Rational Functions
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What rules have they given you for function transformations? (here) How far have you gotten in applying these rules?
Please be complete. Thank you!
Transformation of f(x) = x^2
Question
Identify the function that describes the transformation from the parent function f(x)=1/x
The function has been vertically stretched by a factor of -3, then translated right 6 units and up 5 units?
Example:
The function f(x) = x2 has been dilated by factor 2, reflected in X axis, shifted 3 units left and 4 units down to produce the function g(x).
Determine the rule of g(x).
Answer:
Use your graphics calculator to graph the function: f(x) = x2
Parabola is symmetrical about Y axis with min turning point at (0,0)
Doubling the coefficient of x2 stretches parabolic graph by factor 2 in direction of Y axis.
Rule becomes: f(x) = 2x2
Changing the coefficient of x2 to -2 turns graph upside down.
Rule becomes: f(x) = -2x2
Replacing x with x + 3 shifts the graph 3 units left.
(For horizontal shift plus sign means shift in minus direction)
Rule becomes: f(x) = -2(x + 3)2
Subtracting 4 shifts the graph 4 units down.
(For vertical shift minus sign means shift in minus direction)
Rule becomes: g(x) = -2(x + 3)2 - 4
where g(x) describes the transformation from the parent function f(x) = x2
Use your graphics calculator to graph the function:
g(x) = -2(x + 3)2 - 4
Note the parabola is symmetrical about vertical line x = -3 with max turning point at (-3,-4). It is also narrower than the original graph of f(x).
Check:
Graph of f(x) = x2 passes through point (2,4)
If (2,4) is dilated by factor 2:
(2,4) becomes (2,8)
If (2,8) is reflected in X axis
(2,8) becomes (2,-8)
If (2,-8) is shifted left 3 units
(2,-8) becomes (-1,-8)
If (-1,-8) is shifted down 4 units)
(-1,-8) becomes (-1,-12)
We now need to show that the point (-1,-12) lies on the curve:
g(x) = -2(x + 3)2 - 4
g(-1)
= -2(-1 + 3)2 - 4
= -2(2)2 - 4
= -12
Question
Identify the function that describes the transformation from the parent function f(x)=1/x
The function has been vertically stretched by a factor of -3, then translated right 6 units and up 5 units?
Example:
The function f(x) = x2 has been dilated by factor 2, reflected in X axis, shifted 3 units left and 4 units down to produce the function g(x).
Determine the rule of g(x).
Answer:
Use your graphics calculator to graph the function: f(x) = x2
Parabola is symmetrical about Y axis with min turning point at (0,0)
Doubling the coefficient of x2 stretches parabolic graph by factor 2 in direction of Y axis.
Rule becomes: f(x) = 2x2
Changing the coefficient of x2 to -2 turns graph upside down.
Rule becomes: f(x) = -2x2
Replacing x with x + 3 shifts the graph 3 units left.
(For horizontal shift plus sign means shift in minus direction)
Rule becomes: f(x) = -2(x + 3)2
Subtracting 4 shifts the graph 4 units down.
(For vertical shift minus sign means shift in minus direction)
Rule becomes: g(x) = -2(x + 3)2 - 4
where g(x) describes the transformation from the parent function f(x) = x2
Use your graphics calculator to graph the function:
g(x) = -2(x + 3)2 - 4
Note the parabola is symmetrical about vertical line x = -3 with max turning point at (-3,-4). It is also narrower than the original graph of f(x).
Check:
Graph of f(x) = x2 passes through point (2,4)
If (2,4) is dilated by factor 2:
(2,4) becomes (2,8)
If (2,8) is reflected in X axis
(2,8) becomes (2,-8)
If (2,-8) is shifted left 3 units
(2,-8) becomes (-1,-8)
If (-1,-8) is shifted down 4 units)
(-1,-8) becomes (-1,-12)
We now need to show that the point (-1,-12) lies on the curve:
g(x) = -2(x + 3)2 - 4
g(-1)
= -2(-1 + 3)2 - 4
= -2(2)2 - 4
= -12