The Limit of a Function
Transformation and Symmetry of Functions
Objectives
- Graph function using vertical and horizontal shifts.
- Graph functions using reflection about the x-axis and y-axis.
- Graph functions using compressions and stretches.
- Combine transformations to make a new graph.
Summary
For a function \(y = f(x)\), the general equation \(y = a f(x - h) + k\) is a transformation if \(a, h,\) and \(k\) are real numbers with \(a ≠ 0\). The transformations vary as follows:
Horizontal Translation
Given a function \(f\), a new function \(g(x) = f(x – h)\), where \(h\) is a constant of the function \(f\). When \(h\) is positive, then the graph \(y = f(x)\) will translate h units to the right of the x- axis.When \(h\) is negative, then the graph \(y = f(x)\) will translate \(h\) units to the left of the x-axis.
Figure 1 There is Horizontal Shift of the function \(f(x) = ∛x\) thus \(h = +1\) Note: that the function is shifting left after the addition of 1.
Vertical Translation
Given the function \(f(x)\), a new function \(g(x) = f(x) + k\), where \(k\) is a constant of the function \(f(x)\). When \(k\) is positive, then the graph will translate \(k\) units upward on the y- axis. When \(k\) is negative, then the graph will translate \(k\) units downward on the y-axis.
Figure 2 There is Vertical Shift of the function \(f(x) = ∛x\) thus \(k = 1\) .
Reflection
Given a function \(f(x)\), a new function \(g(x) = - f(x)\) is a vertical reflection of the function \(f(x)\) over the x-axis. Given a function \(f(x)\), a new function \(g(x) = f(-x)\) is a horizontal reflection of the function \(f(x)\) over the y-axis.
Figure 3 You will see the horizontal and vertical reflection of a function \(f(x)\).
Even and Odd Functions (Symmetry)
A function is called an even function if for every input \(x\) ; \(f(x) = f( -x)\). The graph of an even function is symmetric about the y-axis. A function called an odd function if for every input \(x\) ; \(f(x)= -f(-x)\). The graph of an odd function is symmetric about the origin.
Vertical Stretches and Compression
Given a function \(f(x)\), a new function \(g(x) = a f(x)\), where \(a\) is a constant, is a vertical stretch or a vertical compression of the function \(f(x)\).
- If \(a > 1\), then the graph will be stretched.
- If \(0 < a < 1\), then the graph will be compressed.
- If \(a < 0\), then there will be combination of a vertical stretch or compression with a vertical reflection.
Figure 4 The Vertical stretch and compression of a function \(f(x)\).
Horizontal Stretches and Compression
Given a function \(f(x)\), a new function \(g(x) = f(bx)\) where, \(b\) is a constant, is a horizontal stretch or horizontal compression of the function \(f(x)\).
- If \(b > 1\), then the graph will be compressed by \(1/b\).
- If \(0 < b < 1\), then the graph will be stretched by \(1/b\).
- If \(b < 0\)then there will be combination of a horizontal stretch or compression with a horizontal reflection.
Figure 5 The Horizontal stretch and compression of the function \(f(x) = x^2\)