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Transformations of Functions: Shifts, Reflections, and Stretches

This study guide explores the key concepts of function transformations, including shifts, reflections, and vertical stretches. Students will learn to graph parent functions such as f(x) = x^2, f(x) = x^3, and transformations like vertical shifts (up/down), horizontal shifts (left/right), reflections in the x or y-axis, and non-rigid transformations (vertical stretches and shrinks). Examples and exercises are provided to enhance understanding of how to identify and apply these transformations to various functions.

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Transformations of Functions: Shifts, Reflections, and Stretches

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  1. Warm Up • Odd, Even, Neither • 1. f(x) = x3 + 1 • 2. f(x) = x2 + x4 • 3. f(x) = x3 – x

  2. Translations Moving Graphs

  3. The Parent Functions

  4. f(x) = c f(x) = x

  5. f(x) = |x| f(x) =

  6. f(x) = x2 f(x) = x3

  7. Parent Function relative function change? y2 = |x| + 2 y1 = |x| Up 2 y2 = |x| - 3 y1 = |x| Down 3 y2 = |x| + 5 y1 = |x| Up 5 y2 = |x| - 1 y1 = |x| Down 1

  8. Parent Function relative function change? y2 = |x + 2| y1 = |x| left 2 y2 = |x – 3| y1 = |x| right 3 y2 = |x + 5| y1 = |x| left 5 y2 = |x – 1| y1 = |x| right 1

  9. Reflections of the parent function Reflections in the x-axis h(x) = - f(x) Reflections in the y-axis h(x) = f(-x)

  10. Non-rigid Transformations Vertical stretch c >1 h(x) = c f(x) skinnier 0 < c < 1 If c is a fraction Vertical shrink wider

  11. In summary: f(x) + c Vertical shift c units upward Up f(x) - c Vertical shift c units downward Down f(x - c) Horizontal shift c units to the right Right f(x + c) Horizontal shift c units to the left Left

  12. -f(x) Reflection in the x-axis Flipped Up/down f(-x) Reflection in the y-axis Flipped left/right c >1 Vertical stretch c f(x) skinnier 0 < c < 1 Vertical shrink wider

  13. Describe the Shift Left 9 , Down 14 Left 2 , Down 3

  14. Describe the Shift Reflection over x-axis, Right 6 Vertical shrink, Up 7

  15. Write the Equation to this Graph

  16. Write the Equation to this Graph

  17. Write the Equation to this Graph

  18. HOMEWORK PAGE: 74 1-19

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