Vertical and Horizontal Shifting (7.1)
Original Graph f(x) What transformations have occurred to produce the graph g(x) = f(x-3)+1?
Shifted right 3 units and 1 unit up
A reflection in the x axis means we ______?
A reflection in the y axis means we_____?
x axis--> multiply the entire equation by -1.
y axis--> multiply the x's by -1.
y values get larger
this is called a vertical stretch!
When we multiply x by a number > 1 we ____ by that number. This is called a ______ ________.
We divide by that number
Horizontal Compression
Even functions:
Odd functions:
even functions: opposite x's same y's
odd functions: opposite x's opposite y's
Original Graph f(x) What transformations have occurred to produce the graph g(x) = f(x+8)-5?
shifted 8 units left 5 units down
After a reflection in the x-axis, the quadratic function f(x) = -2x2+7x+5
2x2-7x-5
If we multiply the overall function by a number between 0 and 1 --> y values get _____. This is called a ____ ______.
y values get smaller
this is called a vertical compression
When we multiply x by a number between 0 and 1 we ____ by that number. This is called a ______ ________.
divide k (the bottom of the fraction)
horizontal stretch
Axis of symmetry formula
x=-b/2a
Original Graph f(x). What transformations have occurred to produce the graph g(x)= f(x+2) +4
shifted 2 units left and 4 units up
After a reflection in the y-axis, the quadratic function f(x) =-2x2+7x+5 would have what new formula?
-2x2-7x+5
a quadratic function f(x) has a turning point at (6,-9). If h(x) =3f(x) then where does the function h(x) have a turning point?
Multiply 3 by the y value, X value doesn't change.
(6,-9) becomes (6,-27)
The quadratic function g(x) has a turning point at (-12,8). Where would the quadratic function f(x)=g(4x) have a turning point?
What kind of horizontal function is this?
Horizontal compression by a factor of 4, so we divide our x value.
(-12 divided by 4)
New points: (-3,8)
Rate of change equation?
y2-y1/x2-x1
After a reflection in the y-axis, the quadratic function f(x) = 2x2-8x+3 would have what new formula?
2x2+8x+3
what are the transformations that occurred to y=x2
y=1/3x2+5
vertical compression by 1/3 and a vertical shift upwards of 5 units
the domain and range of f(x) are -12 <x<8 and -6<y<10. If g(x) =f(2x)+5, then which of the following are the domain and range of g?
Horizontal compression by a factor of 2. Vertical shift up 5 units.
Domain (x's) get divided by 2 and range gets shifted up 5.
-6<x<4 and -1<y<15
vertical functions only affect the ____ value?
Horizontal functions only affect the ___ value?
vertical (up and down) Y VALUES
horizontal (side to side) X VALUES
The function g(x) is defined by the rule g(x) =-h(x+2). Its zeros of the function (points on the graph) are x=-5 and x=8. What two transformations have occurred and what are the new zeros?
Transformations: reflection on the x axis and shifted two units left. New zeros: x=-7 and x=6
(subtract 2 from both given zeros).
give all the transformations that occur to the function y= lxl
y=2 lxl - 3
vertical stretch by a factor of 2 and vertical shift downward by 3 units.
What type of function is this?
h(x)=f(1/2x)
what would we divide our turning point by to show this transformation on a graph?
Horizontal Stretch
divide the x value by 2
calculate the rate of change for -8<x<4. How would we find the y values?
Find the y values by looking at the graph for f(-8) and f(4). These points are 4 and 0.
Calculate:
f(4)-f(-8)/ 4-(-8)= 4-0/12 = 4/12 =1/3