Objectives• Vertical Shifts Up and Down• Horizontal Shifts to the Right and Left• Reflections of Graphs• Stretching and Shrinking of Graphs

Laws for Graphing Shifts All graphs must be placed on graphing paper. We do not use ruled

paper for graphing. Graphs are to be drawn using a straight edge. Arrows are to be placed on each end of the x and y-axis. Label the x and y-axis. When graphing two functions on the same graph, use different

colored pencils or markers. Label points. Be sure the equation(s) is located on the graph. Keep everything color-coded. Use a straight-edge where needed. Straight lines must not be

sketched. Curved lines are sketched to the best of your ability. Be sure your graph is very neat and professional.

Parent Function

• What is a parent function?• A parent function is the most basic form of a

function. It has the same basic properties as other functions like it, but it has not been transformed in any way.

• Parent functions allow us to quickly tell certain traits of their “children” (which have been transformed).

Parent FunctionsThe six most commonly used parent functions are:

a) Constant Function

b) Linear (Identity) Function

c) Absolute Value Function

d) Square root Function

e) Quadratic Function

f) Cubic Function

Constant Function

f(x) = cwhere c is a constant

Characteristics: f(x)=a is a horizontal line. It is increasing and continuous on its entire domain (-∞, ∞).

𝒚=𝟐𝒚

𝒙

(−5 ,2) (2 , 2)

Linear Function(Identity)

f(x) = x

Characteristics: f(x)=x is increasing and continuous on its entire domain (-∞, ∞).𝒚=𝒙

𝒙

𝒚

(4 , 4)

(−5 ,− 5)

Absolute Value Function

f(x) = │x │

Characteristics:

is a

piecewise function. It decreases on the interval (-∞, 0) and increases on the interval (0, ∞). It is continuous on its entire domain (- ∞, ∞). The vertex of the function is (0, 0).

0 if

0 if )(

xx

xxxxf𝒚=¿ 𝒙∨¿

𝒙

𝒚

(4 , 4)(− 4 , 4)

f(x) = x

Square Root Function

Characteristics: f(x)=√x increases and is continuous on its entire domain [0, ∞). Note: x≥0 for f to be real.

𝒚=√𝒙

(0 , 0)(1 , 1)

(4 , 2)

𝒙

𝒚

Quadratic Functionf(x) = x 2

Characteristics: f(x)=x2 is continuous on its entire domain (-∞, ∞). It is increasing on the interval (0, ∞) and decreasing on the interval (-∞, 0). Its graph is called a parabola, and the point where it changes from decreasing to increasing, (0,0), is called the vertex of the graph.

𝒚=𝒙𝟐

Cubic Function

f(x) = x 3

Characteristics: f(x)=x3 increases and is continuous on its entire domain (-∞, ∞). The point at which the graph changes from “opening downward” to “opening upward” (the point (0,0)) is called the origin.)

(1 , 1)(−1 , −1)

(2 , 8)

(− 2 ,− 8)

𝒚=𝒙𝟑

Vertical Shifts

( )f x cTranslates the function vertically (+ up, - down)

The graph represents the equation f(x)

6

4

2

-2

-4

-6

-5 5

We add or subtract to the output of the function.

Vertical ShiftsGraph

x y

𝒚=𝒙𝟐

𝒚=𝒙𝟐+𝟐What is the shift of the graph

Vertical Shifts

y x 2

y x 2 1

y x 2 2

h(x) = |x| – 4

Example: Use the graph of f (x) = |x| to graph the functions g(x) = |x| + 3 and h(x) = |x| – 4.

f (x) = |x|

x

y

-4 4

4

-4

8

g(x) = |x| + 3

f (x)

f (x) + c

+c

f (x) – c-c

If c is a positive real number, the graph of f (x) + c is the graph of y = f (x) shifted upward c units.

Vertical Shifts

If c is a positive real number, the graph of f (x) – c is the graph of y = f(x) shifted downward c units.

x

y

Neatly graph your parent function (with a colored pencil). Plot some points for reference.

Shift each point from your parent function up c units (if c is positive) or down c units (if c is negative). Use a different colored pencil.

Graphing

Horizontal ShiftsStart with f(x)

( )f x c

Translates the function horizontally (+ left, - right)

6

4

2

-2

-4

-6

-5 5

Add or subtract to the input of the function.

The c is now in parentheses with x.

Horizontal ShiftsGraph

x y

𝒚=𝒙𝟐

𝒚=(𝒙+𝟑 )𝟐

What is the shift of the graph ?

𝒚

𝒙

Horizontal Shifts

y x 2

y (x 1)2

y (x 2)2

Horizontal Shifts

y x 2

y (x 1)2

y (x 2)2

f (x) = x3

h(x) = (x + 4)3

Example: Use the graph of f (x) = x3 to graph

g (x) = (x – 2)3 and h(x) = (x + 4)3 .

x

y

-4 4

4

g(x) = (x – 2)3

x

y

y = f (x) y = f (x – c)

+c

y = f (x + c)

-c

If c is a positive real number, then the graph of f (x – c) is the graph of y = f (x) shifted to the right c units.

Horizontal Shifts

If c is a positive real number, then the graph of f (x + c) is the graph of y = f (x) shifted to the left c units.

Neatly graph your parent function (with a colored pencil). Plot some points for reference.

Shift each point from your parent function to the left c units (if c is positive) or to the right c units (if c is negative). Use a different colored pencil.

Graphing

Horizontal shifts don’t make sense when looking at the equation. Everything is in parentheses. Think opposite direction in your shift.

Reflections about the x-axis

y x 2

What do we know about the x-coordinates in these two graphs?

They are the same.

(1 , 1)

(1 , −1)What do we know about the y-coordinates in these two graphs?

They are the opposite.

𝒚=𝒙𝟐

𝒚=−𝒙𝟐

When the right side of a function is multiplied by , the graph of the new function is the reflection about the x-axis of the graph of the function . The x-axis acts as a mirror of these two functions.

Reflections about the y-axis

When x is replaced with , the graph of the new function is the reflection about the y-axis.

The y-axis acts as a mirror of the two functions.

Use the graph of to obtain the

graph of

f x x

g x x

( )

( )

x f(x) g(x)

-4 ERROR 2

-1 ERROR 1

0 0 0

1 1 ERROR

4 2 ERROR

10 0 10

1

2

3

4

5

(9, 3)

(4, 2)

(1, 1)(-1, 1)

(-4, 2)

(-9, 3)

y xy x

What do we observe about the x-coordinates?

What do we observe about the y-coordinates?

Vertical Stretching of Graphs

x y x y

- 2 4 - 2 8

- 1 1 - 1 2

0 0 0 0

1 1 1 2

2 4 2 8

The x-coordinates stay the same.

The y-coordinates increase by a factor of 2. 2 is the constant c.

Our graph stretches vertically. That is, it moves away from the x-axis to become more vertical.

Vertical Shrinking of Graphs

x y x y

- 2 4 - 2 2

- 1 1 - 1 ½

0 0 0 0

1 1 1 ½

2 4 2 2

The x-coordinates stay the same.

The y-coordinates decrease by a factor of ½. ½ is the constant c.

Our graph shrinks vertically. That is, it moves more towards the x-axis to be less vertical.

Vertical Stretching and Shrinking

If c > 1 then the graph of y = c f (x) is the graph of y = f (x) stretched vertically by c. (Remember: c is the constant)

If 0 < c < 1 then the graph of y = c f (x) is the graph of y = f (x) shrunk vertically by c.

Example: y = 2x2 is the graph of y = x2

stretched vertically by a factor of 2.

– 4

x

y

4

4

y = x2

is the graph of y = x2

shrunk vertically by a factor of .

2

4

1xy

4

1

2

4

1xy

y = 2x2

Horizontal Stretching of Graphs

x y x y

- 2 2 - 2 1

- 1 1 - 1 ½

0 0 0 0

1 1 1 ½

2 2 2 1

The x-coordinates stay the same.

The y-coordinates decrease by a factor of ½, which is the same as .

Our graph stretches horizontally. When the constant is greater than zero and less than one, the graph is stretching horizontally.

½ is the constant, c.

Horizontal Shrinking of Graphs

x y x y

- 2 2 - 2 4

- 1 1 - 1 2

0 0 0 0

1 1 1 2

2 2 2 4

The x-coordinates stay the same.

The y-coordinates increase by a factor of 2, which is the same as .

Our graph shrinks horizontally. When the constant is greater than one, the graph is shrinking horizontally.

2 is the constant, c.

- 4

x

y

4

4

y = |x|

y = |2x|

Horizontal Stretching and Shrinking

If c > 1, the graph of y = f (cx) is the graph of y = f (x) shrunk horizontally by 1/c.

If 0 < c < 1, the graph of y = f (cx) is the graph of y = f (x) stretched horizontally by 1/c.

Example: y = |2x| is the graph of y = |x| shrunk horizontally by 1/2.

xy2

1 is the

graph of y = |x| stretched

horizontally by 2 .

xy2

1

OriginalFunction

NewFunction

How Points in Graph of become

points in new graph

VisualEffect

Shift up by units

Shift down by units

Stretch vertically by

Shrink vertically by

Flip over the x-axis. The x-axis acts as your mirror.

Flip over the y-axis. The y-axis acts as your mirror.

Graphing a New Function from the Original Function