Identify the effect on the graph by replacing A, H and K with a number other than 0 or 1

Question

1. Explanation of the transformation type and how it is identified in the equation (related to A, B, H or K)
2. Equation of an absolute value function that represents the transformation.   Explain how your example has been transformed from the parent function
3. Equation of a quadratic function that represents the transformation.  Explain how your example has been transformed from the parent function

Answer 1

BACKGROUND INFO:

1) First you must know what parent functions are. These are the most basic equations that would be graphed at the origin (0,0) with no movement.

Parent functions:

y= x Linear (straight line)

y = x² Quadratic (u shaped parabola)

y = | x | Absolute Value (v- shaped)

2) Next you must know where a, h, k appear in the equations.

1. a will always be BEFORE the x value
2. h will come after the a value and will be either inside parenthesis or absolute value signs
3. k will be at the end of equation

y = a(x + h)² +k QUADRATIC

y = a | x + h | + k ABSOLUTE VALUE

3) Now we can look at what effects a,h, and k have on a function.

Positive or Negative functions (a values):

1. When the function is Negative you have a REFLECTION over the x -axis. This means your graph opens down or "upside down".
2. When the function is Positive, the graph opens up so there is NO reflection.

A Values:

a values effect the SIZE of the graphed function.

1. They have a scale factor
2. When a is GREATER than 1 (like 1.5, 6, etc) you have a Vertical Stretch [appears taller] & a Horizontal Compression [appears thinner]
3. When a is BETWEEN 0 and 1 (a fraction or decimal) you have a Vertical Compression [appears shorter] and a Horizontal Stretch [appears wider].

H Values:

1. h values cause left or right movement of the graph. This is called a horizontal shift or TRANSLATION.
2. When h is NEGATIVE, the graph moves RIGHT
3. When h is POSITIVE, the graph moves LEFT.

K Values

1. k values cause up and down movement. This is called a vertical shift or TRANSLATION.
2. When k is POSITIVE, the graph moves UP
3. When k is NEGATIVE, the graph moves down.

Without having a graph to accompany your question, I cannot be exactly sure what you are asking, but here are some examples.

4) Transformation of Absolute Value Function

Example 1:

y = | x -4 | - 3

1. The function is POSITIVE . It opens UPWARD so there is NOT a reflection.
2. a = 1 (because there IS no number in front). So there is no change to the size of the graph.
3. h = -4 There is a horizontal shift (translation) 4 units to the RIGHT.
4. k = -3 There is a vertical shift (translation) 3 units DOWN.

Example 2:

y = - 2| x + 3 | + 4

1. a has a negative value. This function is NEGATIVE. It will open down so there is REFLECTION over the x - axis.
2. a = 2 This graph will be VERTICALLY STRETCHED & HORIZONTALLY COMPRESSED by a scale factor of 2.
3. h = 3. There is a horizontal shift 3 units to the LEFT.
4. k = 4. There is a vertical shift 4 units UP.

5) Transformation of Quadratic function

Example: y = - 1/4( x - 5)² + 1

1. a has a negative value. The parabola will open down so there is a REFLECTION across the x- axis.
2. a = 1/4. This is a FRACTION. This graph will be VERTICALLY COMPRESSED and HORIZONTALLY STRETCHED by a scale factor of 1/4 (or 0.25).
3. h = -5. There will be a HORIZONTAL SHIFT 5 units to the RIGHT.
4. k = 1. There will be a VERTICAL SHIFT 1 unit up.