One way to graph functions is to simply plot points. In this section, we cover a method used to quickly sketch graphs related to some basic functions. Here we focus on

**rigid transformations**, that is, transformations that do not change the shape of the graph.

**Vertical Translations:** [

Interactive Graph ]

If

*k* is any positive real number then,

The graph of the basic function

*f*(

*x*) = sqrt(

*x*) follows:

Using this basic graph and the vertical translations described above we can sketch

*f*(

*x*) = sqrt(

*x*) + 2 by shifting all of the points up 2 units. Similarly, graph

*g*(

*x*) = sqrt(

*x*)

− 3 by shifting all points down 3 units.

If

*h* is any positive real number then,

Using the graph of

*f*(

*x*) = sqrt(

*x*) and the horizontal translations described above we can sketch

*f*(

*x*) = sqrt(

*x + *4) by shifting all of the points left 4 units. Similarly, graph

*g*(

*x*) = sqrt(

*x *− 3) by shifting all points right 3 units.

**Reflections:** [

Interactive Graph ]

Given any function

*f*(*x*),

Using the graph of

*f*(

*x*) = sqrt(

*x*) sketch the graph of

*f*(

*x*) = −sqrt(

*x*) by reflecting all of the points about the

*x*-axis. Similarly, graph

*f*(

*x*) = sqrt(−

*x*) by reflecting all of the points about the

*y*-axis.

For the first function *f*(*x*) = −sqrt(*x*) all of the *y*-values are negative which results in a reflection about the *x*-axis. For the second function *f*(*x*) = sqrt(−*x*) all of the *x*-values must be negative thus resulting in a reflection about the *y*-axis.

The -1 indicates a reflection of the graph of the squaring function *f*(*x*) = *x*^2 about the *x*-axis. Be sure to graph the squaring function using a dashed curve because it will be used as a guide and is not the answer. Next, reflect all points about the *x*-axis and draw in the final graph with a solid curve.

**General Steps for Graphing Functions using Transformations**:

2. Identify any reflections first and sketch them using the basic function as a guide.

3. Identify any translations.

4. Use this information to sketch the final graph using a solid curve.

**Graph the function and determine the domain and range**.

**Use the transformations to determine the equation that represents the given function.**
**Example**:

From the general shape of the graph we can determine the basic function and transformations. The graph has the shape of the square root function,

*y* = sqrt(*x*)

Next, notice the reflection about the

*y*-axis,

*y* = sqrt(−*x*)

And finally, we see a shift up 1 unit.

*y* = sqrt(−*x*) + 1

**Example**:

The given function has the general shape of the squaring function (parabola),

*y* = *x^*2

Next, notice the shift right 3 units,

*y* = (*x *− 3)*^*2

And finally, we see a shift down 2 units.

*y* = (*x *− 3)*^*2 − 2

**Example**:

The given function has the general shape of the absolute value function,

*y* = abs(*x*)

Next, notice the reflection about the

*x*-axis,

*y* = −abs(*x*)

Finally we see a shift left 1 unit and down 2 units

*y* = −abs(*x+*1) − 2

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