Shifting, Reflecting, & Stretching Graphs 1.4

Six Most Commonly Used Functions in Algebra Constant f(x) = c Identity f(x) = x Absolute Value f(x) = |x| Square Root f(x) = √x Quadratic f(x) = x 2 Cubic f(x) x 3

Transformations More complicated graphs are created by shifting, reflecting, or stretching these common graphs. Those are called transformations.

Vertical & Horizontal Shifts Let c be a positive real number. Vertical and horizontal shifts in the graph of y = f(x) are represented as follows: Vertical Shift c units upward h(x) = f(x) + c Vertical Shift c units downward h(x) = f(x) - c Horizontal Shift c units to the right h(x) = f(x - c) Horizontal Shift c units to the left h(x) = f(x + c) Remember y = a(x - h) 2 + k where (h, k) is the vertex??

Example 1 Determine the type of shift. A) y = x B) y = (x + 1) 2 C) y = (x - 1) 2 D) y = x 2 – 2 E) y = (x + 2) F) y = (x - 1) 3 - 2

Reflections Reflections in the coordinate axes of the graph of y = f(x) are represented as follows: 1) Reflection in the x-axis h(x) = -f(x) 2) Reflection in the y-axis h(x) = f(-x)

Example 2 The graph of f(x) = x 2 is show. Each of the graphs that follow is a transformation of the graph of f. Find an equation for each function. A)B) C)

Rigid vs. Non-Rigid Transformations Rigid = horizontal shifts, vertical shifts and reflections ONLY CHANGE POSITION in coordinate plane. Non-Rigid = vertical and horizontal stretching or shrinking CHANGE SHAPE of original graph

Vertical and Horizontal Stretching and Shrinking Vertical Stretch y = c ⋅ f(x) c > 1 Vertical Shrink y = c ⋅ f(x) c < 1 Horizontal Stretch y = f(cx) 0 < c < 1 Horizontal Shrink y = f(cx) c > 1

Example 3 A) f(x) = |x| A) f(x) 3|x| B) f(x) = ½ |x| A) f(x) = 2 - x 3 A) f(x) =