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Reflections, Transformations and Interval Notation

Reflections, Transformations and Interval Notation. L. Waihman Revised 2006. Reflections of. The graph of is the reflection of the graph of over the x-axis. The graph of is the reflection of the graph of over the y-axis.

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Reflections, Transformations and Interval Notation

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  1. Reflections, Transformations and Interval Notation L. Waihman Revised 2006

  2. Reflections of • The graph of is the reflection of the graph of over the x-axis. • The graph of is the reflection of the graph of over the y-axis. • The graph of is the reflection of the graph of over the line y=x.

  3. Identify the reflection in each of the graphs below. @ line y=x @ x-axis @ y-axis

  4. Transformations • If is negative, the graph is reflected over the x-axis. • determines the vertical stretch or shrink. If >1, the graph has a vertical stretch by a factor of . If the graph has a vertical shrink by a factor of .

  5. More Transformations • is used to determine the horizontal stretch or shrink of the graph. • If , the graph shrinks horizontally units. • If , the graph stretches horizontally b units. • If b is negative, the graph reflects about the y-axis.

  6. More Transformations • determines the horizontal slide of the graph. If the slide is to the right. If the slide is to the left. • determines the vertical slide of the graph. If the slide is down. If the slide is up.

  7. More Transformations • There is an order of operations to be followed when applying multiple transformations. • This order of transformations is BCAD.

  8. Example: • parent function: quadratic or squaring function • Stretch: Horizontal shrink by • Slide: left 2 unit • Reflection: over the x-axis • Stretch: vertical stretch by a factor of 3 • Slide: up 1 unit

  9. Absolute Value Transformations • Absolute Value of the Function indicates that all the negative y-values must become positive so you need to reflect any parts normally below the x-axis above the x-axis. • Absolute Value of the Argument indicates all negative x-values would produce the same results as the positive counterparts, thus everything to the left of the y-axis would disappear and the remaining graph to the right of the y-axis would reflect across the y-axis.

  10. Absolute Value of the function:

  11. Absolute Value of the Argument:

  12. Absolute Value of the Function and the Argument

  13. Interval Notation • From this point forward we will be using interval notation instead of set notation to describe solutions. • Interval notation is written with either parenthesis or brackets and contains the first element of the interval, the second element. • Parenthesis are used if the element is NOT included in the solution. Brackets are used if the element is included in the solution.

  14. If the domain of a function is , we now describe this as extending from negative infinity to positive infinity – written . Note that infinity cannot be found so it cannot be an included element. • If the domain consisted of the x-values from 0 to infinity with 0 being on the graph, we would write this as .

  15. Interval Notation • If the range is from -3 to infinity with a y-value at -3, we would write the range as . • If the range contained the y-values from 0 to 6 inclusive, we would write the solution as .

  16. Put it all together! • Graph the following function: • List all the transformations including reflections. • Determine the symmetry of the function. • Determine if the function is even, odd or neither. • Identify the domain and range using interval notation.

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