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Functions

Functions. Our objectives: Recognize “ Parent Functions ” Graphically & Algebraically Please take notes and ALWAYS ask questions . Pre- Calculus 2. Do Now (see below). Today’s Agenda. Students Will Be Able To… Define domain and range Recognize Parent Functions.

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Functions

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  1. Functions Our objectives: • Recognize “Parent Functions” • Graphically & Algebraically • Please take notes and ALWAYS ask questions 

  2. Pre-Calculus 2 Do Now (see below) Today’s Agenda Students Will Be Able To… Define domain and range Recognize Parent Functions Exit Ticket – 19-28 all 2. CW Note-taking guide on Parent Functions Today’s Objectives: • Homework: Do NOW: Define domain and range in your notebook. 10 mins .

  3. The following basic graphs will be used extensively in this section. It is important to be able to sketch these from memory.

  4. Constant Functionf(x) = a

  5. Linear functionf(x) = x

  6. quadraticfunction

  7. cubicfunction

  8. Polynomial Function • http://zonalandeducation.com/mmts/functionInstitute/polynomialFunctions/graphs/polynomialFunctionGraphs.html • *zero degree • *first Degree • *second degree • *third degree • Fourth degree

  9. Exponential Functionf(x) = a

  10. Logarithmic Functionf(x)=log x

  11. square rootfunction

  12. cube root function

  13. absolute valuefunction

  14. Rational Functionf(x) =

  15. Reciprocal Functionf(x) =

  16. Inverse Function

  17. Piece-wise Function

  18. Piece-wise Function

  19. We will now see how certain transformations (operations) of a function change its graph. This will give us a better idea of how to quickly sketch the graph of certain functions. The transformations are (1) translations, (2) reflections, and (3) stretching.

  20. Vertical Translation Vertical Translation For b > 0, the graph of y = f(x) + b is the graph of y = f(x) shifted upb units; the graph of y = f(x) b is the graph of y = f(x) shifted downb units.

  21. Horizontal Translation Horizontal Translation For d > 0, the graph of y = f(x  d) is the graph of y = f(x) shifted rightd units; the graph of y = f(x + d) is the graph of y = f(x) shifted leftd units.

  22. Vertical shifts • Moves the graph up or down • Impacts only the “y” values of the function • No changes are made to the “x” values • Horizontal shifts • Moves the graph left or right • Impacts only the “x” values of the function • No changes are made to the “y” values

  23. The values that translate the graph of a function will occur as a number added or subtracted either inside or outside a function.Numbers added or subtractedinside translate left or right, while numbers added or subtractedoutside translate up or down.

  24. Recognizing the shift from the equation, examples of shifting the function f(x) = • Vertical shift of 3 units up • Horizontal shift of 3 units left (HINT: x’s go the opposite direction that you might believe.)

  25. Points represented by (x , y) on the graph of f(x) become If the point (6, -3) is on the graph of f(x), find the corresponding point on the graph of f(x+3) + 2

  26. Combining a vertical & horizontal shift • Example of function that is shifted down 4 units and right 6 units from the original function.

  27. Reflections • The graph of f(x) is the reflection of the graph of f(x) across the x-axis. • The graph of f(x) is the reflection of the graph of f(x) across the y-axis. • If a point (x, y) is on the graph of f(x), then (x, y) is on the graph of f(x), and • (x, y) is on the graph of f(x).

  28. Reflecting • Across x-axis (y becomes negative, -f(x)) • Across y-axis (x becomes negative, f(-x))

  29. Vertical Stretching and Shrinking The graph of af(x) can be obtained from the graph of f(x) by stretching vertically for |a| > 1, or shrinking vertically for 0 < |a| < 1. For a < 0, the graph is also reflected across the x-axis. (The y-coordinates of the graph of y = af(x) can be obtained by multiplying the y-coordinates of y = f(x) by a.)

  30. VERTICAL STRETCH (SHRINK) • y’s do what we think they should: If you see 3(f(x)), all y’s are MULTIPLIED by 3 (it’s now 3 times as high or low!)

  31. Horizontal stretch & shrink • We’re MULTIPLYING by an integer (not 1 or 0). • x’s do the opposite of what we think they should. (If you see 3x in the equation where it used to be an x, you DIVIDE all x’s by 3, thus it’s compressed horizontally.)

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