1. WEEK 6 FUNCTIONS AND GRAPHS TRANSFORMATIONS OF FUNCTIONS

2. OBJECTIVES At the end of this session, you will be able to: Recognize graphs of common functions. Use vertical shifts to graph functions. Use horizontal shifts to graph functions. Use reflections to graph functions. Use vertical stretching and shrinking to graph functions. Graph functions involving a sequence of transformations.

3. INDEX 1. Graphs of Common Functions 2. Vertical Shifts 3. Horizontal Shifts 4. Combining Vertical and Horizontal Shifts 5. Reflections of Graphs 6. Vertical Stretching 7. Vertical Shrinking 8. Sequences of Transformations 9. Summary

4. 1. GRAPHS OF COMMON FUNCTIONS CONSTANT FUNCTION CHARACTERISTICS: Domain: Range: Single number c Constant on Even function IDENTITY FUNCTION CHARACTERISTICS: Domain: Range: Increasing on Odd function STANDARD QUADRATIC FUNCTION CHARACTERISTICS: Domain: Range: Decreasing on Increasing on Even function We have studied graphs of different functions in the previous sections. Now we will summarize graphs of some common functions and their characteristics: 12 -2 -2 1 2 f(x) = c 1 1 2 2 -2 f(x) = x 12 1 2 -2 -2 f(x) = x 2

5. 1. GRAPHS OF COMMON FUNCTIONS(Cont…) STANDARD CUBIC FUNCTION CHARACTERISTICS: Domain: Range: Increasing on Odd function SQUARE ROOT FUNCTION CHARACTERISTICS: Domain: Range: Increasing on Neither even nor odd function ABSOLUTE VALUE FUNCTION CHARACTERISTICS: Domain: Range: Decreasing on Increasing on Even function 12 2 1 -2 12 1 2 -2 12 1 2 -2 f(x) =  x f(x) = |x| f(x) = x 3

6. 2. VERTICAL SHIFTS Now, using the graph of the absolute value function f(x) = |x|, let us plot the graph of the function g(x) = |x| + 3 Comparing the y-coordinates of f and g, we observe that for each x, we add 3 to the corresponding y-coordinate of f. As a result, the graph of g shifts vertically upwards by three units. that is, g(x) = |x| + 3 = f(x) + 3 Thus, the graph of g has the same shape as the graph of f, but it has shifted vertically upwards by three units. As shown in the figure, each point on the graph of g is vertically above, by exactly three units, a corresponding point on the graph of f. Hence, we can generalize: f c c Let f be a function and c a positive real number, then the graph of y = f(x) + c is the graph of y = f(x) shifted c units vertically upward. X Y 1234 2 1 3 4 -2 -3 -4 -2-3-4 5 f(x) = |x| g(x)=|x|+3 Vertical shift xf(x)=|x|(x, f(x))g(x)=|x|+3(x, g(x)) -2|-2| = 2(-2, 2)|-2|+3 = 5(-2, 5) |-1| = 1(-1, 1)|-1|+3 = 4(-1, 4) 0|0| = 0(0, 0)|0|+3 = 3(0, 3) 1|1| = 1(1, 1)|1|+3 = 4(1, 4) 2|2| = 2(2, 2)|2|+3 = 5(2, 5)

7. 2. VERTICAL SHIFTS(Cont…) Next we plot the graph of the function h(x) = |x| - 4 using the graph of f(x) = |x|. Comparing the y-coordinates of f and h, we observe that for each x, we subtract 4 from the corresponding y-coordinate of f. As a result, the graph of h shifts vertically downwards by four units. that is, h(x) = |x| - 4 = f(x) - 4 Thus, the graph of h has the same shape as the graph of f, but it has shifted vertically downwards by four units. As shown in the figure, each point on the graph of h is vertically below, by exactly four units, a corresponding point on the graph of f. Hence, we can generalize: f c c Let f be a function and c a positive real number, then the graph of y = f(x) - c is the graph of y = f(x) shifted c units vertically downward. Y 1234 2 1 3 4 -2 -3 -4 -2-3-4 5 f(x) = |x| h(x)=|x|-4 Vertical shift down xf(x)=|x|(x, f(x))h(x)=|x|-4(x, h(x)) -2|-2| = 2(-2, 2)|-2| - 4 = -2(-2, -2) |-1| = 1(-1, 1)|-1| - 4 = –3(-1, -3) 0|0| = 0(0, 0)|0| - 4 = -4(0, -4) 1|1| = 1(1, 1)|1| - 3 = -2(1, -3) 2|2| = 2(2, 2)|2| - 2 = 0(2, -2)

8. 3. HORIZONTAL SHIFTS The concept of horizontal shifting is similar to that of vertical shifting. Let us see how we can use the graph of the standard quadratic equation f(x) = x 2 to plot a graph of the function g(x) = (x – 2) 2 From the above tables we observe that the y- coordinates for f and g remain the same but there is change in the corresponding x-coordinate. Every point of g is exactly two units to the right of the corresponding point of f. As a result, the graph of g(x) shifts horizontally two units to the right. that is, g(x) = (x – 2) 2 = f(x - 2) X Xf(x) = x 2 (x, f(x)) 0(0) 2 = 0(0, 0) 1(1) 2 = 1(1, 1) 2(2) 2 = 4(2, 4) Y 1234 2 1 3 4 -2 -3 -4 -2-3-4 5 f(x) = x 2 g(x)=(x-2) 2 Horizontal shift xg(x)=(x-2) 2 (x, g(x)) 2(2 - 2) 2 = 0(2, 0) 3(3 - 2) 2 = 1(3, 1) 4(4 - 2) 2 = 4(4, 4)

9. Now we find out that how can we use the graph of f(x) = x 2 to plot a graph of the function h(x) = (x + 2) 2. We observe that: Every point of h is exactly two units to the left of the corresponding point of f, that is, h(x +2) 2 = f(x + 2). The graph of h has the same shape as the graph of f, but has shifted horizontally by two units to the left. In general, we can say that if c is positive, then y = f(x +c) shifts the graph of f to the left by c units, and y = f(x - c) shifts the graph of f to the right by c units. Note: According to convention, we take positive numbers to the right on a number line and negative numbers to the left. The common temptation is to think that f(x + 2) moves f(x) to the right by two units. But this is not so. A positive number causes a horizontal shift to the left and a negative number causes a horizontal shift to the right. 3. HORIZONTAL SHIFTS(Cont…) X Y 1234 2 1 3 4 -2 -3 -4 -2-3-4 5 f(x) = x 2 g(x)=(x+2) 2 Horizontal shift Xf(x) = x 2 (x, f(x)) 0(0) 2 = 0(0, 0) 1(1) 2 = 1(1, 1) 2(2) 2 = 4(2, 4) Xh(x)=(x+2) 2 (x, h(x)) -2(-2 + 2) 2 = 0(-2, 0) (-1 + 2) 2 = 1(-1, 1) 0(0 + 2) 2 = 4(0, 4)

10. 4. COMBINING VERTICAL AND HORIZONTAL SHIFTS Now we use both horizontal and vertical shifts to plot a graph of a function that is a variation of the graph of a function we already know. Let us understand this concept with the help of an example: Using the graph of f(x) =  x we obtain the graph of h(x) = STEP 1: Graph f(x) =  x. We first plot the graph of the square root function f(x) =  x. STEP 2: Graph g(x) = As we subtract 1 from each value of x in the equation g(x) = from the domain of f(x) =  x, we shift the graph of f horizontally one unit to the right. STEP 3: Graph h(x) = Now to obtain the graph of h(x) = we perform a vertical shift. By subtracting 2, we move the graph vertically downwards by two units. Y 1234 2 1 3 4 -2 -3 -4 -2-3-4 5 f(x) =  x h(x)= g(x) = Horizontal shift by 1 Vertical shift by 2 X

11. 5. REFLECTIONS OF GRAPHS A reflection is like a mirror. The line of reflection acts as the mirror and is halfway between the point and its image. The figure shows the graphs of functions f(x) = |x| and g(x) = -|x|. Notice that the green graph is the same as the blue graph folded down across the x-axis. The graph of g is a reflection of the graph of f about the x-axis as g(x) = -|x| = -f(x) That is, for every point (x, y), there is a point (x, -y). The x-axis acts as the line of reflection or the mirror. In general, the graph of y = -f(x) is the graph of y = f(x) reflected about the x-axis. g(x)= -|x| X Y 1234 2 1 3 4 -2 -3 -4 -2-3-4 5 f(x) = |x| x-axis acts as line of reflection

12. 5. REFLECTIONS OF GRAPHS(Cont…) Just as we can reflect the graph about the x-axis, we can also reflect the graph about the y-axis. The figure shows the graphs of the functions f(x) =  x and h(x) =  -x. From the figure we observe that the green graph is the same as the blue graph folded down across the y- axis. The graph of h is a reflection of the graph of f about the x-axis as h(x) =  -x = f(-x) That is, for every point (x, y), there is a point (-x, y). The y-axis acts as the line of reflection or the mirror. In general, the graph of y = f(-x) is the graph of y = f(x) reflected about the y-axis. X Y 1234 2 1 3 4 -2 -3 -4 -2-3-4 5 f(x) =  x h(x)=  -x y-axis acts as line of reflection

13. 6. VERTICAL STRETCHING The transformations we have performed till now do not change the basic shape of the graph. Now we perform certain transformations on the shape of the basic graph wherein we get a graph of a new function by shrinking or stretching the basic graph. Using the graph of f(x) = |x|, let us obtain the graph of g(x) = 3|x|. Let us construct a table showing the coordinates for both f and g. Comparing the y-coordinates for both the functions, we observe that for each x, the y-coordinate of g is 3 times larger than the corresponding y-coordinate of f. As a result, the graph of g is narrower than the graph of f. f(x) = |x| x (x, f(x))g(x)=3|x|(x, g(x)) -2|-2| = 2(-2, 2)3|-2| = 6(-2, 6) |-1| = 1(-1, 1)3|-1| = 3(-1, 3) 0|0| = 0(0, 0)3|0| = 0(0, 0) 1|1| = 1(1, 1)3|1| = 3(1, 3) 2|2| = 2(2, 2)3|2| = 6(2, 6) 1 1 2 2 3 4 3 4 5 -4 -2 -3 -2 -4 -5 X Y g(x) = 3|x|

14. 6. VERTICAL STRETCHING (Cont…) Thus, we can say that the graph of g(x) = 3|x| is obtained by vertically stretching the graph of f(x) = |x|. Hence, we arrive at the following generalization: f c c Let f be a function and c a positive real number greater than 1, then the graph of y = c f(x) is the graph of y = f(x) vertically stretched by multiplying each of its y-coordinates by c. X 1 1 2 2 3 4 3 4 5 -4 -2 -3 -2 -4 -5 f(x) = |x| Y g(x) = 3 |x|

15. 7. VERTICAL SHRINKING Now, using the graph of f(x) = |x|, let us obtain the graph of h(x) = (1/4)|x|. Let us construct a table showing the coordinates for both f and h. We observe from the table that each y-coordinate of h is one- fourth the corresponding y-coordinate of f. The equation of h(x) = (1/4)|x| can be written as h(x) = (1/4)f(x). Thus we can say that the graph of h is obtained by vertically shrinking the graph of f. Hence, we arrive at a generalization: f c c Let f be a function and c a positive integer such that 0< c < 1, then the graph of y = c f(x) is the graph of y = f(x) vertically shrunk by multiplying each of its y-coordinates by c. X 1 1 2 2 3 4 3 4 5 -4 -2 -3 -2 -4 -5 f(x) = |x| Y h(x) =(1/4) |x| xf(x)=|x|(x, f(x))h(x)=(1/4)|x|(x, h(x)) -2|-2|=2(-2, 2)(1/4)|-2|=1/2(-2, ½) |-1|=1(-1, 1)(1/4)|-1|=1/4(-1, ¼) 0|0|=0(0, 0)(1/4)|0|=0(0, 0) 1|1|=1(1, 1)(1/4)|1|=1/4(1, ¼) 2|2|=2(2, 2)(1/4)|2|=1/2(2, ½)

16. 8. SEQUENCES OF TRANSFORMATIONS A function involving more than one transformation can be graphed by performing transformations in the following order: 1. Horizontal shifting 2. Vertical stretching or shrinking 3. Reflection about the x-axis or the y-axis 4. Vertical shifting Example: Using the graph of standard quadratic equation f(x) = x 2, we plot the graph for the function g(x) = -(x - 2) 2 + 3 1 2 3 4 5 -4 -3 -2 12 3 4 -2 -3-4 -5 Horizontal shift right by 2 units 1 2 3 4 5 -4 -3 -2 12 3 4 -2 -3-4 -5 Original Graph of f(x) = x 2 STEP 1 Horizontal shifting Graph of y= (x - 2) 2 STEP 2 Vertical stretching or shrinking No stretching or shrinking is required because the equation y = -(x - 2) 2 + 3 is not multiplied by any constant.

17. 8. SEQUENCES OF TRANSFORMATIONS(Cont…) STEP 3 Reflecting about the x-axis Graph of y = -(x - 2) 2 STEP 4 Vertical Shifting Graph of g(x) = -(x - 2) 2 + 3 Thus, after a sequence of transformations, we obtain the graph of h(x) = -(x - 2) 2 + 3 from the graph of f(x) = x 2. Graph reflected about x- axis 1 2 3 4 5 -4 -3 -2 12 3 4 -2 -3-4 -5 Vertical shift up by 3 units 1 2 3 4 5 -4 -3 -2 12 3 4 -2 -3-4 -5

18. 9. SUMMARY Let us recall what we have learnt so far: VERTICAL SHIFTS Let f be a function and c a positive real number, then the graph of y = f(x) + c is the graph of y = f(x) shifted c units vertically upward. Let f be a function and c a positive real number, then the graph of y = f(x) - c is the graph of y = f(x) shifted c units vertically downward. HORIZONTAL SHIFTS Let f be a function and c a positive real number, then the graph of y = f(x + c) is the graph of y = f(x) shifted to the left by c units. Let f be a function and c a positive real number, then the graph of y = f(x - c) is the graph of y = f(x) shifted to the right by c units. REFLECTIONS OF GRAPHS The graph of y = -f(x) is the graph of y = f(x) reflected about the x-axis. The graph of y = f(-x) is the graph of y = f(x) reflected about the y-axis. VERTICAL STRETCHING AND SHRINKING Let f be a function and c a positive real number greater than 1, then the graph of y = c f(x) is the graph of y = f(x) vertically stretched by multiplying each of its y-coordinates by c. Let f be a function and c is a positive integer such that 0< c < 1, then the graph of y = c f(x) is the graph of y = f(x) vertically shrunk by multiplying each of its y-coordinates by c.