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Graph Transformations: Shifting, Reflecting, and Stretching

Learn how to transform graphs by shifting them vertically and horizontally, reflecting them across axes, and stretching or compressing them vertically and horizontally.

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Graph Transformations: Shifting, Reflecting, and Stretching

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  1. Chapter 3: Functions and GraphsSection 3.4: Graphs and Transformations Essential Question: In the equationg(x) = c[a(x-b)] + dwhat do each of the letters do to the graph?

  2. 3.4: Graphs and Transformations • Parent function: A function with a certain shape that has the simplest rule for that shape. • For example, f(x) = x2 is the simplest rule for a parabola • Any parabola is a transformation of that parent function • All of the following parent functions are on page 173 in your books… there is no need to copy them now. • It’s most important that you get down the words in blue. Everything else is predominately mathematical definition.

  3. 3.4: Graphs and Transformations • Identify the parent function.

  4. 3.4: Graphs and Transformations Constant function Identity Function f(x) = 1 f(x) = x

  5. 3.4: Graphs and Transformations Absolute-value function Greatest Integer Function f(x) = |x| f(x) = [x]

  6. 3.4: Graphs and Transformations Quadratic function Cubic Function f(x) = x2 f(x) = x3

  7. 3.4: Graphs and Transformations Reciprocal function Square Root Function f(x) = 1/x f(x) =

  8. 3.4: Graphs and Transformations Cube root function f(x) = .

  9. 3.4: Graphs and Transformations • Vertical shifts • When a value is added tof(x), the effect is to add thevalue to the y-coordinateof each point, effectivelyshifting the graph up anddown. • If c is a positive number, then: • The graph g(x) = f(x) + c is the graph of f shifted up c units • The graph g(x) = f(x) – c is the graph of f shifted down c units • Numbers adjusted after the parent function affect the graph vertically, as one would expect (+ up, – down)

  10. 3.4: Graphs and Transformations • Horizontal shifts • When a value is added tothe x of a function, the effect is to readjust the graph,effectively shifting the graph left and right. • If c is a positive number, then: • The graph g(x) = f(x+c) is the graph of f shifted c units to the left • The graph g(x) = f(x-c) is the graph of f shifted c units to the right • Numbers adjusted to the x of the parent function [inside a parenthesis] affect the graph horizontally, in reverse of expected values (+ left, – right)

  11. 3.4: Graphs and Transformations • Reflections • Adding a negative sign beforea function reflects the graph about the x-axis. Adding a negative sign before the x in the function reflects the graph aboutthe y-axis. • A negative sign before the function flips up & down(vertically, also called “reflected across the x-axis”) • A negative sign before the x flips left & right (horizontally, also called “reflected across the y-axis”)

  12. 3.4: Graphs and Transformations • Stretches & Compressions (Vertical) • If a function is multiplied by a number, it will stretch or compress the parent function vertically • If c > 1, then the graph g(x) = c • f(x) is the graph of f stretched vertically (away from the x-axis) by a factor of c • If c < 1, then the graph g(x) = c • f(x) is the graph of f compressed vertically (towards the x-axis) by a factor of c • Multiplying the entire function will stretch or compress a function (proportionally) towards or away from the x-axis, as expected (large numbers stretch, small numbers compress)

  13. 3.4: Graphs and Transformations • Stretches & Compressions (Horizontal) • If the x of a function is multiplied by a number, it will stretch or compress the parent function horizontally • If c > 1, then the graph g(x) = f(c • x) is the graph of f compressed horizontally (towardsthe y-axis) by a factor of 1/c • If c < 1, then the graph g(x) = f(c • x) is the graph of f stretched horizontally (away from the y-axis) by a factor of 1/c • Multiplying the x of a function will stretch or compress a function (inversely) away from or towards the y-axis, opposite as expected (large numbers compress by the reciprocal, small numbers stretch by the reciprocal) (x+1)3 (2(x+1))3 (¼(x+1))3

  14. 3.4: Graphs and Functions • Assignment • Page 182 • 1-21, odd problems

  15. Chapter 3: Functions and GraphsSection 3.4: Graphs and TransformationsDay 2 Essential Question: In the equationg(x) = c[a(x-b)] + dwhat do each of the letters do to the graph?

  16. 3.4: Graphs and Transformations • We have our grand equation: g(x) = c[a(x-b)] + d Addition on the outsideshifts the graph vertically (d)(as expected: positive == up, negative == down) Addition on the inside of the parenthesis shifts the graph horizontally (b)(opposite as expected: positive == left, negative == right) A negative on the outsideflips the function vertically (c) A negative on the inside of the parenthesis flips the graph horizontally (a) Multiplication on the outside stretches/compresses the graph vertically (c) (as expected: large numbers == stretch, small numbers == compress) Multiplication on the inside stretches/compresses the graph horizontally (a)(opposite: large numbers == compress by reciprocal, small numbers == stretch by reciprocal)

  17. 3.4: Graphs and Transformations • We have our grand equation: g(x) = c[a(x-b)] + d • Order of application • a (horizontal reflection) • a (horizontal stretch/compression) • b (horizontal shift) • c (vertical reflection) • c (vertical stretch/compression) • d (vertical shift)

  18. 3.4: Graphs and Transformations • Example set #1 – write a rule for the function whose graph can be obtained from the given parent function by performing the given transformation. • Parent function: f(x) = x2Transformations: shift 5 units left and up 4 units • Parent function: f(x) = Transformations: shift 2 units right, stretched vertically by a factor of 2, and shift up 2 units g(x) = (x + 5)2 + 4

  19. 3.4: Graphs and Transformations • Example set #2 – describe a sequence of transformations that transform the graph of the parent function f into the graph of the function g. Do not graph the function. a = -1/2, b = 6 (b is always its opposite), c = -1, d = 0 Horizontal reflection Horizontal stretch by a factor of 2 (horizontal stretches are inverses) Horizontal shift right 6 units Vertical reflection

  20. 3.4: Graphs and Transformations • Example set #2 – describe a sequence of transformations that transform the graph of the parent function f into the graph of the function g. Do not graph the function. a = -2, b = 2, c = 3, d = 0 Horizontal reflection Horizontal compression by a factor of 0.5 Horizontal shift right 2 units Vertical stretch by a factor of 3

  21. 3.4: Graphs and Functions • Assignment • Page 182 • 23-41, odd problems • 35 – 41: Ignore the directions • Instead, identify the parent function and the transformations that occurred to get to the transformed function.

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