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*Graphs of Functions (Part 1) 2.5(1) Even/ odd functions Shifts and Scale Changes Even/ odd functions...*

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Graphs of Functions (Part 1)2.5(1)Even/ odd functions

Shifts and Scale Changes

PODSimplify the following difference quotient, for f(x) = x3.

What does this expression become as h approaches 0?

PODSimplify the following difference quotient, for f(x) = x3.

What does this expression become as h approaches 0?

Even/ odd FunctionsEven functions are reflected over an axis-- which one?

Odd functions are reflected over something else-- what?

The acid test:For even functions, f(-x) = f(x)

For odd functions, f(-x) = -f(x)

(What does that mean in English?)

Write which is which on the parent function handout.

At each table, graph two of the following to see if they are even, odd, or neitherAs you graph each one, test it algebraically.

At each table, graph two of the following to see if they are even, odd, or neitherNotice how p(-x) = p(x), once everything is simplified it is an even function. The graph matches this.

At each table, graph two of the following to see if they are even, odd, or neitherNotice how q(-x) = -q(x), once everything is simplified it is an odd function. The graph matches this.

At each table, graph two of the following to see if they are even, odd, or neitherNotice how k(-x) k(x) or -k(x), once everything is simplified it is neither even nor odd. The graph matches this.

Vertical/ horizontal shiftsStart by graphing y = x2 and then y = x2 + 5.

What do you notice about the relationship between the graphs? How does that compare to the relationship between the equations?

Now, graph y = (x + 5)2.

How do the graph and equation compare to y = x2?

Vertical/ horizontal shiftsy = x2 y = x2 + 5y = (x + 5)2(y 5 = x2)

Shifts and equations in generalA vertical shift of c: y = f(x) + cy - c = f(x)

A horizontal shift of c:y = f(x-c)

How would each of these graphs compare to y = x2

y = x2 + 6x + 9 y = x2 - 6x + 9 y = x2 +3?

Vertical/ horizontal scale changes (Vertical/ horizontal stretching and compressing)Start by graphing y = sin x

Then, at tables graph one of these:y = 3sin x y = 1/3 sin (x)y = sin (3x)y = -3sin(x)

What do you notice about the relationship between the graphs? Between the equations?

Vertical/ horizontal scale changes (Vertical/ horizontal stretching and compressing)y = sin xy = 3sin x

What do you notice about the relationship between the graphs? Between the equations?

Vertical/ horizontal scale changes (Vertical/ horizontal stretching and compressing)y = sin xy = sin (3x)

What do you notice about the relationship between the graphs? Between the equations?

Vertical/ horizontal scale changes (Vertical/ horizontal stretching and compressing)y = sin xy = (1/3)sin x

What do you notice about the relationship between the graphs? Between the equations?

Vertical/ horizontal scale changes (Vertical/ horizontal stretching and compressing)y = sin xy = -3sin x

What do you notice about the relationship between the graphs? Between the equations?

Scale changes and equations in general

A vertical stretch of c:y = cf(x)y/c = f(x)

A horizontal stretch of c: y = f(x/c)

Were used to thinking of expansion with values of c greater than 1. How would we achieve a compression?

Scale changes and equations in general

If c is negative, the graph will

reflect over the y- axis when multiplied by x.

reflect over the x-axis when multiplied by y.

The bottom lineIf you change the equation, you do the opposite to the graph.

When you change a graph, everything changes: intercepts, asymptotes, holes, domain, and range.

Scale change and translationMany graphs have a combination of scale change and translation (multiplication and addition). In that case, just like PEMDAS, you multiply then add scale change then translation in each direction.

Scale change and translationTry it with a point. Give the coordinates of the point (3, -1) after the function undergoes the transformation y = 2f(x+5) 5. Remember, scale change first.

Scale change and translationTry it with a point. Give the coordinates of the point (3, -1) after the function undergoes the transformation y = 2f(x+5) 5.

(-2, -7)

Try it, if theres timeIn groups, find a parent (tool kit) function that is even, or one that is odd. Graph it on your calculators.

Shift it 3 units up and 4 units to the left. What is the new equation? Graph that to test your work.

Stretch your original graph horizontally by a factor of 2, and reflect it over the x-axis. What is the new equation? Graph that to test your work.

For a Take a Chance Award, come up to demonstrate your work.