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Reflections and Symmetry. Lesson 5.2. Across the x-axis. Across the y-axis. Flipping the Graph of a Function. Given the function below We wish to manipulate it by reflecting it across one of the axes. Flipping the Graph of a Function. Consider the function

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flipping the graph of a function

Across the x-axis

Across the y-axis

Flipping the Graph of a Function
  • Given the function below
    • We wish to manipulate it by reflecting it across one of the axes
flipping the graph of a function3
Flipping the Graph of a Function
  • Consider the function
    • f(x) = 0.1*(x3 - 9x2 + 5) : place it in y1(x)
    • graphed on the window   -10 < x < 10  and  -20 < y < 20
flipping the graph of a function4
Flipping the Graph of a Function
  • specify the following functions on the Y= screen:
    • y2(x) = y1(-x)                dotted style
    • y3(x) = -y1(x)                thick style
  • Predict which of these will rotate the function
    • about the x-axis
    • about the y-axis
flipping the graph of a function5

use -f(x)

use f(-x)

Flipping the Graph of a Function
  • Results
  • To reflect f(x) in the x-axis       or rotate about
  • To reflect f(x) in the y-axis         or rotate about

Spreadsheet Demo

even and odd functions
Even and Odd Functions
  • If  f(x) = f(-x)  the graph is symmetric across the y-axis
  • It is also an even function
even and odd functions7
Even and Odd Functions
  • If f(x) = -f(x) the graph is symmetric across the x-axis
  • But ... is it a function ??
even and odd functions8
Even and Odd Functions
  • A graph can be symmetric about a point
    • Called point symmetry
  • If f(-x) = -f(x) it is symmetric about the origin
  • Also an odd function
applications
Applications
  • Consider a frozen yam placed into a hot oven.  Think what the graph of the temperature would look like.  Sketch the graph of the temperature of the yam.  It is frozen at 0 degrees Fahrenheit and the oven is at 300 degrees Fahrenheit.

This will be both a flip and a shift of an exponential function

applications10
Applications
  • This is the function
    • f(x) = 300 - 300(0.97)t
  • It has been flipped about the y-axis
  • Then it has been shifted up
  • Which part did the shift?
  • Which part did theflip?
reflecting in the line y x
Reflecting in the Line y = x
  • Given the function below:
  • For each (x,y) shown, reverse the values to get (y,x)
  • Plot the (y,x) values and connect the points
reflecting in the line y x12
Reflecting in the Line y = x
  • Results
  • Note: it is not a function.
reflecting in the line y x13
Reflecting in the Line y = x
  • Try it for this graph … will the result be a function or not?
assignment
Assignment
  • Lesson 5.2
  • Page 209
  • Exercises 1 – 31 odd