Section 3.5 Graphing Techniques: Transformations

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1 Addition Shifts Subtraction Inside Horizontal Outside Vertical Left Right Up Down (Add inside) (Subtract inside) (Add Outside) (Subtract Outside) Transformation Multiplication Compressions Stretches Inside Horizontal Outside Vertical Compressions Stretches Compressions Stretches (Negative Outside) Opposite About x-axis (Replace by ) Reflections About y-axis (Replace by ) (Negative Inside) Horizontal and Vertical Shifts If is a positive real number, then A function is the graph of shifted horizontally left units. A function is the graph of shifted horizontally right units. A function is the graph of shifted vertically up units. A function is the graph of shifted vertically down units. Horizontal Shifts Vertical Shifts Cheon-Sig Lee Page 1

2 Stretches and Compressions A function is the graph of compressed vertically if. A function is the graph of stretched vertically if. A function is the graph of compressed horizontally if. A function is the graph of stretched horizontally if., 1, 0 1, 0 1, 1 Vertical Compressions/Stretches, 1, 0 1, 1, 0 1 Horizontal Compressions/Stretches Reflection A function is the graph of the reflected about the x-axis. A function is the graph of the reflected about the y-axis. Reflection about x-axis Reflection about y-axis Cheon-Sig Lee Page 2

3 Increasing and Decreasing For in an open interval, A function is increasing if. A function is decreasing if. Local Maxima and Minima A function has a local maximum,, at if for all x in an open interval and. A function has a local minimum,, at if for all x in an open interval and. Exercises 1. (Solution 1) Reflection about x-axis Shift up 1 unit (Solution 2) Reflection about x-axis Shift left 2 units 2 2 Cheon-Sig Lee Page 3

4 3. (Solution 3) Reflection about x-axis Reflection about y-axis (Solution 4) 1 is the graph of shifted left 1 unit. Thus, 1 is the graph of shifted left 1 unit. For using a graphing calculator, the window is better to be fixed; 12, 12, 2 by 12, 12, 2 for this problem. The domain is 1, because the graph starts at 1. The Range is 0, because the minimum for is (Solution 5) Vertical Stretch 5 Window: 3, 11, 1 by 1, 12, 1 Cheon-Sig Lee Page 4

5 Section 3.5 Graphing Technique es: Transformations 6. (Solution 6.a) Adjust the window; 4, 4,,1 by 12, 12, 2 (Solution 6.b) (Solution 6.c) 2nd TRACE Cheon-Sig Lee Page 5

6 Section 3.5 Graphing Technique es: Transformations (c) Continued (Solution 6.c) Continued 2nd TRACE (Solution 6.e) x-intercept 2 is the graph of shifted left 2 units, so does x-intercept. Because x-intercept of was 3, 0, 3, the x-intercepts of 2 is 3 2, 0 2,3 2,, 3..73, 10.39, , (Solution 6.e) Minimum/ Maximum, Because 2 is the graph of shifted left, x-values are shifted left 2 units but Minima remain the same. Because 2 is the graph of shifted left, x-values are shifted left 2 units but Maxima remain the same. Cheon-Sig Lee Page 6

(6.e) Continued 3.73, 10.39, 10.39 0.27, 10.39, 10.39 6 3.73 0.27 2 (Solution 6.

Because 2 is shifted left 2 units, intervals that 2 is increasing are also shifted left 2 units.

Because 2 is shifted left 2 units, intervals that 2 is increasing is also shifted left 2 units,

f) x-intercepts/ Minimum/ Maximum 3 is the graph of stretched vertically by a factor of 3.

7 (6.e) Continued 3.73, 10.39, , 10.39, (Solution 6.e) Increasing/ Decreasing Intervals that is increasing are 4, and, 4. Because 2 is shifted left 2 units, intervals that 2 is increasing are also shifted left 2 units. Thus, 6, 3.73 and 0.27, 2. Intervals that is decreasing is,. Because 2 is shifted left 2 units, intervals that 2 is increasing is also shifted left 2 units, Thus, 3.73, , 31.17, 10.39, 31.17, (Solution 6.f) x-intercepts/ Minimum/ Maximum 3 is the graph of stretched vertically by a factor of 3. Because the graph is stretched vertically, x-intercepts remain the same and the maxima and minima are stretched vertically by a factor of 3. Minimum:, 10.39, , Maximum:, 10.39, , Cheon-Sig Lee Page 7

Thus, intervals that 3 is increasing are same as those of and intervals that 3 is increasing are also same

Because reflected about y-axis, x-intercepts of is opposite of x-intercepts in., 10.39, 10.

8 (6.f) Continued 3 (Solution 6.f) Increasing/ Decreasing Because 3 is stretched vertically, x-values remain the same. Thus, intervals that 3 is increasing are same as those of and intervals that 3 is increasing are also same as those of. (Solution 6.g ) 3 is the graph of reflected about y-axis. Because reflected about y-axis, x-intercepts of is opposite of x-intercepts in., 10.39, , 10.39, Because is reflected about y-axis, x-values of are opposite to x-values in. Intervals that is decreasing are intervals that is increasing and intervals that is increasing are intervals that is increasing. Cheon-Sig Lee Page 8

Section 5.2 Graphs of the Sine and Cosine Functions

A Periodic Function and Its Period Section 5.2 Graphs of the Sine and Cosine Functions A nonconstant function f is said to be periodic if there is a number p > 0 such that f(x + p) = f(x) for all x in