Math-3 Lesson 1-7 Analyzing the Graphs of Functions

Transcription

1 Math- Lesson -7 Analyzing the Graphs o Functions

2 Which unctions are symmetric about the y-axis? cosx sin x x We call unctions that are symmetric about the y -axis, even unctions.

3 Which transormation is it? g Parent unction k ( x VSF= Right ) h j 5 j ( x) Relect (x-axis) Down 5 Relect (y-axis)

4 Why does relection across the y-axis not change the square unction? Parent unction x-y pairs: - is the input but the unction changes the sign o the input to its opposite sign j ( x) Relect (y-axis) x - - (x) -x (-x) -(-) = - -

5 graphically: To relect a point across the y-axis, we multiply the x-value by -. j ( x) (-, ) (, ) I all x-values multiplied by - whole graph relected across y-axis Relect (y-axis) -x (-x) -(-)= -(-)= -()= -()= - -()= -

6 Even unction: is symmetric about the y -axis) Algebraically: replace x with -x j ( x) ( x) ( x)( x) Replacing x with -x results in the square unction simpliying to the un-relected equation. equation exactly the same or both cases, you can t tell i it has been relected across the y-axis symmetric about the y-axis.

7 Your turn: Test the ollowing unction to see i it is even. ( x) ( x) ( x) x ( x) The original equation looks exactly like the equation that has been relected across the y-axis symmetric about the y-axis.

8 Your turn: Test the ollowing unction to see i it is even. ( x ) g ( x) (( x) ) ( )( x ) ( ) ( x ( x ) ) Factor out - (still inside the square unction) Power o a Power Property ( x) Not even since relection across y-axis is not the same as the original equation (or graph)

9 Draw the shape o each unction rom memory cosx sin x x Which o them are even unctions?

10 Odd unctions Describe algebraically how the parent unction should be transormed in order to relect it across the x-axis. x g (x-y pairs: multiply y-values by -) g( x) x Describe algebraically how the parent unction should be transormed in order to relect it across the y-axis. k ( x) (x-y pairs: multiply x-values by -) g( x) ( x) x What do you notice about the two relections? ( x)

11 Odd unctions: relection across x-axis looks exactly like a relection across y-axis. Which o them are odd unctions? cosx sin x x

12 Test the ollowing unction to see i it is odd. x ( x ) Relect across x-axis (multiply parent by -) ( x) ( x) ( ) x Relect across y-axis (replace x with -x Power o a power property ( x) ( x) Relection across x-axis same as relection across y-axis odd

13 Your turn: Test the ollowing unction to see i it is odd. x ( x) x x ( x) Relection across x-axis not the same as relection across y-axis not odd

14 Analyzing Functions Graphically ( x ) What is the average rate o change between x = and? Means what is the slope o the graph between the two points? (, y ) and (, y) slope y m x

15 Analyzing Functions Graphically ( x ) What is the average rate o change between x = and? Means what is the slope o the graph between the two points (, ()) and (, ())? () (() ) () ( ) () (() ) () ( ) 5 slope y m x

16 Vocabulary A unction is increasing at a point i the slope o a tangent line at that point on the graph is positive A unction is decreasing at a point i the slope o a tangent line at that point on the graph is negative. This unction is increasing at (, ) This unction is decreasing at (, )

17 Analyzing Functions Graphically 9 Where is the unction increasing? ( x ) The slope o a tangent line at any point on the graph or the interval x = (, ) is positive (x) is increasing on x (, ) What about when x =? The slope o a tangent line at x = is zero (not increasing).

18 Analyzing Functions Graphically 9 Where is the unction decreasing? ( x ) The slope o a tangent line at any point on the graph or the interval x = (-, ) is negative (x) is decreasing on x (, ) What about when x =? The slope o a tangent line at x = is zero (not decreasing).

19 Analyzing Functions Graphically Vocabulary Extrema: a point on a graph whose tangent line has a slope o zero. Does the graph have any extrema? Yes. The slope o a tangent line at (, ) is zero (slope changes rom negative to positive at x = ). You can think o extrema an points on the graph that are peaks or valleys.

20 Extrema: a point on a graph whose tangent line has a zero slope. We classiy extrema by their y-values. Absolute minimum (maximum): an extrema whose y-value is the smallest (largest) y-value or the entire unction. relative maximum (minimum): an extrema whose y-value is the greater than (less than) the y-value o points near it. This graph has: One relative minimum, One absolute minimum One relative maximum

21 Analyzing Functions Graphically 9 Is the unction even? ( x ) means, is the graph symmetrical about the y-axis? NO

22 Analyzing Functions Graphically ( x ) How does it relate to its parent unction? 5 The graph o ( x ) is the parent unction translated to the right by

23 Analyzing Functions Graphically ( x ) What is the end behavior o the graph? Up on right, up on let As x gets bigger, y gets bigger. as x, y As x gets smaller, y gets bigger. as x, y We call this ininity notation.

24 Analyzing Functions Graphically 9 What is the domain o the graph? ( x ) Means what values o x are ound in the graph? Except or some special unctions, all values o x are used We say the domain is all real numbers.

25 Analyzing Functions Graphically 9 What is the range o the graph? ( x ) Means what values o y are ound in the graph? The smallest y-value o this graph is zero, and it goes upward rom there. We say the range is: y

26 Analyzing the graph. Where is the unction increasing?. Where is the unction decreasing?. Is the unction even?. Are there any extrema? I so, what type are they? 5. How does it relate to its parent unction? 6. What is the end behavior o the graph? as x, y? as x, y? 7. What is the domain o the graph? 8. What is the range o the graph? 9. What is the average rate o change between two given values o x?

27 . Where is the unction increasing? When. Is the unction even? x the graph is increasing.. Where is the unction decreasing? When the graph is decreasing. x NO (not symmetrical about the y-axis). Where and what type extrema are there? 5. How does it relate to its parent unction? Parent translated let, down 6. What is the end behavior o the graph? as x, y as x, y Absolute Minimum at (-, -) 7. What is the domain o the graph? All real numbers 8. What is the range o the graph? y