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Understanding Parent Functions and Their Transformations in Mathematics

This presentation covers the essential concepts of parent equations and their transformations, including linear, quadratic, exponential, logarithmic, and trigonometric functions. Each parent function has a distinctive graph and real-world applications. We will explore vertical and horizontal translations, vertical and horizontal stretches and compressions, as well as reflections across the axes. By mastering these transformations, you can deeply understand how different functions behave and interact, paving the way for more advanced mathematical studies.

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Understanding Parent Functions and Their Transformations in Mathematics

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Presentation Transcript

  1. Parent Equation General Forms Transforming

  2. Parent Equation • The simplest form of any function • Each parent function has a distinctive graph • We will summarize these in the next few slides

  3. Constant • f(x)=a; where a is any number

  4. Linear • f(x)=x

  5. Absolute Value

  6. Exponential Value

  7. Logarithmic • y=lnx

  8. Square Root/Radical

  9. Cube Root

  10. Quadratic • f(x)=x2

  11. Cubic • f(x)=x3

  12. Reciprocal • Same as a Rational Graph

  13. Rational

  14. Sine • y=sinx

  15. Cosine • y=cosx

  16. Tangent • y=tanx

  17. Constant Function • f(x)=a; where a is any number • Domain: all real numbers • Range: a

  18. Linear Function • f(x)=x • Domain: all real numbers • Range: all real numbers

  19. Transformations Linear and Quadratic

  20. Vertical Translations • Positive Shift (Shift up) • Form: y=f(x)+b where b is the shift up • Negative Shift (Shift down) • Form: y=f(x)-b where b is the shift down

  21. Horizontal Translations • Shift to the right • Form: y=f(x-h) • The negative makes you think left, but actually means right here • Shift to the left • Form y=f(x+h) • This would shift to the left of the origin

  22. Vertical Stretch and Compression • If y=f(x), then y=af(x) gives a vertical stretch or compression of the graph of f • If a>1, the graph is stretched vertically by a factor of a • If a<1, the graph is compressed vertically by a factor of a

  23. Horizontal Stretch and Compression • If y=f(x),then y=f(bx) gives a horizontal stretch or compression of the graph of f • If b>1, the graph is compressed horizontally by a factor of 1/b • If b<1, the graph is stretched horizontally by a factor of 1/b

  24. Reflection • If y=f(x), then y=-f(x) gives a reflection of the graph f across the x axis • If y=f(x), then y= f(-x) gives a reflection of the graph f across the y axisl

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