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}
Example 1 :
Graph the functions f(x) = x^{3} and g(x) = ^{3}√x on the same coordinate plane. Find f ◦ g and graph it on the plane as well. Explain your results.
Solution :
f(x) = x^{3} and g(x) = ^{3}√x
f ◦ g (x) = f [ g (x) ]
= f (x^{1/3})
= (x^{1/3})^{3}
f ◦ g (x) = x
y = x^{3} if x = -2, then y = -8 if x = -1, then y = -1 if x = 0, then y = 0 if x = 1, then y = 1 if x = 2, then y = 8 |
y = x^{1/3} if x = -8, then y = -2 if x = -1, then y = -1 if x = 0, then y = 0 if x = 1, then y = 1 if x = 8, then y = 2 |
f(x) = x^{3}
Let y = x^{3}
Let us find the inverse function, for that we have to solve for x.
x = y^{1/3}
Now we have to replace "x" by "f^{-1}(x)" and "y" by "x".
f^{-1}(x) = x^{1/3}
By finding inverse of the given function, we get the other function.
Note : The graph of y = f^{−1}(x) is the reflection of the graph of f in y = x.
Example 2 :
Write the steps to obtain the graph of the function y = 3(x − 1)^{2} + 5 from the graph y = x^{2}
Solution :
Step 1 :
By graphing the curve y = x^{2}, we get a open upward parabola with vertex (0, 0).
Step 2 :
Here 1 is subtracted from x, so we have to shift the graph of y = x^{2}, 1 unit to the right side.
Step 3 :
The positive number 3 is multiplied by (x-1) which is greater than 1, so we have to compress the curve y = (x-1)^{2} towards y-axis.
Step 4 :
5 is added to the function, so we have to move the graph of y = 3(x-1)^{2}, 5 units to the left side.
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