september 4, 2012. general graph shapes think about splitting up the domain draw both separately...

MA 107 RecitationSeptember 4, 2012

Piecewise Functions• 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

Piecewise Functions• For example, what does this graph look like?

Piecewise Functions• Graph each separately

first if you need to

Piecewise Functions

Piecewise Functions

Piecewise Functions• 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).

Piecewise Functions• What would the equation be for this graph?

(-4,-4)

(-1,2)

(-1,1)

(1,1)

(3,-1)

Piecewise Functions• Tackle each section

independently• Three pieces in the

domain:

(-4,-4)

(-1,2)

(-1,1)

(1,1)

(3,-1)

Piecewise Functions• 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)

Piecewise FunctionsFirst piece [-4,-1) Linear Slope = rise/run

= (2 - - 4)/(-1 - - 4)= (6)/(3) = 2

Intercept:y = mx+b(2) = (2)(-1)+b4=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)

Piecewise FunctionsSecond 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)

Piecewise FunctionsThird piece (1,3] Linear Slope = (-1 – 1)/(3-1)

= -2/2 = - 1 Intercept:

y = mx+b(1)=(-1)(1) + b2 = 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)

Piecewise FunctionsPut it all together:

(-4,-4)

(-1,2)

(-1,1)

(1,1)

(3,-1)

WebAssign HW 4

Graph Adjustments• 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.

Graph Adjustments• 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 (could be any

number > 1)• 0.5*f(x)• Compresses vertically by a factor of 2

(any fraction between 0 and 1)• -f(x)• Reflection over the x axis

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

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

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

Graph Adjustments• 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:

Graph Adjustments• 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:

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

Graph Adjustments• Horizontal Adjustments (usually backwards from

what you expect)• f(x + c) • Moves graph left c units

• f(x – c)• Moves graph right c units

• f(2*x)• Compresses horizontally by a factor of (1/2) (could be any number > 1)

• f(0.5*x)• Stretches by a factor of 2 (any fraction between 0 and 1)

• f(-x)• Reflection over the y axis

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

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

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

Graph Adjustments• 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)

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

Graph Adjustments• 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

Graph Adjustments• Usually the situation ends up being a combination

of both, with adjustments being made from a basic function that you are already familiar with.