Michael Bowen's VC Course Pages

# Tutorial

## Transformations of Functions

### Function Notation

In the discussion that follows, we will be writing expressions such as $f(x - 4)$ or $f(2x)$. We need to understand how these expressions differ in meaning from $f(x)$.

Suppose that $f(x) = {x^2} - 4x + \dfrac{5}{{{x^3}}}$, and that we would like to find the value of $f(x - 4)$. We first replace each occurrence of $x$ in the original function by an empty set of parentheses. This includes the $x$ inside the $f(x)$ expression on the left side of the equation, resulting in $$f\big( {\;\;\;} \big) = {\big( {\;\;\;} \big)^2} - 4\big( {\;\;\;} \big) + \frac{5}{{{{\big( {\;\;\;} \big)}^3}}}.$$ Note that exponents originally applied to the variable $x$ always appear outside the empty set of parentheses. Next, we fill each empty set of parentheses with an appropriate expression. We copy this expression from the inside of the parentheses located immediately adjacent to the name of the function. So if we wish to evaluate $f(x - 4)$, then each pair of empty parentheses will be "stuffed" with the expression $(x - 4).$ The initial result will be $$f\left( {x - 4} \right) = {\left( {x - 4} \right)^2} - 4\left( {x - 4} \right) + \frac{5}{{{{\left( {x - 4} \right)}^3}}}$$ Depending on the circumstances, this expression may be left as it is, or it may need to be simplified. Applying the FOIL technique or the distributive property as needed, the first step toward simplification would be $$f\left( {x - 4} \right) = {x^2} - 8x + 16 - 4x + 16 + \frac{5}{{{x^3} - 12{x^2} + 48x - 64}},$$ and the final result, after combining like terms, would be $$f\left( {x - 4} \right) = {x^2} - 12x + 32 + \frac{5}{{{x^3} - 12{x^2} + 48x - 64}}.$$

The table below shows examples of values of some function expressions for specific functions. Where two expressions are given, the first is the original one obtained using the $f\big( {\;\;\;} \big)$ procedure described above, and the second is the same result after it has been simplified.

Overview: This table describes the specific expressions derived from generic function expressions for several common types of functions. It contains eight columns. First row: Column headers. Second and subsequent rows: Generic and specific function expressions. Column one of these rows contains a generic function expression. Columns two through eight of these rows contain the corresponding specific function expressions.
$f\big( \;\;\; \big)$ $f(x) = 3x$
(odd function)
$f(x) = {x^2}$
(even function)
$f(x) = {x^3}$
(odd function)
$f(x) = \lvert {x} \rvert$
(even function)
$f(x) = \sqrt x$
(no symmetry)
$f(x) = \sqrt[3]{x}$
(odd function)
$f(x) = \frac{1}{x}$
(odd function)
$f(x - 4)$ $3\left( {x - 4} \right)$ ${\left( {x - 4} \right)^2}$ ${\left( {x - 4} \right)^3}$ $\lvert {x - 4} \rvert$ $\sqrt {x - 4}$ $\sqrt[3]{{x - 4}}$ $\frac{1}{{x - 4}}$
$f(x) - 4$ $3x - 4$ ${x^2} - 4$ ${x^3} - 4$ $\lvert x \rvert - 4$ $\sqrt x - 4$ $\sqrt[3]{x} - 4$ $\frac{1}{x} - 4$
$- f(x)$ $- 3x$ $- {x^2}$ $- {x^3}$ $- \lvert x \rvert$ $- \sqrt x$ $- \sqrt[3]{x}$ $- \frac{1}{x}$
$f( - x)$ $\begin{split} 3\big( { - x} \big) \\ = - 3x\end{split}$ $\begin{split} \big( { - x} \big)^2 \\ = x^2\end{split}$ $\begin{split} \big( { - x} \big)^3 \\ = - x^3\end{split}$ $\begin{split} \lvert { - x} \rvert \\ = \lvert x \rvert\end{split}$ $\sqrt { - x}$ $\begin{split} \sqrt[3]{ - x} \\ = - \sqrt[3]{x}\end{split}$ $\begin{split} \tfrac{1}{\big( { - x} \big)} \\ = - \tfrac{1}{x}\end{split}$
Note (previous two rows) that for odd functions, $f( - x) = - f(x)$, and for even functions, $f( - x) = f(x)$.
$4f(x)$ $\begin{split} 4\left( {3x} \right) \\ =12x\end{split}$ $4{x^2}$ $4{x^3}$ $4\lvert x \rvert$ $4\sqrt x$ $4\sqrt[3]{x}$ $\frac{4}{x}$
$f(4x)$ $\begin{split} 3\left( {4x} \right) \\ =12x\end{split}$ $\begin{split} \left( {4x} \right)^2 \\ = 16x^2\end{split}$ $\begin{split} \left( {4x} \right)^3 \\ = 64x^3\end{split}$ $\begin{split} \lvert {4x} \rvert \\ = 4\lvert x \rvert\end{split}$ $\begin{split} \sqrt {4x} \\ = 2 \sqrt{x}\end{split}$ $\sqrt[3]{{4x}}$ $\frac{1}{{4x}}$

### Introduction to Transformations

Transformations of functions allow one to obtain the graph of a function quickly and efficiently if it can be identified as being related to a basic function whose graph is well known. This is because there is a basic visual resemblance between the graph of a basic function and those of its transformations. Although there are many types of transformations, we shall concern ourselves with three basic types: shifts, also known as translations; reflections; and deformations.

#### Shifts (Translations)

Shifts involve moving the graph of a basic function vertically, horizontally, or both. Shifts change the position of a graph, but not its orientation, shape (proportions), or size.

#### Reflections

Reflections involve creating a mirror image of a function's graph about the $x$- or $y$-axes, or both. Reflections change the graph's orientation (except in certain cases when the original graph has a symmetry property; see the next paragraph). They often (though not always) result in a change of the graph's position. However, reflections never change the shape (proportions) or size of a graph.

Due to the symmetry of even functions about the $y$-axis, if you reflect an even function about the $y$-axis, you will not be able to tell that a change occurred (the graph will look exactly the same). Similarly, if a curve has symmetry about the $x$-axis, reflecting it about the $x$-axis will not result in a visible change. However, since curves with $x$-axis symmetry necessarily fail the vertical line test, they are not functions, so most textbooks ignore them, even though the transformation rules still apply to such curves.

#### Deformations

Deformations involve the horizontal or vertical stretching or shrinking (compression) of a graph. Deformations do not change a graph's position or orientation; rather, they change the graph's aspect ratio and thus its shape, causing it to appear shorter, taller, narrower, or wider than the graph of the basic function to which it is related.

#### Multiple Transformations

More complex transformations may be achieved by performing two or more of the above transformations in sequence. Therefore, they are often referred to as "multiple transformations" or "sequential transformations".

A small set of relatively simple rules governs how certain modifications to the definitions of basic functions are expressed visually when the modified functions are graphed. These rules are clarified in the following discussion, which more closely examines each type of transformation described above. The rules work in a consistent manner when applied to the definition of any function whose domain and range consist of the set of real numbers, or any subset thereof. They can be even used on graphs of curves that are not functions (for example, circles; these can be graphed and transformed, but the graphs do not pass the vertical line test).

### Shifts or Translations

The effect of a simple shift or translation is to cause the graph of the translated function to move vertically or horizontally relative to the graph of the basic function to which it is related. For example, the graph of the basic function $$f(x) = {x^2}$$ is an upward-opening parabola whose vertex lies at the origin (the point $(0,0)$) of a Cartesian coordinate system. To move this graph vertically upward by $3$ units (so that the vertex lies at $(0,3)$) without rotating or deforming it, we change the function to $$g(x) = {x^2} + 3 = f(x) + 3$$ (We use the $g(x)$ notation for this shifted graph to distinguish it from the original $f(x)$ function.) To move this graph horizontally rightward by $4$ units (so that the vertex lies at $(4,0)$), we change the function to $$h(x) = {(x - 4)^2} = f(x - 4)$$ In the following instructions, we assume that $f(x)$ represents the original function, and $g(x)$ represents the shifted or translated function.

#### Recipe for vertical shifts

1. If the equation is written using function ($f(x)$) notation, replace "$f(x)$" with "$y$".
1. To translate the graph of this function vertically upward by $c$ units, replace "$y$" with "$(y - c)$".
2. To translate a function vertically downward, use a negative number for $c$, then follow the procedure for an upward shift. (For example, to move the graph $2$ units downward, use $c = - 2$.)
2. After making the substitution, solve the resulting expression for $y$ (that is, isolate the term $y$ on one side of the resulting equation).
3. Finally, replace "$y$" with "$g(x)$".
##### Example:

Find the function $g(x)$ that moves the graph of the square root function $f(x) = \sqrt x$ vertically downward by $4$ units.

1. Substitute "$y$" for "$f(x)$": $$y = \sqrt x$$
2. Replace "$y$" with "$(y - c)$": $$y - c = \sqrt x$$ Since the shift is downward, set $c = - 4$ and substitute: $$y - ( - 4) = \sqrt x$$
3. Simplify and solve for $y$: $$y + 4 = \sqrt x \to y = \sqrt x - 4$$
4. Replace "$y$" with "$g(x)$": $$g(x) = \sqrt x - 4$$ This is the equation of the shifted function. Since $f(x) = \sqrt x$, we may also write this in the form $$g(x) = f(x) - 4$$ which shows how this downward shift may be written generically in terms of the original and shifted functions $f(x)$ and $g(x)$.

#### Recipe for horizontal shifts

1. If the equation is written using function ($f(x)$) notation, replace "$f(x)$" with "$y$".
1. To translate the graph of this function horizontally rightward by $c$ units, replace "$x$" with "$(x - c)$".
2. To translate a function horizontally leftward, use a negative number for $c$, then follow the procedure for a rightward shift. (For example, to move the graph $2$ units leftward, use $c = - 2$.)
2. If necessary, simplify the resulting equation.
3. Finally, replace "$y$" with "$g(x)$".
##### Example:
Find the function $g(x)$ that moves the graph of the square root function $f(x) = \sqrt x$ horizontally leftward by $4$ units.
1. Substitute "$y$" for "$f(x)$": $$y = \sqrt x$$
2. Replace "$x$" with "$(x - c)$": $$y = \sqrt {x - c}$$ Since the shift is leftward, set $c = - 4$, then substitute and simplify: $$y = \sqrt {x - ( - 4)}$$
3. Simplify: $$y = \sqrt {x + 4}$$
4. Replace "$y$" with "$g(x)$": $$g(x) = \sqrt {x + 4}$$ This is the equation of the shifted function. Since $f(x) = \sqrt x$, we may also write this in the form $$g(x) = f(x + 4)$$ which shows how this leftward shift may be written generically in terms of the original and shifted functions $f(x)$ and $g(x)$.

#### The Bottom Line on Shifts:

• To translate the graph of a function vertically by $c$ units, replace "$y$" with "$(y - c)$"; use positive $c$ for upward shifts and negative $c$ for downward shifts. If $f(x)$ is the original function and $g(x)$ is the shifted function, then $g(x) = f(x) + c$.
• To translate the graph of a function horizontally by $c$ units, replace "$x$" with "$(x - c)$"; use positive $c$ for rightward shifts and negative $c$ for leftward shifts. If $f(x)$ is the original function and $g(x)$ is the shifted function, then $g(x) = f(x - c)$.
• If necessary, solve for $y$ or $g(x)$ when finished.

Additional examples of shifts for a generic function, including a combination of two shifts, are depicted in the following diagram. Pay attention to the fact that each shifted graph has exactly the same shape and size as the original, but a different position. Note: when shifting a graph horizontally or vertically, it is easiest to pick specific dots on the original graph, shift each dot by the same number of grid squares, then re-create the shifted graph by connecting the shifted dots.

### Reflections

The effect of a reflection is to cause the graph of the reflected function to flip (rotate) either vertically or horizontally relative to the graph of the basic function to which it is related. The flip usually also results in a displacement of the graph across one axis, unless the original function was symmetric about the $y$-axis and the rotation also occurs about this axis. For example, the graph of the basic function $$f(x) = \sqrt x$$ is an increasing function whose graph (in Cartesian coordinates) lies entirely in the first quadrant, with the exception of a single point that coincides with the origin. To reflect this graph about the $y$-axis without deforming it so that the reflection lies in the second quadrant, we change the function to $$g(x) = \sqrt { - x}$$ To reflect this graph about the $x$-axis without deforming it so that the reflection lies in the fourth quadrant, we change the function to $$h(x) = - \sqrt x$$

Note that both types of reflections involve the insertion of a negative sign into the function. There are subtle distinctions between the locations of the insertions that distinguish between a reflection about the $y$-axis and a reflection about the $x$-axis. We discuss these distinctions in more detail below.

As in the instructions for translations, we assume that $f(x)$ represents the original function, and $g(x)$ represents the reflected function.

#### Recipe for reflections about the $y$-axis

1. If the equation is written using function ($f(x)$) notation, replace "$f(x)$" with "$y$".
2. To reflect the graph of this function about the $y$-axis, replace "$x$" with "$( - x)$" everywhere it occurs in the equation, then simplify the result, being sure to keep $y$ isolated.
3. Finally, replace "$y$" with "$g(x)$".
##### Example:
Find the function $g(x)$ that reflects the graph of the quadratic function $f(x) = {x^2}$ about the $y$-axis.
1. Substitute "$y$" for "$f(x)$": $$y = {x^2}$$
2. Replace "$x$" with "$( - x)$", then simplify: $$y = {( - x)^2} \to y = {x^2}$$
3. Replace "$y$" with "$g(x)$": $$g(x) = {x^2}$$ This is the equation of the reflected function. In this case, the original function was symmetric about the $y$-axis, so the reflected function $g(x)$ is the same as the original function $f(x)$. This will always happen if (and only if) $f(x)$ is an even function. The generic relation between any function $f(x)$ and its reflection about the $y$-axis $g(x)$ may be written $$g(x) = f( - x)$$

#### Recipe for reflections about the $x$-axis

1. If the equation is written using function ($f(x)$) notation, replace "$f(x)$" with "$y$".
2. To reflect the graph of this function about the $x$-axis, replace "$y$" with "$( - y)$", then solve the resulting equation for $y$, usually by multiplying through both sides of the equation by $- 1$.
3. Finally, replace "$y$" with "$g(x)$".
##### Example:
Find the function $g(x)$ that reflects the graph of the quadratic function $f(x) = {x^2}$ about the $x$-axis.
1. Substitute "$y$" for "$f(x)$": $$y = {x^2}$$
2. Replace "$y$" with "$( - y)$", then isolate $y$: $$- y = {x^2} \to y = - {x^2}$$
3. Replace "$y$" with "$g(x)$": $$g(x) = - {x^2}$$ This is the equation of the reflected function. The generic relation between any function $f(x)$ and its reflection about the $x$-axis $g(x)$ may be written $$g(x) = - f(x)$$

#### The Bottom Line on Reflections:

• To reflect the graph of a function about the $\boldsymbol y$-axis, replace "$x$" with "$( - x)$". If $f(x)$ is the original function and $g(x)$ is the reflected function, then $g(x) = f(- x)$. Even functions are unchanged by this type of reflection.
• To reflect the graph of a function about the $\boldsymbol x$-axis, replace "$y$" with "$( - y)$". If $f(x)$ is the original function and $g(x)$ is the reflected function, then $g(x) = - f(x)$.
• If necessary, solve for $y$ or $g(x)$ when finished.

Additional examples of reflections for a generic function, including a combination of two reflections, are depicted in the following diagram. Pay attention to the fact that each reflected graph is the same size as the original, but has a different orientation. Note: when reflecting a graph about either axis, it is easiest to pick specific dots on the original graph, reflect each dot so it appears the same number of grid squares from the reflection axis as the original dot (but on the opposite side of the reflection axis from the original), then re-create the reflected graph by connecting the reflected dots.

### Deformations

The effect of a deformation is to cause the graph of the deformed function to expand (stretch) or contract (shrink), either vertically or horizontally, relative to the graph of the basic function to which it is related. The deformation results in a change to the function's aspect ratio, causing the visual appearance of the deformed graph to become "stretched" or "squashed". The effect is similar to what one might see when viewing one's reflection in a curved mirror, such as might be found in a carnival "fun house".

To deform a function's graph, we multiply either $x$ or $y$ by a positive number $c$ whose value is other than $1$. (In principle, a negative number could also be used, but this would introduce a reflection in addition to the deformation; see above.) The type of deformation obtained (e.g., shrink or stretch) depends on the value of $c$ and on whether it multiplies $x$ or $y$.

The next two diagrams illustrate the effects of deformations. The first is devoted to vertical and horizontal stretches; the second to vertical and horizontal shrinks. Both diagrams employ the curve $$f(x) = {x^3}$$ as the basic function from which to derive all the deformations. To demonstrate how each deformation affects the domain or range of the derived functions, we limit the domain of $f(x)$ to the closed interval $[ - 2,2]$. In addition, we specify the multiplicative constant $c$ to have a value of either $2$ or $\frac{1}{2}$, which makes it easy to visualize the effects of each deformation on the functions' graphs; each stretch depicted expands the graph by a factor of $2$, and each shrink depicted contracts the graph by the same factor (since multiplying by $\frac{1}{2}$ is equivalent to dividing by $2$).

#### Stretches

The diagram appearing below will serve as the basis for the discussion of stretch deformations that follows.

##### Basic function

The basic function $$f(x) = {x^3}$$ limited to the domain $[ - 2,2]$, is depicted as the yellow trace above.

##### Vertical stretch

The light blue trace appearing in the above diagram $$g(x) = 2{x^3}$$ represents a vertical stretch deformation of this function by a factor of $c = 2$.

For a vertical stretch, $c$ represents the factor by which the vertical aspect of the graph is expanded. Also note that for any vertical stretch, the domains of both the original and stretched functions are the same. However, the range of the stretched function is different from that of the original function, since the stretched function's graph extends to larger (more positive) $y$ values at its upper end, and smaller (more negative) $y$ values at its lower end, compared to the basic function.

The form of the equation for a generic vertical stretch of any function is $$g(x) = cf(x)$$ where $c > 1$.

##### Horizontal stretch

The dark blue trace appearing in the above diagram $$h(x) = \frac{1}{8}{x^3}$$ represents a horizontal stretch deformation of this function, again by a factor of $2$. However, in this case, $c = \frac{1}{2}$.

For a horizontal stretch, $c$ represents the reciprocal of the factor by which the horizontal aspect of the graph is expanded. Also note that for any horizontal stretch, the ranges of both the original and stretched functions are the same. However, the domain of the stretched function is different from that of the original function, since the stretched function's graph extends to larger (more positive) $x$ values at its right end, and smaller (more negative) $x$ values at its left end, compared to the basic function.

The form of the equation for a generic horizontal stretch of any function is $$h(x) = f(cx)$$ where $0 < c < 1$.

In the specific case depicted here, we obtain the equation of the horizontally stretched function $h(x)$ discussed above from this generic form as follows: $$h(x) = f(cx) = f\left( {\frac{1}{2}x} \right) = {\left( {\frac{1}{2}x} \right)^3} = {\left( {\frac{1}{2}} \right)^3}{x^3} = \frac{1}{8}{x^3}$$

##### Combined stretch

The pink trace appearing in the above diagram $$k(x) = \frac{1}{4}{x^3}$$ represents a combination of horizontal and vertical stretches of this function, each by a factor of 2. The domain of this function matches the domain of the horizontally stretched function (dark blue trace), and the range of this function matches the range of the vertically stretched function (light blue trace).

#### Shrinks

The diagram appearing below will serve as the basis for the discussion of shrink deformations that follows.

##### Basic function

The basic function $$f(x) = {x^3}$$ limited to the domain $[ - 2,2]$, is depicted as the yellow trace above. It is identical to the basic function employed in the description of vertical and horizontal stretches that appears above, and the graph uses the same scale as in the preceding diagram.

##### Vertical shrink

The light blue trace appearing in the above diagram $$g(x) = \frac{1}{2}{x^3}$$ represents a vertical shrink deformation of this function by a factor of $c = \frac{1}{2}$. As was true for a vertical stretch, $c$ represents the factor by which the vertical aspect of the graph is changed (contracted if $0 < c < 1$, or expanded if $c > 1$). As was also true for a vertical stretch, the domains of both the original and deformed functions are the same, but the range of the deformed function is different from that of the original function, since the shrunken function's graph extends to smaller (less positive) $y$ values at its upper end, and larger (less negative) $y$ values at its lower end, compared to the basic function.

The form of the equation for a generic vertical shrink of any function is $$g(x) = cf(x)$$ where $0 < c < 1$.

This functional form is the same as for a vertical stretch; the only difference is that the multiplicative constant $c$ takes on a different (larger) set of values for a vertical stretch than for a vertical shrink.

##### Horizontal shrink

The dark blue trace appearing in the above diagram $$h(x) = 8{x^3}$$ represents a horizontal shrink deformation of this function, again by a factor of $\frac{1}{2}$. However, in this case, $c = 2$. For a horizontal shrink, $c$ represents the reciprocal of the factor by which the horizontal aspect of the graph is contracted. As for the horizontal stretch, the ranges of both the original and shrunken functions are the same. However, the domain of the shrunken function is different from that of the original function, since the shrunken function's graph extends to smaller (less positive) $x$ values at its right end, and larger (less negative) $x$ values at its left end, compared to the basic function.

The form of the equation for a generic horizontal shrink of any function is $$h(x) = f(cx)$$ where $c > 1$.

In the specific case depicted here, we obtain the equation of the horizontally compressed function $h(x)$ discussed above from this generic form as follows: $$h(x) = f(cx) = f(2x) = {(2x)^3} = {2^3}{x^3} = 8{x^3}$$

##### Combined shrink

The pink trace appearing in the above diagram $$k(x) = 4{x^3}$$ represents a combination of horizontal and vertical shrinks of this function, each by a factor of $\frac{1}{2}$. The domain of this function matches the domain of the horizontally shrunken function (dark blue trace), and the range of this function matches the range of the vertically shrunken function (light blue trace).

#### Is a vertical stretch the same as a horizontal shrink (and vice versa)?

In many cases, the graph obtained by shrinking a function horizontally is identical to the graph obtained by stretching it vertically. For example, if the domain is taken as $( - \infty ,\infty )$, a vertical stretch of the basic function $f(x) = {x^3}$ by a factor of $8$ gives rise to the deformed function $g(x) = 8{x^3}$, which is the same as the function obtained by shrinking the basic function horizontally by a factor of $\frac{1}{2}$. However, there is more to defining a function than merely stating its equation; we must also specify its domain (either explicitly or implicitly). As we can see by re-examining the above examples, the domain of a function that is obtained by a vertical stretch or shrink is always the same as the domain of the basic function from which it is derived (although the range may be different). However, the domain of a function that is obtained by a horizontal stretch or shrink is often not the same as the domain of the basic function (although the range will always be the same).

As an example, consider a basic function defined as $F(x) = {x^3}$, but with a domain of $[ - 2,2]$. A vertical stretch by a factor of $8$ would yield the deformed function $$G(x) = 8{x^3}{\text{ with domain }}[ - 2,2]$$ whereas a horizontal shrink by a factor of $\frac{1}{2}$ would yield the deformed function $$H(x) = 8{x^3}{\text{ with domain }}[ - 1,1]$$ which has the same equation as $G(x)$, but a different domain. Therefore, $G(x)$ and $H(x)$ are technically different functions, because they have different domains, even though they share the same equation. If you are having trouble understanding why $G(x)$ and $H(x)$ are different, construct a graph of each one. If you carefully limit the domains as described above, you will see that the graphs are not the same.

#### The Bottom Line on Deformations:

• To stretch the graph of a function vertically by a positive factor of $c$, replace "$y$" with "$\left(\tfrac{1}{c}\;y\right)$"; if $\lvert c \rvert < 1$, the result is a vertical compression rather than a vertical stretch. If $f(x)$ is the original function and $g(x)$ is the deformed function, then $g(x) = c \cdot f(x)$, regardless of whether the transformation is a vertical stretch or a vertical compression.
• To stretch the graph of a function horizontally by a positive factor of $c$, replace "$x$" with "$\left(\tfrac{1}{c}\;x\right)$"; if $\lvert c \rvert < 1$, the result is a horizontal compression rather than a horizontal stretch. If $f(x)$ is the original function and $g(x)$ is the deformed function, then $g(x) = f \left(\tfrac{1}{c}\;x\right)$, regardless of whether the transformation is a horizontal stretch or a horizontal compression.
• If $c$ is negative, you are attempting to combine a deformation with a reflection. Try to separate the $-1$ from the $\lvert c \rvert$ so that you are not doing both at once.
• If necessary, solve for $y$ or $g(x)$ when finished.

### Multiple Transformations

More complicated transformations may be carried out by combining two or more of the simple transformations described above. With multiple transformations, the order in which they are carried out is usually important.

#### Order of Operations for Transformations

If you examine the function to be graphed, transformations from the basic underlying function must be carried out in the same order as dictated by standard order of operations.

##### Example:

Suppose it is desired to draw the graph of $$g(x) = -5\left( -4x + 8 \right)^2 - 3.$$ The basic function underlying this curve is the parabola $f(x) = x^2$, so the transformed curve will also be a parabola. The transformations needed to convert $f(x)$ into $g(x)$ include two shifts (horizontal and vertical), two reflections (about the $x$-axis and about the $y$-axis), and two deformations. When the variable $x$ presents with a coefficient, as in this case, the necessary transformations are easier to see and understand if we factor out this coefficient from the interior of the expression. The rewritten version of $g(x)$ would be $$g(x) = -5\left[ -4 \left( x - 2 \right) \right]^2 - 3.$$

If we were to evaluate this function (say, to calculate $g(1)$ by plugging (1) in for $x$), the necessary calculations (think in terms of baby steps here, and follow order of operations strictly) would be:

1. Subtract $2$ from $x$ (since this operation is in the innermost parentheses, it has the highest order-of-operations priority).
2. Multiply the previous result by $4$ (don't worry about the negative yet).
3. Multiply the previous result by $-1$ (changing the $4$ into $-4$; you'll see in a minute why this part is separate).
4. The inside of the square brackets are now simplifed to a single number, so square the previous result (here is where the basic $f(x) = x^2$ function is applied).
5. Multiply the previous result by $5$ (don't worry about the negative yet).
6. Multiply the previous result by $-1$ (changing the $5$ into $-5$; again, you'll see in a minute why this part is separate).
7. Subtract $3$ from the previous result (to obtain the final value of the function).

In terms of transformations, these same steps correspond respectively to:

1. Shifting the graph horizontally $2$ squares to the right (since changing $x$ to $x-2$ represents a horizontal shift).
2. Shrinking the graph horizontally by a factor of $4$ (since multiplying $x$ by a factor greater than $1$ represents a horizontal compression).
3. Reflecting the graph about the $y$-axis (since multiplying $x$ by $-1$ represents reflection about the $y$-axis; separating the compression in the previous step from the reflection in this step is why we multiplied the $4$ separately from the $-1$ above).
4. No transformation; this is where the basic underlying function plays its role in converting $x$ to $y$ (as a general rule, no graphical transformation is associated with the step that involves evaluation of the basic function).
5. Stretching the graph vertically by a factor of $5$.
6. Reflecting the graph about the $x$-axis (since multiplying $y$ by a factor greater than $1$ represents a vertical stretch; separating the stretch in the previous step from the reflection in this step is why we multiplied the $5$ separately from the $-1$ above).
7. Shifting the graph vertically $3$ squares down (since changing $y$ to $y-3$ represents a vertical shift).

The numbered list presented just above therefore clearly indicates the specific order in which the transformations must be carried out. Any deviation from this order is likely to result in an erroneous graph.

##### Graphed Examples:

The legend in the diagram below documents the types of transformations that were combined to achieve each graph depicted on the coordinate grid. The generic functional forms are also provided in the legend.

Michael Bowen's VC Course Pages: Tutorial, Transformations of Functions