1 / 21

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.

belser
Download Presentation

Graph Transformations: Shifting, Reflecting, and Stretching

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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.

More Related