Week 5: Transformations – Day 5
Each function g(x) below is a transformation of a basic function f(x). Say what f(x) is, and describe the transformation (sometimes there is more than one transformation). For extra practice graph both g and f on the same set of axes.
1. g(x) = (x – 5)2
f(x) = x2 and the transformation is a horizontal shift 5 units to the right.
2. g(x) = -|x| + 1
f(x) = |x| and there are two transformations: reflection in the x-axis and a vertical shift up 1 unit.
3. g(x) = sqrt(-x) (square root of -x)
f(x) = sqrt(x) and the transformation is reflection in the y-axis.
4. g(x) = 3(x + 1) – 4
There is some leeway here about what is f(x). If you say:
f(x) = 3x – 4 ( a line of slope 3 passing through (0, 4)) then the transformation is a simple horizontal shift 1 unit to the left.
If you say f(x) = 3x, there are two transformations: vertical shift down by 4 units, horizontal shift left by 1 unit.
If you say f(x) = x, there are three transformations: vertical shift down by 4 units, horizontal shift left by 1 unit, and a vertical stretch by 3 units.
5. g(x) = – (x + 1)
f(x) = x and there are two transformations: horizontal shift left by 1 unit and reflection in the x-axis.
Week 5: Transformations – Day 4
If f(x) is a function, the function g(x) = f(-x) is the reflection of f in the y-axis. In all of the problems below, g(x) = f(-x).
1. If f(x) = x – 1, what is the formula for g(x)? Sketch the graphs of f and g on the same set of axes.
g(x) = -x – 1
2. If f(x) = |x – 1|, what is the formula for g(x)? Sketch the graphs of f and g on the same set of axes.
g(x) = |-x – 1| = |x + 1|
3. If f(x) = x2 + 1, what is the formula for g(x)? Sketch the graphs of f and g on the same set of axes.
g(x) = (-x)2 + 1 = x2 + 1 = f(x)
This is an example of an even function, where f(x) = f(-x), and its graph is symmetric about the y-axis.
4. If f(x) = -(x – 1)2, what is the formula for g(x)? Sketch the graphs of f and g on the same set of axes.
g(x) = -(-x – 1)2 = -(x + 1)2
5. If f(x) = |x + 2|, what is the formula for g(x)? Sketch the graphs of f and g on the same set of axes.
g(x) = |-x + 2| = |2 – x| = |x – 2|
Week 5: Transformations – Day 3
What is the formula of the function with the given horizontal shift?
1. The function g(x) is obtained from f(x) = |x| by shifting to the right 5 units.
What is the formula for g(x)?
g(x) = |x – 5|
2. g(x) is obtained from f(x) = x3 by shifting to the left 1 unit.
What is the formula for g(x)?
g(x) = (x + 1)3
3. g(x) is obtained from f(x) = x2 – 7x + 1 by shifting to the right 3 units.
What is the formula for g(x)?
g(x) = (x – 3)2
4. g(x) is obtained from f(x) = (x – 10)4 + 9 by shifting to the left 2 units.
What is the formula for g(x)?
g(x) = (x + 2 – 10)4 + 9
g(x) = (x – 8)4 + 9
5. g(x) is obtained from f(x) = |x – 3| + x by shifting to the right by 1 unit.
What is the formula for g(x)?
g(x) = |x – 1 – 3| + (x – 1) = |x – 4| + x – 1
Week 5: Transformations – Day 2
Rewrite the following functions with the indicated shift.
The transformed function will be called g(x).
1. Shift up 2 units: f(x) = 2x – 4
The transformed function is g(x) = 2x – 4 + 2 = 2x – 2
2. Shift down 3 units: f(x) = 4x3
g(x) = 4x3 – 3
3. Shift up 10 units: f(x) = sqrt(x) (f(x) = square root of x).
g(x) = sqrt(x) + 10
4. Shift up 5 units: f(x) = 1/x
g(x) = 1/x + 10 = (1 + 10x) / x
5. Shift down 100 units: f(x) = x1.5
g(x) = x1.5 – 100
Week 5: Transformations – Day 1
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