Functions and Graphs

1
Functions and Graphs

• Chapter 2

2
2.2 The Graph of a Function
3
Range

• The possible outputs of a function.
• A list of the y values that are possible.

4
Finding Range Graphically

• Look at the graph and find the lowest y value,
  how low does the graph go?
• Find the highest y value, how high does the graph
  go?
• The range will be those 2 values and all the ones
  in between unless the graph has holes in it.

5
Finding Range Graphically

• The table can help you discover holes in the
  graph, values of y that are not graphed, and
  highest/lowest y values.
• The Max and Min functions on the calculator can
  also help you find the highest and lowest y
  values.

6
TBLSET

• Press 2ND and then WINDOW.
• Enter the value that the table should start with
  by typing in the number next to TblStart and
  press enter.
• Enter a value next to delta Tbl that represent
  the amount by which the chart should count and
  press enter.
• Pressing 2nd graph will display the table.

7
Example--Range

• (2x)/(x2 4)
• Graphing this function and zooming in gets the
  following picture.
• It looks like all y values are being graphed, so
  the range would be all reals. Note that the
  vertical lines are really asymptotes and are not
  part of the graph.

8
Example--Range

• (x)/sqrt(x 4)
• Looking at the domain tells us that x has to be
  greater than 4. The graph looks like it graphs
  infinite y values then dips down to a positive
  number and then goes back up.

9
Example--Range

• (x)/sqrt(x 4)
• Note from the calculator table as x gets large so
  does y, so the graph definitely goes back up.

10
Example--Range

• (x)/sqrt(x 4)
• Note from the calculator tables that y gets
  larger and larger as x gets closer and closer to
  4. As x gets larger than 4, notice that y
  decreases until about x8 and then increases. Is
  4 the lowest y?

11
Example--Range

• (x)/sqrt(x 4)
• Use the minimum function on the calc to discover
  that 4 is the lowest y between x6 and x10.

12
Odd Symmetry

• Symmetry with respect to the origin.
• Algebraically--If (x,y) is on the graph, then
  (-x,-y) is on the graph or f(-x)-f(x)

13
Odd Symmetry

• Graphically--
• Graph folds over the x-axis and then the y-axis
  to match up.

14
Even Symmetry

• Symmetry with respect to the y-axis.
• Algebraically
• If (x,y) is on the graph, then (-x,y) is on the
  graph or
• f(x) f(-x).

15
Even Symmetry

• Graphically
• Graph folds over the y-axis to match up.

16
X-axis Symmetry

• Symmetry with respect to the x-axis.
• If (x,y) is on the graph, then (x,-y) is on the
  graph.
• Graph folds over the x-axis to match up.

17
Determining Symmetry Algebraically

• Take the original function and plug in x
  wherever you see an x and simplify.
• If the answer to 1 is the same as the original
  function, then the function has even symmetry.

18
Determining Symmetry Algebraically

• 3. If not, then take the original function and
  multiply it through by a negative one.
• 4. If the results from 1 and 3 are equal, then
  the function has odd symmetry.
• 5. Otherwise it has neither type of symmetry.

19
Symmetry Example
20
Symmetry Example
21
Symmetry Example
22
Domain,Range and Symmetry from a Graph Example

• Find f(0) and f(6).
• f(0)0 and f(6)0
• Find f(2) and f(-2).
• f(2)-2 and f(-2)1
• 3. Is f(3) positive or negative?
• negative
• 4. Is f(-1) positive or negative?
• positivie

23
Domain,Range and Symmetry from a Graph Example

• For what numbers x is f(x)0?
• 0,4,6
• For what numbers x is f(x) lt 0?
• 0x4
• What is the domain of f?
• -4x6
• What is the range of f?
• -2y5

24
Domain,Range and Symmetry from a Graph Example

• What are the x-intercepts?
• (0,0),(4,0),(6,0)
• What are the y-intercepts?
• (0,0)
• How often does the line y-1 intersect the graph?
• Twice
• How often does the line x1 intersect the graph?
• once
• For what value of x does f(x) -2?
• X2

25
Domain, Range and Symmetry from a Graph Example 2

• The graph is a function since it passes the
  vertical line test.
• What is the domain?
• -pxp
• What is the range?
• -1y1
• What are the intercepts?
• (-p,0),(p,0), and (0,0)
• What type of symmetry does the graph have?
• Odd Symmetry

26
Domain, Range and Symmetry from a Graph Example 2

• The graph is a function since it passes the
  vertical line test.
• What is the domain?
• 0x4
• What is the range?
• 0y3
• What are the intercepts?
• (0,0)
• What type of symmetry does the graph have?
• Neither odd nor even nor x-axis.