Unit 6, Lesson 6
Symmetry in Equations
Algebra 2
6.1
Warm-up
What is the same? What is different?
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What is the same? What is different?
Here is a graph of for . Draw the graph for , and be prepared to explain your reasoning if:
is even
A graph of function f on a coordinate plane. X axis from negative 4 to 4, by 1’s. Y axis from negative 4 to 4, by 1’s. From left to right, the function begins in the second quadrant, moves downward and to the right, crossing the x axis at approximately negative 4, continues downward reaching minimum at negative 3 point 4 comma negative 3 point 0. It then curves up crossing the x axis at negative 2 until it reaches negative 1 point 4 comma 0 point 4. It curves back down until it reaches about 0 point 5 comma 0.
is odd
A graph of function f on a coordinate plane. X axis from negative 4 to 4, by 1’s. Y axis from negative 4 to 4, by 1’s. From left to right, the function begins in the second quadrant, moves downward and to the right, crossing the x axis at approximately negative 4, continues downward reaching minimum at negative 3 point 4 comma negative 3 point 0. It then curves up crossing the x axis at negative 2 until it reaches negative 1 point 4 comma 0 point 4. It curves back down until it reaches about 0 point 5 comma 0.
is neither even nor odd
A graph of function f on a coordinate plane. X axis from negative 4 to 4, by 1’s. Y axis from negative 4 to 4, by 1’s. From left to right, the function begins in the second quadrant, moves downward and to the right, crossing the x axis at approximately negative 4, continues downward reaching minimum at negative 3 point 4 comma negative 3 point 0. It then curves up crossing the x axis at negative 2 until it reaches negative 1 point 4 comma 0 point 4. It curves back down until it reaches about 0 point 5 comma 0.
Remember the even function with this graph from earlier?
An equation for is . Since we already know is even, we also know that the output at and is the same for any value of in the domain of . Said another way, for all inputs . If we didn't know was even, we could check by using as the input.
For example, since ,
which shows the function is even.
Let's look at a different function. Consider the function defined as . Using as the input, we have:
This means is odd since .