# Week 5: Transformations – Day 5

Mixup of basic function transformations:

– to reflect across the x-axis, take the negative of the function: – f(x)

– to reflect across the y-axis, change the input to -x: f(-x)

– to shift upward vertically by C: add C: f(x) + C

– to shift to the right horizontally by A: replace x by x – A: f(x – A)

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

2. g(x) = -|x| + 1

3. g(x) = sqrt(-x) (square root of -x)

4. g(x) = 3(x + 1) – 4

5. g(x) = – (x + 1)

# Week 5: Transformations – Day 4

Reflections in the y-axis

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.

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.

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.

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.

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.

# Week 5: Transformations – Day 3

Horizontal Function Shifts

The graph of a function is shifted horizontally by a if you replace x by “x – a”.

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)?

2. g(x) is obtained from f(x) = x3 by shifting to the left 1 unit.

What is the formula for g(x)?

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)?

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)?

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)?

# Week 5: Transformations – Day 2

Transformations and functions: vertical shift

The graph of a function y = f(x) is vertically shifted H units up when you add H to the function.

For example, if f(x) = x2 and g(x) = x2 + 1, then the graph of g(x) is the same as the graph of f(x), shifted up one unit.

Rewrite the following functions with the indicated shift. Check the graphs with a graphing program such as desmos.com:

1. Shift up 2 units: f(x) = 2x – 4

2. Shift down 3 units: f(x) = 4x3

3. Shift up 10 units: f(x) = sqrt(x) (f(x) = square root of x).

4. Shift up 5 units: f(x) = 1/x

5. Shift down 100 units: f(x) = x1.5

# Week 5: Transformations – Day 1

For each diagram, copy and draw an axis of symmetry. There is at least one axis in each figure.

Example: The square below has a red line where one axis of symmetry is. 1. 2. 3. 4. 5. 