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This guide covers key concepts in graph transformations, including reflections across the x-axis, y-axis, and the line y=x. It explains vertical and horizontal stretches, shrinks, and slides. The material also introduces interval notation as a method for expressing domains and ranges instead of set notation, including the use of parentheses and brackets. Additionally, examples illustrate how to identify symmetries of functions, determine whether they are even, odd, or neither, and properly express their domain and range.
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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. • The graph of is the reflection of the graph of over the line y=x.
Identify the reflection in each of the graphs below. @ line y=x @ x-axis @ y-axis
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 .
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.
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.
More Transformations • There is an order of operations to be followed when applying multiple transformations. • This order of transformations is BCAD.
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
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.
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.
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 .
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 .
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.
