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Lab 2, Shifting of Graphs, Activity 2.1, Shifting of graphs : f (x) + a, Steps, 1. Draw the graph of f (x) = x2, 2. Create a number slider a with increment 0.1, 3. Draw the graph of g(x) = f (x) + a (Input:f+a), 4. Observe how the graph of g(x) changes according to a, 5. Create input boxes for editing function ‘f’ and slider ‘a’, 6. Do the above observations for different functions such as, |x|, [x], x3 etc, 7. Save this as Activity 2.1, Apply trace to the graph and change value of slider using, Animation to get a pattern., [Right click → trace on, Right click on the slider and turn on animation]., To erase the pattern , press ctrl+F
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Observations, f(x) f(x)+a Domain Range Domain Range of Shifting, of f(x) of f(x) of f(x) + a f(x) + a, f(x) Shifted a units, x2 x2 +a, R, [0,∞), R, [a,∞), verticaliy upwords, , x2, , x2 -a, , R, , [0,∞), , R, , [-a,∞), , f(x) Shifted a units, verticaliy downwords, , ׀x׀ ׀x׀+a, , R, , [0,∞), , R, , [a,∞), , f(x) Shifted a units, verticaliy upwords, , ׀x׀ ׀x׀-a, , R, , [0,∞), , R, , [-a,∞), , f(x) Shifted a units, verticaliy downwords, , [x] [x]+a, , R, , Z, , R, , Z, , f(x) Shifted a units, verticaliy upwords, , [x] [x]-a, , R, , Z, , R, , Z, , f(x) Shifted a units, verticaliy downwords, , x3, , x3 +a, , R, , R, , R, , R, , f(x) Shifted a units, verticaliy upwords, , x3, , x3 -a, , R, , R, , R, , R, , f(x) Shifted a units, verticaliy downwords, , 1 . Graph of f(x)+a is obtained by shifting the graph of f(x) by ‘a’, , units vertically upwards if ‘a’ is positive and vertically, downwards if ‘a’ is negative., 2 . Realise that vertical shift doesn’t change the domain and may, change the range., 3, , . f(x)+a represents a family of curves.
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Activity 2.2, Shifting of graphs : f (x + a), Steps, 1. Open a new GeoGebra window., 2. Draw the graph of f(x) = x2 ., 3. Create a number slider a with increment 0.1, 4. Draw the graph of g(x) = f(x + a)., (Input: f(x+a)), 5. Create input boxes for editing function f and slider a., 6. Observe how the graph of g(x) changes according to a., 7. Generalise the above observations with different functions, such as |x|, [x], x3 etc, 8. Use the animation option to change the slider and apply, trace to the graph, 9. Save this as Activity 2.2
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Observations, f(x) f(x+a) Domain Range Domain Range of Shifting, of f(x) of f(x) of f(x + a) f(x + a), f(x) shifted a units, x2 (x+a)2 R, [0,∞), R, [0,∞), parallel to x axis, towards left, , x2, , (x-a)2, , R, , [0,∞), , R, , [0,∞), , f(x) shifted a units, parallel to x axis, towards right, , ׀x׀ ׀x+a׀, , R, , [0,∞), , R, , [0,∞), , f(x) shifted a units, parallel to x axis, towards left, , ׀x׀ ׀x-a׀, , R, , [0,∞), , R, , [0,∞), , f(x) shifted a units, parallel to x axis, towards rights, , [x] [x+a], , R, , Z, , R, , Z, , f(x) shifted a units, parallel to x axis, towards left, , [x] [x-a], , R, , Z, , R, , Z, , f(x) shifted a units, parallel to x axis, towards right, , x3, , (x+a)3, , R, , R, , R, , R, , f(x) shifted a units, parallel to x axis, towards left, , x3, , (x-a)3, , R, , R, , R, , R, , f(x) shifted a units, parallel to x axis, towards right, , 1. Graph of f(x + a) is obtained by shifting the graph of f(x), by ‘a’ units parallel to x axis towards left if ‘a’ is positive, and towards right if ‘a’ is negative., 2 . Realise that horizontal shift doesn’t change the range and may, change the domain., 3. f(x+a) represents a family of curves.
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Activity 2.3, Reflection of a Graph: -f(x), Steps, 1. Open a new GeoGebra window., 2 Draw the graph of f(x) = x2, 3. Draw the graph of g(x) = −f(x) (Input: -f), 4. Compare the graphs of f(x) and g(x)., 5. Create an input box for f and change the function to, i) x2 + 2, , ii) x2 − 1, , iii) |x| − 1, , iv) |x − 1|, , v) [x], vi) x2 + 2x + 1, vii)1/x, 6. Compare the graphs of f and g in each case., Write your findings., 7. Save this as Activity 2.3
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Observations, f(x), , g(x)= -f(x) Domain, of f(x), x2, -x2, R, x2+2 -(x2+2), R, x2-1 -(x2-1), R, , Range Domain Range of, of f(x) of g(x) g(x), [0,∞), R, (-∞,2], [2,∞), R, (-∞,-2], [-1,∞), R, (-∞,-1], , |x|-1, , - (|x|-1), , R, , [-1,∞), , R, , (-∞,1], , |x -1| -(|x -1|), , R, , [0,∞), , R, , (-∞,0], , R, R, , Z, (-∞,-1], , [x], -[x], R, Z, x2+2x -(x2+2x+1) R, [-1,∞), +1, www.rrvgirls.com, 1/x, -1/x, R-{0} R-{0}, , R-{0}, , R-{0}, , 1 . Graph of −f (x) is obtained by reflecting the graph of f(x) on, x axis., 2 . Realise that reflection on x axis doesn’t change the domain, and may change the range
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Activity 2.4, Reflection of a Graph: f(-x), Steps, 1. Open new GeoGebra window, 2. Draw the graph of f (x) = x3, 3. Draw the graph of g(x) = f (−x) (Input: f(-x)), 4. Compare the graph of f (x) and g(x), 5. Create an input box for f and change the function to, i) 1/x, ii) [x], iii) |x|, iv) x2, v) (x − 2)2 www.rrvgirls.com, 6. Compare the graphs of f and g in each case., Write your findings., 7. Save this file as Activity 2.4
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Observations, f(x), x3, 1/x, [x], |x|, x2, , g(x)= f(-x) Domain, of f(x), R, - x3, -1/x, R-{0}, [-x], R, |x|, R, x2, , (x − 2)2 (x + 2)2, , R, R, , Range Domain Range of g(x), of f(x) of g(x), R, R, R, R-{0} R-{0} R-{0}, Z, R, Z, [0,∞) R, [0,∞), [0,∞), , R, , [0,∞), , R, , [0,∞), , www.rrvgirls.com, , [0,∞), , 1. Graph of f(−x) is obtained by reflecting the graph of f(x) on y, axis., 2. Realise that reflection on x axis doesn’t change the range and, may change the domain, 3. . If f is even, f (−x) = f (x), which shows that reflection on y, axis doesn’t change the graph. So the graph of an even, function is symmetric with respect to y axis., eg. x2 , |x| + 2 ,|x| etc., . If f is odd, f (−x) = −f (x) ⇒ −f (−x) = f (x) which shows, graph of −f (−x) is obtained by reflecting the graph of f (x) on, x axis and then on y axis. So the graph of an odd function is, symmetric about the origin., eg. X3 , 1/x , etc., . There are functions which are neither odd nor even., eg. x3 + 1, (x − 2)2, etc.
