The following are what we call The Parent Functions

f(x) = c f(x) = x Constant Function Linear Function

f(x) = f(x) = |x| Absolute Value Function Square Root Function

f(x) = x 3 f(x) = x 2 Quadratic Function Cubic Function

What happens when we change the equations of these parent functions?

Mother Functionrelative functionchange? y 1 = |x| y 2 = |x| + 2 y 2 = |x| - 3 y 2 = |x| + 5 y 1 = |x| y 2 = |x| - 1 Up 2 Up 5 Down 3 Down 1 Now, make a conclusion…

Mother Functionrelative functionchange? y 1 = |x| y 2 = |x + 2| y 2 = |x – 3| y 2 = |x + 5| y 1 = |x| y 2 = |x – 1| left 2 left 5 right 3 right 1 Now, make a conclusion…

In summary: f(x) + c f(x) - c f(x + c) f(x - c) Vertical shift c units upward Vertical shift c units downward Horizontal shift c units to the left Horizontal shift c units to the right

Left 9, Down 14 Left 2, Down 3

Write the Equation to this Graph

-f(x) f(-x) Reflection in the x-axis Reflection in the y-axis What did the negative on the outside do? What do you think the negative on the inside will do? Study tip: If the sign is on the outside it has “x”-scaped

What word do we use to describe what as happened to these functions? These have all been examples of rigid transformations, because they do not change shape.

Nonrigid Transformations What is this? This is a transformation that causes distortion to the original graph! What do we do to an equation to make this happen? We put a number in front or on the inside of the mother function.

Nonrigid Transformations h(x) = c f(x) c >1 0 < c < 1 Vertical stretch Vertical shrink Closer to y-axis Closer to x-axis

h(x) = f(cx) c >1 Horizontal compression Closer to y-axis h(x) = f(cx) 0 < c < 1 Horizontal stretch Closer to x-axis

Reflection over x-axis, Right 6 Vertical shrink, Up 7