each time

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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 below:

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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.

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Absolute Value Problem Graph the following functions: a) y = |x| + 5 b) y = |x + 5| y x y x

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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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Warm up The domain of a function is its a)y-values b) x-values c) intercepts The range of a function is its a) y-values b) x-values c) intercepts.

Warm up The domain of a function is its a)y-values b) x-values c) intercepts The range of a function is its a) y-values b) x-values c) intercepts.

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