Reflection :
A reflection is the mirror image of the graph where line l is the mirror of the reflection.
Here f' is the mirror image of f with respect to l. Every point of f has a corresponding image in f'. Some useful reflections of y = f(x) are
(i) The graph y = -f(x) is the reflection of the graph of f about the x-axis.
(ii) The graph y = f(-x) is the reflection of the graph of f about the y-axis.
(iii) The graph of y = f^{−1}(x) is the reflection of the graph of f in y = x.
Translation :
A translation of a graph is a vertical or horizontal shift of the graph that produces congruent graphs.
The graph of
y = f(x + c), c > 0 causes the shift to the left.
y = f(x − c), c > 0 causes the shift to the right.
y = f(x) + d, d > 0 causes the shift to the upward.
y = f(x) − d, d > 0 causes the shift to the downward.
Consider the functions:
(i) f(x) = |x| (ii) f(x) = |x − 1| (iii) f(x) = |x + 1|
Dilation :
Dilation is also a transformation which causes the curve stretches (expands) or compresses (contracts). Multiplying a function by a positive constant vertically stretches or compresses its graph; that is, the graph moves away from x-axis or towards x-axis.
If the positive constant is greater than one, the graph moves away from the x-axis. If the positive constant is less than one, the graph moves towards the x-axis.
Consider the functions:
(i) f(x) = x^{2} (ii) f(x) = (1/2) x^{2} (iii) f(x) = 2x^{2}
Example 1 :
For the curve y = x^{3} given in the figure shown below, draw
(i) y = −x^{3}
(ii) y = x^{3} + 1
(iii) y = x^{3} − 1
(iv) y = (x + 1)^{3} with the same scale.
Solution :
(i) y = −x^{3}
To find the graph of y = −x^{3}, we have to use the concept reflection.
Let f(x) = x^{3}
f(-x) = (-x)^{3}
f(-x) = -x^{3}
f(-x) = -f(x)
So, the reflection is about x-axis.
(ii) y = x^{3} + 1
To graph the above function, we have to use the concept translation. That is, move the curve 1 unit upward.
(iii) y = x^{3} − 1
To graph the above function, we have to use the concept translation. That is, move the curve 1 unit downward.
(iv) y = (x + 1)^{3}
To graph the above function, we have to use the concept translation. That is, move the curve 1 unit left side.
Example 2 :
For the curve y = x^{1/3} given in the following figure, draw
(i) y = -x^{1/3}
(ii) y = x^{1/3} + 1
(iii) y = x^{1/3} - 1
(iv) y = (x + 1)^{1/3}
Solution :
(i) y = -x^{1/3}
Here we have negative sign in front of x^{1/3}
f(x) = x^{1/3}
f(-x) = (-x)^{1/3}
f(-x) = -x^{1/3}
f(-x) = -f(x)
So, the reflection is about x - axis.
(ii) y = x^{1/3} + 1
Since 1 is added with the given function, we have to move the curve 1 unit upward.
(iii) y = x^{1/3} - 1
Since 1 is subtracted from the given function, we have to move the curve 1 unit downward.
(iv) y = (x + 1)^{1/3}
Since 1 is added to x in the given function, we have to move the curve 1 unit left side.
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