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Your image is being transformed by the mirror – more precisely, it's being reflected. Transformations like this happen every day and every morning in our world, as well as in the much less chaotic and confusing world of Calculus.
Throughout calculus, you will be asked to transform and translate functions. What does this mean, exactly? It means taking one function and applying changes to it to create a new function. This is how graphs of functions can be transformed into different ones to represent different functions!
In this article, you will explore function transformations, their rules, some common mistakes, and cover plenty of examples!
It'd be a good idea to have a good grasp of the general concepts of various types of functions before taking a dive into this article: make sure to first read the article on Functions!
- Function transformations: meaning
- Function transformations: rules
- Function transformations: common mistakes
- Function transformations: order of operations
- Function transformations: transformations of a point
- Function transformations: examples
Function Transformations: Meaning
So, what are function transformations? So far, you have learned about parent functions and how their function families share a similar shape. You can further your knowledge by learning how to transform functions.
Function transformations are the processes used on an existing function and its graph to give you a modified version of that function and its graph that has a similar shape to the original function.
When transforming a function, you should usually refer to the parent function to describe the transformations performed. However, depending on the situation, you might want to refer to the original function that was given to describe the changes.
Fig. 1.
Function Transformations: Rules
As illustrated by the image above, function transformations come in various forms and affect the graphs in different ways. That being said, we can break down the transformations into two major categories:
Horizontal transformations
Vertical transformations
Any function can be transformed, horizontally and/or vertically, via four main types of transformations:
Horizontal and vertical shifts (or translations)
Horizontal and vertical shrinks (or compressions)
Horizontal and vertical stretches
Horizontal and vertical reflections
Horizontal transformations only change the \(x\)-coordinates of functions. Vertical transformations only change the \(y\)-coordinates of functions.
Function Transformations: Rules Breakdown
You can use a table to summarize the different transformations and their corresponding effects on the graph of a function.
| Transformation of \( f(x) \), where \( c > 0 \) | Effect on the graph of \( f(x) \) |
| \( f(x)+c \) | Vertical shift up by \(c\) units |
| \( f(x)-c \) | Vertical shift down by \(c\) units |
| \( f(x+c) \) | Horizontal shift left by \(c\) units |
| \( f(x-c) \) | Horizontal shift right by \(c\) units |
| \( c \left( f(x) \right) \) | Vertical stretch by \(c\) units, if \( c > 1 \)Vertical shrink by \(c\) units, if \( 0 < c < 1 \) |
| \( f(cx) \) | Horizontal stretch by \(c\) units, if \( 0 < c < 1 \)Horizontal shrink by \(c\) units, if \( c > 1 \) |
| \( -f(x) \) | Vertical reflection (over the \(\bf{x}\)-axis) |
| \( f(-x) \) | Horizontal reflection (over the \(\bf{y}\)-axis) |
Horizontal Transformations – Example
Horizontal transformations are made when you act on a function's input variable (usually \(x\)). You can
add or subtract a number from the function's input variable, or
multiply the function's input variable by a number.
Here is a summary of how horizontal transformations work:
Shifts – Adding a number to \(x\) shifts the function to the left; subtracting shifts it to the right.
Shrinks – Multiplying \(x\) by a number whose magnitude is greater than \(1\) shrinks the function horizontally.
Stretches – Multiplying \(x\) by a number whose magnitude is less than \(1\) stretches the function horizontally.
Reflections – Multiplying \(x\) by \(-1\) reflects the function horizontally (over the \(y\)-axis).
Horizontal transformations, except reflection, work the opposite way you'd expect them to!
Consider the parent function from the image above:
\[ f(x) = x^{2} \]
This is the parent function of a parabola. Now, say you want to transform this function by:
- Shifting it to the left by \(5\) units
- Shrinking it horizontally by a factor of \(2\)
- Reflecting it over the \(y\)-axis
How can you do that?
Solution:
- Graph the parent function.
Fig. 2. A graph of the parent function of a parabola.
- Write the transformed function.
- Start with the parent function:
- \( f_{0}(x) = x^{2} \)
- Add in the shift to the left by \(5\) units by putting parentheses around the input variable, \(x\), and putting \(+5\) within those parentheses after the \(x\):
- \( f_{1}(x) = f_{0}(x+5) = \left( x+5 \right)^{2} \)
- Next, multiply the \(x\) by \(2\) to shrink it horizontally:
- \( f_{2}(x) = f_{1}(2x) = \left( 2x+5 \right)^{2} \)
- Finally, to reflect over the \(y\)-axis, multiply \(x\) by \(-1\):
- \( f_{3}(x) = f_{2}(-x) = \left( -2x+5 \right)^{2} \)
- So, your final transformed function is:
- \( \bf{ f(x) } = \bf{ \left( -2x + 5 \right)^{2} } \)
- Start with the parent function:
- Graph the transformed function, and compare it to the parent to make sure the transformations make sense.
Fig. 3. The graphs of the parent function of a parabola (blue) and its transformation (green).- Things to note here:
- The transformed function is on the right due to the \(y\)-axis reflection performed after the shift.
- The transformed function is shifted by \(2.5\) instead of \(5\) due to the shrinking by a factor of \(2\).
Vertical Transformations – Example
Vertical transformations are made when you act on the entire function. You can either
add or subtract a number from the entire function, or
multiply the entire function by a number.
Unlike horizontal transformations, vertical transformations work the way you expect them to (yay!). Here is a summary of how vertical transformations work:
Shifts – Adding a number to the entire function shifts it up; subtracting shifts it down.
Shrinks – Multiplying the entire function by a number whose magnitude is less than \(1\) shrinks the function.
Stretches – Multiplying the entire function by a number whose magnitude is greater than \(1\) stretches the function.
Reflections – Multiplying the entire function by \(-1\) reflects it vertically (over the \(x\)-axis).
Again, consider the parent function:
\[ f(x) = x^{2} \]
Now, say you want to transform this function by
- Shifting it up by \(5\) units
- Shrinking it vertically by a factor of \(2\)
- Reflecting it over the \(x\)-axis
How can you do that?
Solution:
- Graph the parent function.
Fig. 4. A graph of the parent function of a parabola.
- Write the transformed function.
- Start with the parent function:
- \( f_{0}(x) = x^{2} \)
- Add in the shift up by \(5\) units by putting \(+5\) after \( x^{2} \):
- \( f_{1}(x) = f_{0}(x) + 5 = x^{2} + 5 \)
- Next, multiply the function by \( \frac{1}{2} \) to compress it vertically by a factor of \(2\):
- \( f_{2}(x) = \frac{1}{2} \left( f_{1}(x) \right) = \frac{x^{2}+5}{2} \)
- Finally, to reflect over the \(x\)-axis, multiply the function by \(-1\):
- \( f_{3}(x) = -f_{2}(x) = - \frac{x^{2}+5}{2} \)
- So, your final transformed function is:
- \( \bf{ f(x) } = \bf{ - \frac{x^{2}+5}{2} } \)
- Start with the parent function:
- Graph the transformed function, and compare it to the parent to make sure the transformations make sense.
Fig. 5. The graphs of a parent function of a parabola (blue) and its transformation (green).
Function Transformations: Common Mistakes
It is tempting to think that the horizontal transformation of adding to the independent variable, \(x\), moves the function's graph to the right because you think of adding as moving to the right on a number line. This, however, is not the case.
Remember, horizontal transformations move the graph the opposite way you expect them to!
Let's say you have the function, \( f(x) \), and its transformation, \( f(x+3) \). How does the \(+3\) move the graph of \( f(x) \)?
Solution:
- This is a horizontal transformation because the addition is applied to the independent variable, \(x\).
- Therefore, you know that the graph moves opposite to what you'd expect.
- The graph of \( f(x) \) is moved to the left by 3 units.
Why are Horizontal Transformations the Opposite of what is Expected?
If horizontal transforms are still a bit confusing, consider this.
Look at the function, \( f(x) \), and its transformation, \( f(x+3) \), again and think about the point on the graph of \( f(x) \) where \( x = 0 \). So, you have \( f(0) \) for the original function.
- What does \(x\) need to be in the transformed function so that \( f(x+3) = f(0) \)?
- In this case, \(x\) needs to be \(-3\).
- So, you get: \( f(-3+3) = f(0) \).
- This means you need to shift the graph left by 3 units, which makes sense with what you think of when you see a negative number.
When identifying whether a transformation is horizontal or vertical, keep in mind that transformations are only horizontal if they are applied to \(x\) when it has a power of \(1\).
Consider the functions:
\[ g(x) = x^{3} - 4 \]
and
\[ h(x) = (x-4)^{3} \]
Take a minute to think about how these two functions, with respect to their parent function \( f(x) = x^{3} \), are transformed.
Can you compare and contrast their transformations? What do their graphs look like?
Solution:
- Graph the parent function.
Fig. 6. The graph of the parent cubic function.
- Determine the transformations indicated by the \( g(x) \) and \( h(x) \).
- For \( g(x) \):
- Since \(4\) is subtracted from the entire function, not just the input variable \(x\), the graph of \( g(x) \) shifts vertically down by \(4\) units.
- For \( h(x) \):
- Since \(4\) is subtracted from the input variable \(x\), not the entire function, the graph of \( h(x) \) shifts horizontally to the right by \(4\) units.
- For \( g(x) \):
- Graph the transformed functions with the parent function and compare them.
Fig. 7. the graph of the parent cubic function (blue) and two of its transformations (green, pink).
Let's look at another common mistake.
Expanding on the previous example, now consider the function:
\[ f(x) = \frac{1}{2} \left( x^{3} - 4 \right) + 2 \]
At first glance, you might think this has a horizontal shift of \(4\) units with respect to the parent function \( f(x) = x^{3} \).
This is not the case!
While you might be tempted to think so due to the parentheses, the \( \left( x^{3} - 4 \right) \) does not indicate a horizontal shift because \(x\) has a power of \(3\), not \(1\). Therefore, \( \left( x^{3} - 4 \right) \) indicates a vertical shift of \(4\) units down with respect to the parent function \( f(x) = x^{3} \).
To get the complete translation information, you must expand and simplify:
\[ \begin{align}f(x) &= \frac{1}{2} \left( x^{3} - 4 \right) + 2 \\&= \frac{1}{2} x^{3} - 2 + 2 \\&= \frac{1}{2} x^{3}\end{align} \]
This tells you that there is, in fact, no vertical or horizontal translation. There is only a vertical compression by a factor of \(2\)!
Let's compare this function to one that looks very similar but is transformed much differently.
| \( f(x) = \frac{1}{2} \left( x^{3} - 4 \right) + 2 = \frac{1}{2} x^{3} \) | \( f(x) = \frac{1}{2} (x - 4)^{3} + 2 \) |
| vertical compression by a factor of \(2\) | vertical compression by a factor of \(2\) |
| no horizontal or vertical translation | horizontal translation \(4\) units right |
| vertical translation \(2\) units up |
Fig. 8. the graph of the parent cubic function (blue) and two of its transformations (green, pink).
You have to ensure the coefficient of the \(x\) term is factored out fully to get an accurate analysis of the horizontal translation.
Consider the function:
\[ g(x) = 2(3x + 12)^{2} +1 \]
At first glance, you might think this function is shifted \(12\) units to the left with respect to its parent function, \( f(x) = x^{2} \).
This is not the case! While you might be tempted to think so due to the parentheses, the \( (3x + 12)^{2} \) does not indicate a left shift of \(12\) units. You must factor out the coefficient on \(x\)!
\[ g(x) = 2(3(x + 4)^{2}) + 1 \]
Here, you can see that the function is actually shifted \(4\) units left, not \(12\), after writing the equation in the proper form. The graph below serves to prove this.
Fig. 9. Ensure you fully factor out the coefficient of \(x\) to get an accurate analysis of the horizontal transformations.
Function Transformations: Order of Operations
As with most things in math, the order in which transformations of functions are done matters. For instance, considering the parent function of a parabola,
\[ f(x) = x^{2} \]
If you were to apply a vertical stretch of \(3\) and then a vertical shift of \(2\), you would get a different final graph than if you were to apply a vertical shift of \(2\) and then a vertical stretch of \(3\). In other words,
\[ \begin{align}2 + 3f(x) &\neq 3(2 + f(x)) \\2 + 3(x^{2}) &\neq 3(2 + x^{2})\end{align} \]
The table below visualizes this.
| A vertical stretch of \(3\), then a vertical shift of \(2\) | A vertical shift of \(2\), then a vertical stretch of \(3\) |
|
|
Function Transformations: When does the Order Matter?
And as with most rules, there are exceptions! There are situations where the order doesn't matter, and the same transformed graph will be generated regardless of the order in which the transformations are applied.
The order of transformations matters when
there are transformations within the same category (i.e., horizontal or vertical)
but are not the same type (i.e., shifts, shrinks, stretches, compressions).
What does this mean? Well, look the example above again.
Do you notice how the transformation (green) of the parent function (blue) looks quite different between the two images?
That is because the transformations of the parent function were the same category (i.e., vertical transformation), but were a different type (i.e., a stretch and a shift). If you change the order in which you perform these transformations, you get a different result!
So, to generalize this concept:
Say you want to perform \( 2 \) different horizontal transformations on a function:
No matter which \( 2 \) types of horizontal transformations you choose, if they are not the same (e.g., \( 2 \) horizontal shifts), the order in which you apply these transforms matters.
Say you want to perform \( 2 \) different vertical transformations on another function:
No matter which \( 2 \) types of vertical transformations you choose, if they are not the same (e.g., \( 2 \) vertical shifts), the order in which you apply these transforms matters.
Function transformations of the same category, but different types do not commute (i.e., the order matters).
Say you have a function, \( f_{0}(x) \), and constants \( a \) and \( b \).
Looking at horizontal transformations:
- Say you want to apply a horizontal shift and a horizontal stretch (or shrink) to a general function. Then, if you apply the horizontal stretch (or shrink) first, you get:\[ \begin{align}f_{1}(x) &= f_{0}(ax) \\f_{2}(x) &= f_{1}(x+b) = f_{0} \left( a(x+b) \right)\end{align} \]
- Now, if you apply the horizontal shift first, you get:\[ \begin{align}g_{1}(x) &= f_{0}(x+b) \\g_{2}(x) &= g_{1}(ax) = f_{0}(ax+b)\end{align} \]
- When you compare these two results, you see that:\[ \begin{align}f_{2}(x) &\neq g_{2}(x) \\f_{0} \left( a(x+b) \right) &\neq f_{0}(ax+b)\end{align} \]
Looking at vertical transformations:
- Say you want to apply a vertical shift and a vertical stretch (or shrink) to a general function. Then, if you apply the vertical stretch (or shrink) first, you get:\[ \begin{align}f_{1}(x) &= af_{0}(x) \\f_{2}(x) &= b+f_{1}(x) = b+af_{0}(x)\end{align} \]
- Now, if you apply the vertical shift first, you get:\[ \begin{align}g_{1}(x) &= b+f_{0}(x) \\g_{2}(x) &= ag_{1}(x) = a \left( b+f_{0}(x) \right)\end{align} \]
- When you compare these two results, you see that:\[ \begin{align}f_{2}(x) &\neq g_{2}(x) \\b+af_{0}(x) &\neq a \left( b+f_{0}(x) \right)\end{align} \]
The order of transformations does not matter when
- there are transformations within the same category and are the same type, or
- there are transformations that are different categories altogether.
What does this mean?
If you have a function that you want to apply multiple transformations of the same category and type, the order does not matter.
You can apply horizontal stretches/shrinks in any order and get the same result.
You can apply horizontal shifts in any order and get the same result.
You can apply horizontal reflections in any order and get the same result.
You can apply vertical stretches/shrinks in any order and get the same result.
You can apply vertical shifts in any order and get the same result.
You can apply vertical reflections in any order and get the same result.
If you have a function that you want to apply transformations of different categories, the order does not matter.
You can apply a horizontal and a vertical transformation in any order and get the same result.
Function transformations of the same category and same type do commute (i.e., the order does not matter).
Say you have a function, \( f_{0}(x) \), and constants \( a \) and \( b \).
- If you want to apply multiple horizontal stretches/shrinks, you get:\[ \begin{align}f_{1}(x) &= f_{0}(ax) \\f_{2}(x) &= f_{1}(bx) \\&= f_{0}(abx)\end{align} \]
- The product \(ab\) is commutative, so the order of the two horizontal stretches/shrinks does not matter.
- If you want to apply multiple horizontal shifts, you get:\[ \begin{align}f_{1}(x) &= f_{0}(a+x) \\f_{2}(x) &= f_{1}(b+x) \\&= f_{0}(a+b+x)\end{align} \]
- The sum \(a+b\) is commutative, so the order of the two horizontal shifts does not matter.
- If you want to apply multiple vertical stretches/shrinks, you get:\[ \begin{align}f_{1}(x) &= af_{0}(x) \\f_{2}(x) &= bf_{1}(x) \\&= abf_{0}(x)\end{align} \]
- The product \(ab\) is commutative, so the order of the two vertical stretches/shrinks does not matter.
- If you want to apply multiple vertical shifts, you get:\[ \begin{align}f_{1}(x) &= a + f_{0}(x) \\f_{2}(x) &= b + f_{1}(x) \\&= a + b + f_{0}(x)\end{align} \]
- The sum \(a+b\) is commutative, so the order of the two vertical shifts does not matter.
Let's look at another example.
Function transformations that are different categories do commute (i.e., the order does not matter).
Say you have a function, \( f_{0}(x) \), and constants \( a \) and \( b \).
- If you want to combine a horizontal stretch/shrink and a vertical stretch/shrink, you get:\[ \begin{align}f_{1}(x) &= f_{0}(ax) \\f_{2}(x) &= bf_{1}(x) \\&= bf_{0}(ax)\end{align} \]
- Now, if you reverse the order in which these two transformations are applied, you get:\[ \begin{align}g_{1}(x) &= bf_{0}(x) \\g_{2}(x) &= g_{1}(ax) \\&= bf_{0}(ax)\end{align} \]
- When you compare these two results, you see that:\[ \begin{align}f_{2}(x) &= g_{2}(x) \\bf_{0}(ax) &= bf_{0}(ax)\end{align} \]
So, is there a correct order of operations when applying transformations to functions?
The short answer is no, you can apply transformations to functions in any order you wish to follow. As you saw in the common mistakes section, the trick is learning how to tell which transformations have been made, and in which order, when going from one function (usually a parent function) to another.
Function Transformations: Transformations of Points
Now you are ready to transform some functions! To start, you will try to transform a point of a function. What you will do is move a specific point based on some given transformations.
If the point \( (2, -4) \) is on the function \( y = f(x) \), then what is the corresponding point on \( y = 2f(x-1)-3 \)?
Solution:
You know so far that the point \( (2, -4) \) is on the graph of \( y = f(x) \). So, you can say that:
\[ f(2) = -4 \]
What you need to find out is the corresponding point that is on \( y = 2f(x-1)-3 \). You do that by looking at the transformations given by this new function. Walking through these transformations, you get:
- Start with the parentheses.
- Here you have \( (x-1) \). → This means you shift the graph to the right by \(1\) unit.
- Since this is the only transformation applied to the input, you know there are no other horizontal transformations on the point.
- So, you know the transformed point has an \(x\)-coordinate of \(3\).
- Apply the multiplication.
- Here you have \( 2f(x-1) \). → The \(2\) means you have a vertical stretch by a factor of \(2\), so your \(y\)-coordinate doubles to \(-8\).
- But, you are not done yet! You still have one more vertical transformation.
- Apply the addition/subtraction.
- Here you have the \(-3\) applied to the entire function. → This means you have a shift down, so you subtract \(3\) from your \(y\)-coordinate.
- So, you know the transformed point has a \(y\)-coordinate of \(-11\).
- Here you have the \(-3\) applied to the entire function. → This means you have a shift down, so you subtract \(3\) from your \(y\)-coordinate.
So, with these transformations done to the function, whatever function it may be, the corresponding point to \( (2, -4) \) is the transformed point \( \bf{ (3, -11) } \).
To generalize this example, say you are given the function \( f(x) \), the point \( (x_0, f(x_0)) \), and the transformed function\[ g(y) = af(x = by+c)+d,\]what is the corresponding point?
First, you need to define what the corresponding point is:
It's the point on the transformed function's graph such that the \(x\)-coordinates of the original and the transformed point are related by the horizontal transformation.
So, you need to find the point \((y_0, g(y_0))\) such that
\[x_0 = by_0+c\]
To find \(y_0\), isolate it from the above equation:
\[y_0 = \frac{x_0-c}{b}\]
To find \(g(y_0)\), plug in \(g\):
\[g(y_0) = af(x = by_0+c)+d = af(x_0)+d\]
Bottom line: to find the \(x\)-component of the transformed point, solve the inverted horizontal transformation; to find the \(y\)-component of the transformed point, solve the vertical transformation.
Function Transformations: Examples
Now let's look at some examples with different types of functions!
Exponential Function Transformations
The general equation for a transformed exponential function is:
\[ f(x) = a(b)^{k(x-d)}+c \]
Where,
\[ a = \begin{cases}\mbox{vertical stretch if } a > 1, \\\mbox{vertical shrink if } 0 < a < 1, \\\mbox{reflection over } x-\mbox{axis if } a \mbox{ is negative}\end{cases} \]
\[ b = \mbox{the base of the exponential function} \]
\[ c = \begin{cases}\mbox{vertical shift up if } c \mbox{ is positive}, \\\mbox{vertical shift down if } c \mbox{ is negative}\end{cases} \]
\[ d = \begin{cases}\mbox{horizontal shift left if } +d \mbox{ is in parentheses}, \\\mbox{horizontal shift right if } -d \mbox{ is in parentheses}\end{cases} \]
\[ k = \begin{cases}\mbox{horizontal stretch if } 0 < k < 1, \\\mbox{horizontal shrink if } k > 1, \\\mbox{reflection over } y-\mbox{axis if } k \mbox{ is negative}\end{cases} \]
Let's transform the parent natural exponential function, \( f(x) = e^{x} \), by graphing the natural exponential function:
\[ f(x) = -e^{2(x-1)}+3. \]
Solution:
- Graph the parent function.
Fig. 12. Graph of function \(e^x\).
- Determine the transformations.
Start with the parentheses (horizontal shifts)
Here you have \(f(x) = e^{(x-1)}\), so the graph shifts to the right by \(1\) unit.
Fig. 13. Graph of the function \(e^x\) and its transformation.
Apply the multiplication (stretches and/or shrinks)
Here you have \( f(x) = e^{2(x-1)} \), so the graph shrinks horizontally by a factor of \(2\).
Fig. 14. The graph of the parent natural exponential function (blue) and the first two steps of the transform (yellow, purple).
Apply the negations (reflections)
Here you have \( f(x) = -e^{2(x-1)} \), so the graph is reflected over the \(x\)-axis.
Fig. 15. The graph of the parent natural exponential function (blue) and the first three steps of the transform (yellow, purple, pink)
Apply the addition/subtraction (vertical shifts)
Here you have \( f(x) = -e^{2(x-1)} + 3 \), so the graph is shifted up by \(3\) units.
Fig. 16. The graph of the parent natural exponential function (blue) and the steps to get the transform (yellow, purple, pink, green).
Graph the final transformed function.
Fig. 17. The graphs of the parent natural exponential function (blue) and its transform (green).
Logarithmic Function Transformations
The general equation for a transformed logarithmic function is:
\[ f(x) = a\mbox{log}_{b}(kx+d)+c. \]
Where,
\[ a = \begin{cases}\mbox{vertical stretch if } a > 1, \\\mbox{vertical shrink if } 0 < a < 1, \\\mbox{reflection over } x-\mbox{axis if } a \mbox{ is negative}\end{cases} \]
\[ b = \mbox{the base of the logarithmic function} \]
\[ c = \begin{cases}\mbox{vertical shift up if } c \mbox{ is positive}, \\\mbox{vertical shift down if } c \mbox{ is negative}\end{cases} \]
\[ d = \begin{cases}\mbox{horizontal shift left if } +d \mbox{ is in parentheses}, \\\mbox{horizontal shift right if } -d \mbox{ is in parentheses}\end{cases} \]
\[ k = \begin{cases}\mbox{horizontal stretch if } 0 < k < 1, \\\mbox{horizontal shrink if } k > 1, \\\mbox{reflection over } y-\mbox{axis if } k \mbox{ is negative}\end{cases} \]
Let's transform the parent natural log function, \( f(x) = \text{log}_{e}(x) = \text{ln}(x) \) by graphing the function:
\[ f(x) = -2\text{ln}(x+2)-3. \]
Solution:
- Graph the parent function.
Fig. 18. The graph of the parent natural logarithm function.
- Determine the transformations.
Start with the parentheses (horizontal shifts)
Here you have \( f(x) = \text{ln}(x+2) \), so the graph shifts to the left by \(2\) units.
Fig. 19. The graphs of the parent natural logarithm function (blue) and the first step of the transform (green)
Apply the multiplication (stretches and/or shrinks)
Here you have \( f(x) = 2\text{ln}(x+2) \), so the graph stretches vertically by a factor of \(2\).
Fig. 20. The graphs of the parent natural logarithm function (blue) and the first two steps of the transform (green, pink).
Apply the negations (reflections)
Here you have \( f(x) = -2\text{ln}(x+2) \), so the graph reflects over the \(x\)-axis.
Fig. 21. The graphs of the parent natural logarithm function (blue) and the first three steps of the transform (green, purple, pink).
Apply the addition/subtraction (vertical shifts)
Here you have \( f(x) = -2\text{ln}(x+2)-3 \), so the graph shifts down \(3\) units.
Fig. 22. The graphs of the parent natural logarithm function (blue) and the steps to get the transform (yellow, purple, pink, green)
- Graph the final transformed function.
Fig. 23. The graphs of the parent natural logarithm function (blue) and its transform (green
Rational Function Transformations
The general equation for a rational function is:
\[ f(x) = \frac{P(x)}{Q(x)} ,\]
where
\[ P(x) \mbox{ and } Q(x) \mbox{ are polynomial functions, and } Q(x) \neq 0. \]
Since a rational function is made up of polynomial functions, the general equation for a transformed polynomial function applies to the numerator and denominator of a rational function. The general equation for a transformed polynomial function is:
\[ f(x) = a \left( f(k(x-d)) + c \right), \]
where,
\[ a = \begin{cases}\mbox{vertical stretch if } a > 1, \\\mbox{vertical shrink if } 0 < a < 1, \\\mbox{reflection over } x-\mbox{axis if } a \mbox{ is negative}\end{cases} \]
\[ c = \begin{cases}\mbox{vertical shift up if } c \mbox{ is positive}, \\\mbox{vertical shift down if } c \mbox{ is negative}\end{cases} \]
\[ d = \begin{cases}\mbox{horizontal shift left if } +d \mbox{ is in parentheses}, \\\mbox{horizontal shift right if } -d \mbox{ is in parentheses}\end{cases} \]
\[ k = \begin{cases}\mbox{horizontal stretch if } 0 < k < 1, \\\mbox{horizontal shrink if } k > 1, \\\mbox{reflection over } y-\mbox{axis if } k \mbox{ is negative}\end{cases} \]
Let's transform the parent reciprocal function, \( f(x) = \frac{1}{x} \) by graphing the function:
\[ f(x) = - \frac{2}{2x-6}+3. \]
Solution:
- Graph the parent function.
Fig. 24. The graph of the parent rational function.
- Determine the transformations.
Start with the parentheses (horizontal shifts)
- To find the horizontal shifts of this function, you need to have the denominator in standard form (i.e., you need to factor out the coefficient of \(x\)).
- So, the transformed function becomes:\[ \begin{align}f(x) &= - \frac{2}{2x-6}+3 \\&= - \frac{2}{2(x-3)}+3\end{align} \]
- Now, you have \( f(x) = \frac{1}{x-3} \), so you know the graph shifts right by \(3\) units.
Apply the multiplication (stretches and/or shrinks) This is a tricky step
Here you have a horizontal shrink by a factor of \(2\) (from the \(2\) in the denominator) and a vertical stretch by a factor of \(2\) (from the \(2\) in the numerator).
Here you have \( f(x) = \frac{2}{2(x-3)} \), which gives you the same graph as \( f(x) = \frac{1}{x-3} \).
The graphs of the parent rational function (blue) and the first step of the transform (fucsia).
Fig. 25.
Apply the negations (reflections)
Here you have \( f(x) = - \frac{2}{2(x-3)} \), so the graph reflects over the \(x\)-axis.
The graphs of the parent rational function (blue) and the first three steps of the transform (yellow, purple, pink).
Fig. 26.
Apply the addition/subtraction (vertical shifts)
Here you have \( f(x) = - \frac{2}{2(x-3)} + 3 \), so the graph shifts up \(3\) units.
Fig. 27. The graphs of the parent rational function (blue) and the steps to get the transform (yellow, purple, pink, green).
- Graph the final transformed function.
- The final transformed function is \( f(x) = - \frac{2}{2(x-3)} + 3 = - \frac{2}{2x-6} + 3 \).
Fig. 28. The graphs of the parent rational function (blue) and its transform (green).
Function Transformations – Key takeaways
- Function transformations are the processes used on an existing function and its graph to give us a modified version of that function and its graph that has a similar shape to the original function.
- Function transformations are broken down into two major categories:
Horizontal transformations
- Horizontal transformations are made when we either add/subtract a number from a function's input variable (usually x) or multiply it by a number. Horizontal transformations, except reflection, work in the opposite way we'd expect them to.
- Horizontal transformations only change the x-coordinates of functions.
Vertical transformations
Vertical transformations are made when we either add/subtract a number from the entire function, or multiply the entire function by a number. Unlike horizontal transformations, vertical transformations work the way we expect them to.
- Vertical transformations only change y-coordinates of functions.
Any function can be transformed, horizontally and/or vertically, via four main types of transformations:
Horizontal and vertical shifts (or translations)
Horizontal and vertical shrinks (or compressions)
Horizontal and vertical stretches
Horizontal and vertical reflections
- When identifying whether a transformation is horizontal or vertical, keep in mind that transformations are only horizontal if they are applied to x when it has a power of 1.
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Frequently Asked Questions about Function Transformations
What are transformations of a function?
Transformations of a function, or function transformation, are the ways we can change a function's graph so that it becomes a new function.
What are the 4 transformations of a function?
The 4 transformations of a function are:
- Horizontal and vertical shifts (or translations)
- Horizontal and vertical shrinks (or compressions)
- Horizontal and vertical stretches
- Horizontal and vertical reflections
How do you find the transformation of a function at a point?
To find the transformation of a function at a point, follow these steps:
- Choose a point that lies on the function (or use a given point).
- Look for any Horizontal Transformations between the original function and the transformed function.
- Horizontal Transformations are what the x-value of the function is changed by.
- Horizontal Transformations only affect the x-coordinate of the point.
- Write the new x-coordinate.
- Look for any Vertical Transformations between the original function and the transformed function.
- Vertical Transformations are what the entire function is changed by.
- Vertical Transformation only affect the y-coordinate of the point.
- Write the new y-coordinate.
- With both the new x- and y-coordinates, you have the transformed point!
How to graph exponential functions with transformations?
To graph an exponential function with transformations is the same process to graph any function with transformations.
Given an original function, say y = f(x), and a transformed function, say y = 2f(x-1)-3, let's graph the transformed function.
- Horizontal transformations are made when we either add/subtract a number from x, or multiply x by a number.
- In this case, the horizontal transformation is shifting the function to the right by 1.
- Vertical transformations are made when we either add/subtract a number from the entire function, or multiply the entire function by a number.
- In this case, the vertical transformations are:
- A vertical stretch by 2
- A vertical shift down by 3
- In this case, the vertical transformations are:
- With these transformations in mind, we now know that the graph of the transformed function is:
- Shifted to the right by 1 unit compared to the original function
- Shifted down by 3 units compared to the original function
- Stretched by 2 units compared to the original function
- To graph the function, simply choose input values of x and solve for y to get enough points to draw the graph.
What is an example of a transformed equation?
An example of a transformed equation from the parent function y=x2 is y=3x2 +5. This transformed equation undergoes a vertical stretch by a factor of 3 and a translation of 5 units up.
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