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Reflections and Symmetry

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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Reflections and Symmetry

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  1. Reflections and Symmetry Lesson 5.2

  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

  3. 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

  4. 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

  5. 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

  6. Even and Odd Functions • If  f(x) = f(-x)  the graph is symmetric across the y-axis • It is also an even function

  7. Even and Odd Functions • If f(x) = -f(x) the graph is symmetric across the x-axis • But ... is it a function ??

  8. 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

  9. 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

  10. 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?

  11. 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

  12. Reflecting in the Line y = x • Results • Note: it is not a function.

  13. Reflecting in the Line y = x • Try it for this graph … will the result be a function or not?

  14. Assignment • Lesson 5.2 • Page 209 • Exercises 1 – 31 odd

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