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

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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.

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