Precalc – 1.5 Shifting, reflecting, and stretching graphs

DO NOW! Sketch the graphs of: If you don’t KNOW some of them, try plotting some points!! f(x) = x f(x) = x 2 f(x) = x 3 f(x) = xf(x) = sin(x)f(x) = e x

The basic graphs f(x) = x 2 f(x) = x 3 f(x) = x f(x) = sin(x)f(x) = e x

Less Basic Graphs f(x) = x 2 We can make a new function, based on the old one. I can do so by multiplying or adding new numbers. Call the new function h(x). Here are some examples. h(x) = f(x+1)  h(x) = (x+1) 2 h(x) = f(x)+ 1  h(x) = (x) h(x) = 2f(x)  h(x) = 2x 2 h(x) = 2f(x+1)  h(x) = 2(x+1) 2

Transformation As you change the functions, the graph moves in predictable ways. As groups, you will investigate exactly what those changes are.

Each group: Enter the functions into the graphing calculator For all graphs of the same type, either graph them on the calculator at the same time or sketch them onto paper ***make sure you know which graph represents which function Take note of what the changes in algebra make happen in the different graphs Write a few sentences explaining your findings

Vertical and Horizontal Shifts Move C units up Move C units down Move C units right Move C units left

Reflecting Reflect across the x-axis Reflect across the y-axis

Nonrigid transformations (STRETCHING) Stretch the graph wider Shrink the graph more narrow

This picture is f(x). Sketch the other pictures. y = f(x) + 2 y = f(x +2) y = f(x – 2) y = -f(x) y = f(-x)

Write the equations for the functions matching these descriptions. Also graph them. The shape of x 3, but moved 10 units to the right and 2 units down The shape of sin(x) but stretched out twice as wide and moved half a unit up The shape of x 2, but flipped across the y- axis and moved 3 units to the left The shape of of x 2, but moved three units to the right and then flipped across the x- axis

What is different about these? (Draw and write the equations.) Shape of x 2, but moved 3 units right and flipped across the x-axis Shape of x 2, but flipped across the x-axis and then moved 3 units right

Write the equation. Based on y=x 2

Describe the transformation that occurs. Sketch the graph.

COMPREHENSION QUESTIONS Consider the basic function f(x)=x. What sort of transformation is f(x)=x+2? Which way is the graph moving? When you add 2 outside of the function (for example, f(x) = sinx+2), are you moving the y-intercept up? Are you moving other points up?

HOMEWORK! Pg 158 # 6abd, 9, 13-15, 19-22, 35-37