September 4, 2012

General graph shapes Think about splitting up the domain Draw both separately before you put them together if it helps you to visualize it better Can be infinitely many pieces, usually you’ll only have to deal with 3 or less Usually we notate piecewise functions with a large curly bracket, and each piece of the function needs to have the values of x for which it is defined to the right hand side of the equation Remember that colored circles indicate that the value is included in the domain (square bracket), while an open circle indicates that it is not, but values can get infinitely closer to that value

For example, what does this graph look like?

Graph each separately first if you need to

We can also look at a graph of a piecewise function and be asked to come up with its equation, if it is made up of simple line segments with which we are familiar (review the types of lines from last week).

What would the equation be for this graph? (-4,-4) (-1,2) (-1,1) (1,1) (3,-1)

Tackle each section independently Three pieces in the domain: (-4,-4) (-1,2) (-1,1) (1,1) (3,-1)

Tackle each section independently Three pieces in the domain: [-4,-1) [-1,1] (1,3] **Note: the 1 can be considered in either the second or third domain, but not both.** (-4,-4) (-1,2) (-1,1) (1,1) (3,-1)

First piece [-4,-1)  Linear  Slope = rise/run = ( )/( ) = (6)/(3) = 2  Intercept: y = mx+b (2) = (2)(-1)+b 4=b  y = 2x + 4 Thus we have the first part of the function:  2x + 4, -4 <= x < -1 (-4,-4) (-1,2) (-1,1) (1,1) (3,-1)

Second piece [-1,1]  Quadratic (x^2)  Does not appear to have any adjustments  Thus we have the second part of the function:  x^2, -1 <= x <=1 (-4,-4) (-1,2) (-1,1) (1,1) (3,-1)

Third piece (1,3]  Linear  Slope = (-1 – 1)/(3-1) = -2/2 = - 1  Intercept: y = mx+b (1)=(-1)(1) + b 2 = b  Thus we have the last part of the function:  - x + 2, 1 < x < 3 (-4,-4) (-1,2) (-1,1) (1,1) (3,-1)

Put it all together: (-4,-4) (-1,2) (-1,1) (1,1) (3,-1)

Horizontal changes happen “within” the function, and they’re usually the opposite of what you think would happen. Vertical changes happen “outside” the function, and they are usually the direction you would expect. There are a total of 10 different things that we can do to a graph to manipulate it, and theoretically we could do all these things to the same function.

Vertical Adjustments f(x) + c Moves graph up c units f(x) – c Moves graph down c units 2*f(x) Stretches vertically by a factor of 2 Stretches vertically by a factor of 2 (could be any number > 1) 0.5*f(x) Compresses vertically by a factor of 2 Compresses vertically by a factor of 2 (any fraction between 0 and 1) -f(x) Reflection over the x axis

Vertical Example For example, let’s look at f(x) = x^2

Vertical Example For example, let’s look at f(x) = x^2 This is g(x) = f(x) + 2, which shifts up 2 units:

Vertical Example For example, let’s look at f(x) = x^2 This is g(x) = f(x) - 2, which shifts down 2 units:

Vertical Example For example, let’s look at f(x) = x^2 This is g(x) = 2*f(x), which stretches vertically by a factor of 2:

Vertical Example For example, let’s look at f(x) = x^2 This is g(x) = 0.5*f(x), which compresses vertically by a factor of 2:

Vertical Example For example, let’s look at f(x) = x^2 This is g(x) = -f(x), which flips over the axis:

(usually backwards from what you expect) Horizontal Adjustments (usually backwards from what you expect) f(x + c) left Moves graph left c units f(x – c) right Moves graph right c units f(2*x) Compresses horizontally by a factor of (1/2) Compresses horizontally by a factor of (1/2) (could be any number > 1) f(0.5*x) Stretches by a factor of 2 Stretches by a factor of 2 (any fraction between 0 and 1) f(-x) Reflection over the y axis

Horizontal Example Let’s think about f(x) = sqrt(x).

Horizontal Example Let’s think about f(x) = sqrt(x). Here’s g(x) = sqrt(x-2), which shifts right 2 units

Horizontal Example Let’s think about f(x) = sqrt(x). Here’s g(x) = sqrt(x+2), which shifts left 2 units

Horizontal Example Let’s think about f(x) = sqrt(x). Here’s g(x) = sqrt(2*x), which compresses by a factor of (1/2)

Horizontal Example Let’s think about f(x) = sqrt(x). Here’s g(x) = sqrt(0.5x), which stretches by a factor of 2

Horizontal Example Let’s think about f(x) = sqrt(x). Here’s g(x) = sqrt(-x), which causes it to flip over the y axis

Usually the situation ends up being a combination of both, with adjustments being made from a basic function that you are already familiar with.