Here we are going to see, transformation of graphs of modulus function.
Reflection :
A reflection is the mirror image of the graph where line l is the mirror of the reflection.
(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.
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.
Problem 1 :
From the curve y = |x|, draw
(i) y = |x − 1| + 1
(ii) y = |x + 1| − 1
(iii) y = |x + 2| − 3
Solution :
First let us draw the graph of y = |x|, for any negative values of x, we will get positive values of y.
So, the graph will be
(i) y = |x − 1| + 1
First 1 is subtracted from x in the modulus function, so we have to move the curve 1 unit to the right side.
1 is added to |x - 1|, so we have to move the graph of y = |x - 1| 1 unit upward direction.,
Problem 2 :
From the curve y = sin x, draw y = sin|x| (Hint: sin(−x) = −sin x.)
Solution :
Graph of y = sin x
If x = 0 y = sin 0 y = 0 |
If x = π/2 y = sin π/2 y = 1 |
If x = π y = sin π y = 0 |
If x = 3π/2 y = sin 3π/2 y = -1 |
If x = 2π y = sin 2π y = 0 |
(0, 0) (π/2, 1) (π, 0) (3π/2, - 1) and (2π, 0)
To draw the graph of y = sin|x|, let us use the hint
sin(−x) = −sin x
If x = 0 y = sin 0 y = 0 |
If x = -π/2 y = -sin π/2 y = -1 |
If x = π/2 y = sin π/2 y = 1 |
If x = -π y = -sin π y = 0 |
If x = π y = sin π y = 0 |
If x = -3π/2 y = -sin 3π/2 y = -1 |
If x = 3π/2 y = sin 3π/2 y = -1 |
If x = -2π y = -sin 2π y = 0 |
If x = 2π y = sin 2π y = 0 |
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