Here we are going to see, how to graph the function using transformations.
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}
From the curve y = x, draw
(i) y = −x
(ii) y = 2x
(iii) y = x + 1
(iv) y = (1/2)x + 1
(v) 2x + y + 3 = 0.
Solution :
(i) y = −x
Given function :
y = x
put x = -x
y = -x
y = - f(x)
Hence the graph y = -x is the reflection of the graph y =x.
(ii) y = 2x
Here the constant 2 is multiplied with x, so we have to perform dialation.
Since the positive constant is greater than one, the graph moves away from the x-axis 2 units.
(iii) y = x + 1
Since 1 is added to the function, we have to translate the graph of y = x 1 unit upward.
(iv) y = (1/2)x + 1
Step 1 :
Since 1/2 is multiplied by x, we have to perform translation. If the positive constant which is multiplied by x is less than one, the graph moves towards the x-axis.
Step 2 :
1 is added to f(x), we have to move the graph 1 units upward.
(v) 2x + y + 3 = 0
y = -2x - 3
First let us consider y = -2x, it is the reflection of y = 2x about x axis.
Now we have to to subtract 3 from -2x, so we have move the curve 3 units below.
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