**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) = x^{2} 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) = x^{2} + 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 = x^{2} + 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) = x^{3} 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) = x^{2} – 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) = 4x^{3}

g(x) = 4x^{3} – 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) = x^{1.5}

g(x) = x^{1.5} – 100

**Week 5: Transformations – Day 1**

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