Graphs of Functions (Part 1)2.5(1)

Graphs of Functions (Part 1)2.5(1)

Even/ odd functions

Shifts and Scale Changes

Even/ odd functions

Shifts and Scale Changes

PODPODSimplify the following difference quotient,

for f(x) = x3.

What does this expression become as h approaches 0?

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

What does this expression become as h approaches 0?

h

xfhxf )()(

PODPODSimplify the following difference quotient,

for f(x) = x3.

What does this expression become as h approaches 0?

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

What does this expression become as h approaches 0?

22 33 hxhx

Even/ odd FunctionsEven/ 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.

Even 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 neither

At each table, graph two of the following to see if they are even, odd, or neither

As you graph each one, test it algebraically.

As you graph each one, test it algebraically.

6)(

)(

472)(

523)(

2

23

35

24

xxt

xxxk

xxxxq

xxxp

1)(

)4()(

4)(

)(

3

2

2

3

xxm

xxh

xxg

xxf

At each table, graph two of the following to see if they are even, odd, or neither

At each table, graph two of the following to see if they are even, odd, or neither

Notice how p(-x) = p(x), once everything is simplified– it is an even function. The graph matches this.

Notice how p(-x) = p(x), once everything is simplified– it is an even function. The graph matches this.

)(

523

5)(2)(3)(24

24

xp

xx

xxxp

At each table, graph two of the following to see if they are even, odd, or neither

At each table, graph two of the following to see if they are even, odd, or neither

Notice how q(-x) = -q(x), once everything is simplified– it is an odd function. The graph matches this.

Notice how q(-x) = -q(x), once everything is simplified– it is an odd function. The graph matches this.

)(

)472(

472

)(4)(7)(2

)(4)(7)(2)(

35

35

35

35

xq

xxx

xxx

xxx

xxxxq

At each table, graph two of the following to see if they are even, odd, or neither

At each table, graph two of the following to see if they are even, odd, or neither

Notice how k(-x) ≠ k(x) or -k(x), once everything is simplified– it is neither even nor odd. The graph matches this.

Notice how k(-x) ≠ k(x) or -k(x), once everything is simplified– it is neither even nor odd. The graph matches this.

23

23 )()()(

xx

xxxk

Vertical/ horizontal shiftsVertical/ horizontal shifts

Start 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?

Start 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 shiftsVertical/ horizontal shifts

y = x2 y = x2 + 5 y = (x + 5)2

(y – 5 = x2)y = x2 y = x2 + 5 y = (x + 5)2

(y – 5 = x2)

Shifts and equations in generalShifts and equations in general

A 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?

A 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)

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?

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)

Vertical/ horizontal scale changes (Vertical/ horizontal stretching and compressing)

y = sin x y = 3sin x

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

y = sin x 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)

Vertical/ horizontal scale changes (Vertical/ horizontal stretching and compressing)

y = sin x y = sin (3x)

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

y = sin x y = sin (3x)

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

Vertical/ horizontal scale changes (Vertical/ horizontal stretching and compressing)

Vertical/ horizontal scale changes (Vertical/ horizontal stretching and compressing)

y = sin x y = (1/3)sin x

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

y = sin x y = (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)

Vertical/ horizontal scale changes (Vertical/ horizontal stretching and compressing)

y = sin x y = -3sin x

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

y = sin x y = -3sin x

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

Scale changes and equations in general

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)

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

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

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

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

Scale changes and equations in general

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.

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 line…The bottom line…

If you change the equation, you do the opposite to the graph.

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

If 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 translationScale change and translation

Many 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.

Many 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 translationScale change and translation

Try 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.

Try 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 translationScale change and translation

Try 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 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 there’s timeTry it, if there’s 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.

In 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.