Functions

FunctionsOur objectives:

• Recognize “Parent Functions”– Graphically & Algebraically

– Please take notes and ALWAYS ask questions

Today’s Objectives:

Today’s Today’s AgendaAgenda

.

Homework:

The following basic graphs will be used extensively in this section. It is important to be able to sketch these from memory.

Constant Functionf(x) = a

Linear function f(x) = x

quadratic function

2)( xxf

3)( xxf

cubic function

Polynomial Function

• http://zonalandeducation.com/mmts/functionInstitute/polynomialFunctions/graphs/polynomialFunctionGraphs.html

• *zero degree• *first Degree• *second degree• *third degree• Fourth degree

Exponential Functionf(x) = a x

Logarithmic Functionf(x)=log xa

xxf )(

square root function

cube root function

3( )f x x

xxf )(absolute value function

Rational Functionf(x) =

)2)(1(

)2)(1( 2

xx

xx

Reciprocal Functionf(x) =

x

1

Inverse Function

Piece-wise Function

Piece-wise Function

•

We will now see how certain transformations (operations) of a

function change its graph. This will give us a better idea of how to

quickly sketch the graph of certain functions. The transformations are (1) translations, (2) reflections, and

(3) stretching.

Vertical Translation

Vertical TranslationFor b > 0,the graph of y = f(x) + b is the graph of y = f(x) shifted up b units;

the graph of y = f(x) b is the graph of y = f(x) shifted down b units.

2( )f x x 2( ) 3f x x

2( ) 2f x x

Horizontal Translation

Horizontal TranslationFor d > 0,

the graph of y = f(x d) is the graph of y = f(x) shifted right d units;

the graph of y = f(x + d) is the graph of y = f(x) shifted left d units.

2( )f x x

22y x 2

2y x

• Vertical shifts– Moves the graph up or

down– Impacts only the “y”

values of the function– No changes are made to

the “x” values

• Horizontal shifts– Moves the graph left or

right– Impacts only the “x”

values of the function– No changes are made to

the “y” values

The values that translate the graph of a function will occur as a number added or

subtracted either inside or outside a function.

Numbers added or subtracted inside translate left or right, while numbers

added or subtracted outside translate up or down.

( )y f x d b

Recognizing the shift from the equation, examples of shifting the function f(x) =

• Vertical shift of 3 units up

• Horizontal shift of 3 units left (HINT: x’s go the opposite direction that you might believe.)

3)(,)( 22 xxhxxf

22 )3()(,)( xxgxxf

2x

Points represented by (x , y) on the graph of f(x) become

, for the function ( )x d y b f x d b

)1,3(

If the point (6, -3) is on the graph of f(x),find the corresponding point on the graph of f(x+3) + 2

Combining a vertical & horizontal shift

• Example of function that is shifted down 4 units and right 6 units from the original function.

( ) 6

)

4

( ,

g x x

f x x

Reflections• The graph of f(x) is the reflection of the graph of

f(x) across the x-axis.

• The graph of f(x) is the reflection of the graph of f(x) across the y-axis.

• If a point (x, y) is on the graph of f(x), then (x, y) is on the graph of f(x), and • (x, y) is on the graph of f(x).

Reflecting

• Across x-axis (y becomes negative, -f(x))

• Across y-axis (x becomes negative, f(-x))

Vertical Stretching and ShrinkingThe graph of af(x) can be obtained from the graph of f(x) by

stretching vertically for |a| > 1, orshrinking vertically for 0 < |a| < 1.

For a < 0, the graph is also reflected across the x-axis.

(The y-coordinates of the graph of y = af(x) can be obtained by multiplying the y-coordinates of y = f(x) by a.)

VERTICAL STRETCH (SHRINK)

• y’s do what we think they should: If you see 3(f(x)), all y’s are MULTIPLIED by 3 (it’s now 3 times as high or low!)

2( ) 3 4f x x

2( ) 4f x x

21( ) 4

2f x x

Horizontal stretch & shrink• We’re MULTIPLYING by

an integer (not 1 or 0).• x’s do the opposite of

what we think they should. (If you see 3x in the equation where it used to be an x, you DIVIDE all x’s by 3, thus it’s compressed horizontally.)

2( ) (3 ) 4g x x

2( ) 4f x x

21( ) ( ) 4

3f x x