How do you write an equation for a translation of an absolute value function?

The general equation for a translated absolute value function is y = a|x - h| + k, where (h, k) represents the vertex of the absolute value graph and 'a' determines the direction and steepness.

Here's a breakdown of how each part of the equation affects the graph's translation:

• a:
  • If a > 0, the graph opens upwards (V-shape).
  • If a < 0, the graph opens downwards (inverted V-shape).
  • The larger the absolute value of 'a', the steeper the graph.
• h: Represents the horizontal shift.
  • If h > 0, the graph shifts right by h units.
  • If h < 0, the graph shifts left by |h| units. Note the negative sign in the equation (x - h) causes the shift direction to be counterintuitive. For instance, y = |x - (-2)| which simplifies to y = |x + 2|, shifts the graph to the left by 2 units.
• k: Represents the vertical shift.
  • If k > 0, the graph shifts up by k units.
  • If k < 0, the graph shifts down by |k| units.

Example:

Let's say you want to translate the basic absolute value function, y = |x|, 3 units to the right and 2 units up.

• h = 3 (shift right)
• k = 2 (shift up)
• Assume a = 1 (no stretching or reflection)

The equation for the translated function would be: y = |x - 3| + 2. The vertex of this graph would be at the point (3, 2).

Summary Table:

Transformation	Equation Change	Effect on Graph
Horizontal Shift	y = |x - h|	Shifts the graph left or right. h > 0 shifts right; h < 0 shifts left.
Vertical Shift	y = |x| + k	Shifts the graph up or down. k > 0 shifts up; k < 0 shifts down.
Vertical Stretch/Compression/Reflection	y = a|x|	Stretches or compresses vertically. If a < 0, reflects over the x-axis.

By understanding how 'a', 'h', and 'k' affect the absolute value function, you can easily write equations for translations and transformations of the basic graph.