4. Transformations of Linear Functions
Sign In
An error ocurred, try again later!
Chapter 3
4.
Transformations of Linear Functions
Dylan and Emily dive into the world of linear functions, aiming to design a sports video game. They learn about translating linear functions to adjust the size of the playing court. Translations, both vertical and horizontal, allow them to move functions without altering their shape. Emily, an avid jogger, uses linear functions to track her jogging progress. One day, she invites Heichi for a jog and decides to give him a head start, showcasing the real-world application of translating functions. Through these experiences, they discover that linear transformations are essential tools for various scenarios, from game design to tracking physical activities.
Show less Show more expand_more 
Lesson Settings & Tools
| | 10 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Transformations of Linear Functions
Slide of 10
All linear functions share the characteristic of having a constant rate of change. It is often required to relate different linear functions by means of transformations, which will be discussed in this lesson.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Share
Report error
Translate
Explore
Translating a Linear Function
Consider the linear function f(x)=2x+1. The applet below shows how transformations can be used to translate the graph of the given function. Two points of the graph will be shown to keep track of the transformation.
Share
Report error
Translate
Discussion
Function Families and Parent Functions
Functions can be grouped according to their characteristics.
A function family is a group of functions that have similar characteristics. A parent function is the most basic function in a function family. The most common parent functions are those of linear functions, absolute value functions, quadratic functions, and exponential functions.
Share
Report error
Translate
Discussion
Translation
In the previous applet, the whole graph was moved without changing its shape or orientation by changing the h and k values. This is known as a translation.
A translation of a function is a transformation that moves a function graph in some direction, without any rotation, shrinking, or stretching. A function's graph is vertically translated by adding a number to — or subtracting from — the function rule.
g(x)=f(x) ± k
Every point of the graph of y=f(x) will be moved up or down by k units, depending on the sign of k.
g(x)=f(x± h)
Every point on the graph of y=f(x) will be moved to the left or to the right by h units, depending on the sign of h.
| Translations of f(x) | |
|---|---|
| Vertical Translations | Translation up k units, k>0 y=f(x)+k |
| Translation down k units, k>0 y=f(x)-k | |
| Horizontal Translations | Translation to the right h units, h>0 y=f(x-h) |
| Translation to the left h units, h>0 y=f(x+h) | |
Why
Why the Graph is Translated?Consider adding a number k to a function rule.
g(x) = f(x) + k
If k is positive, this operation increases the value of the output for every x, moving the graph upward. Similarly, if k is negative, then the graph is moved downward, because the output of the function is decreased.
Now consider subtracting a number h from the input. g(x) = f(x - h) If h is positive, then the value of the input is reduced. Therefore, greater x-values are needed to obtain the original output, leading to a translation to the right. In contrast, when h is negative, the input value is increased. This means that smaller inputs are needed to obtain the original output. This leads to a translation to the left.
Share
Report error
Translate
Example
Adjusting a Savings Plan
Dylan wants to buy a new set of canvases. He plans to save $2 each day, starting tomorrow. His math teacher told him that linear functions can be used to make a savings plan and provided him with the following graph.
Here, x is the number of days that have passed since Dylan started saving and f(x) is the total amount saved.
a Dylan plans to ask his parents for some money for his initial savings. If they give him $2, write the function rule that adjusts to this new plan and graph it along with the original function.
b Instead of asking for money, Dylan decides that he will go out tomorrow to buy a new brush for $2, meaning that he will not be able to start saving until the day after tomorrow. Write the function that adjusts to this new plan and graph it along with the original function.
Answer
a Function: g(x)=2x+2
Graph:
b Function: g(x)=2(x-1)
Graph:
Hint
a Dylan will start with $2 and will still be saving $2 each day.
b Dylan will have to delay his original plan by 1 day.
Solution
a Dylan's original savings plan was modeled by the following linear function with the assumption that he would not have any initial savings.
f(x)=2x If his parents give him $2, Dylan will have $2 more than he originally planned on each day. Since each y-value represents his savings on a particular day, each point of the graph has to be translated 2 units up. Therefore, translate 2 units up the graph of the original savings plan.
The expression for the new savings plan can be found by adding 2 to the function rule. f(x) + 2=2x + 2 Knowing this, the function of the new savings plan can be written. g(x)=2x+2 Finally, both functions will be graphed on the same coordinate plane.
b If Dylan goes tomorrow to buy a new brush for $2, he will not be able to save money that day and his savings plans will be delayed by 1 day. Therefore, he can draw the graph of his saving plans starting at x=1 instead of starting at x=0. This means the graph of the original savings plan has to be translated 1 unit to the right.
The expression for the new savings plan can be found by subtracting 1 from the input of the function. f(x - 1)=2(x - 1) Knowing this, the function of the new savings plan can be written. g(x)=2(x-1) Finally, both functions will be graphed on the same coordinate plane.
Share
Report error
Translate
Example
Keeping Track of a Jogging Course
Every Monday afternoon, Emily likes to go jogging in the park next to her job. She uses the pine tree seen in the diagram as her starting point to keep track of her progress.
Following her physics teacher's advice, Emily wrote and graphed a linear function d(t) that models the distance in meters to the right of the pine tree that she has jogged. Here, t is the time in seconds since the start of her run.
One day, she decided to invite Heichi to join her. Since Heichi is not used to jogging, Emily said that she will give him a head start. Emily will start 10 seconds later, and also she will start from the fountain, which she knows is 5 meters away from the pine.
Write the function rule g(t) that models Emily's jog, taking into account the advantage that she will give to Heichi.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["t"],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"g(t)=","formTextAfter":null,"answer":{"text":["2.5(t-10)-5","2.5t-30"]}}
Hint
Both a vertical and a horizontal translation will be needed.
Solution
Emily's jog is normally modeled by the following function.
d(t)=2.5t
Here, t is the time in seconds since Emily started jogging. Since she is giving Heichi a head start, she will start jogging 10 seconds after Heichi starts jogging. This means that at t=10, her distance jogged will be 0 meters, so the graph has to pass through (10,0). This can be done by translating the original graph 10 units to the right.
Furthermore, she will start from the fountain, 5 meters left of the pine tree, which is Heichi's starting point. Therefore, at t=10, Emily will be 5 meters to the left of the pine tree. This can be thought as her distance to the tree being -5 meters, so the graph has to pass through (10,-5). To achieve this, translate the previous graph 5 units down.
Therefore, the function g(t) that models Emily's new jog is obtained by translating d(t) 10 units to the right and 5 units down. g(t)=d(t-10)-5 ⇓ g(t)=2.5(t-10)-5
Share
Report error
Translate
Example
Early Stage of Video Game Design
Dylan and Emily are designing a sports video game. In order to draw different playing courts, they need to be able to freely move linear functions according to a function rule that is to be interpreted by a computer.
In this stage of design, they are interested in translating f(x). This will let them adjust the size of the playing court.
a Draw the graph of f(x) translated 1 unit to the left.
b Draw the graph of f(x) translated 2 units down.
c Draw the graph of f(x) translated 1 unit to the right and 3 units up.
Answer
a Graph:
b Graph:
c Graph:
Hint
a Two points define a unique line. This means that two points can be translated to find the translation of the linear function.
b Use two appropriate points of the original function as a reference for the translation.
c The translations can be done each one at a time.
Solution
a Two points can be used to define a unique line. This fact can be used to help with the translation of the graph. Start by identifying two arbitrary points in the graph of f(x).
Since the translation is 1 unit to the left, these two points will be translated in this way. The translated linear function can be drawn using the new points.
A better view of the translated graph can be taken by removing the two points.
b Two points will be used as a reference again for this translation. This time the translation is two units down, so it might be better to choose a different pair of points.
The translated function is shown in the following graph.
c This time two translations are involved. These can be done one at a time. One way of doing it is by starting by the translation by one unit to the right, then translating the new line up by three units. The same result can be obtained by starting with the vertical translation followed by the horizontal one.
Finally, the points are removed to take a better look at the graph.
Share
Report error
Translate
Pop Quiz
Identifying Horizontal Translations
The linear functions shown in the following diagram are related by a translation. Be careful with the sign of h.
Share
Report error
Translate
Pop Quiz
Identifying Vertical Translations
The linear functions shown in the following diagram are related by a translation.
Share
Report error
Translate
Closure
Relating Vertical and Horizontal Translations of Linear Functions
Horizontal and vertical translations of linear functions have the particular characteristic of being related to each other. In the following applet, a horizontal translation can be done to the linear function by dragging the red point. The blue point will keep track of the vertical translation of the function.
Share
Report error
Translate
Transformations of Linear Functions
Exercises
Consider the following functions. f(x) &= 2x+1 g(x) &= 2x+4 Which of the following describes the transformation of function f to function g? A. & Vertical translation3 units down B. & Vertical translation3 units up C. & Horizontal translation3 units to the left D. & Horizontal translation3 units to the right
{"type":"choice","form":{"alts":["A","B","C","D"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}
security
report
code
security
report
code
Let's begin by rewriting the function g in terms of the function f. This can be done by rewriting the constant term 4 as 1+3. From there, we can identify the function rule of f(x).
The function g is of the form g(x)=f(x)+k, which is a vertical translation. Since k=3, this is a vertical translation of 3 units up. Therefore, the correct option is B. Finally, we will take a look at the graph of both functions.
As we can see, the graph of g(x) is the same as f(x), just moved up three units. Please note that the transformation can also be seen as a horizontal translation 2 units to the left, but this option is not available, so the vertical translation up is the correct answer.
security
report
code
E
Hint & Answer
S
Solution
Consider the following functions. f(x) &= 2x+1 g(x) &= 2(x+2)+1 Which of the following describes the transformation of function f to function g? A. & Vertical translation2 units down B. & Vertical translation2 units up C. & Horizontal translation2 units to the left D. & Horizontal translation2 units to the right
{"type":"choice","form":{"alts":["A","B","C","D"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":2}
security
report
code
security
report
code
Let's begin by noting that f evaluated at x+2 equals g(x). f(x)=2x+1 ⇓ f( x+2) = 2( x+2)+1 This means that the right-hand side expression is equal to g(x). That is, g is of the form g(x)=f(x+h), which is a horizontal translation. g(x)=f(x+2) Since h>0, this is a horizontal translation of h= 2 units to the left. Therefore, the correct option is C. Finally, we will take a look at the graph of both functions.
As we can see, the graph of g(x) is the same as the graph of f(x), just moved two units to the left. Please note that the transformation can also be seen as a vertical translation 4 units up, but this option is not available as a potential answer.
security
report
code
E
Hint & Answer
S
Solution
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"g(x)=","formTextAfter":null,"answer":{"text":["2x-3"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"h(x)=","formTextAfter":null,"answer":{"text":["2x-7"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"m(x)=","formTextAfter":null,"answer":{"text":["2x-8"]}}
security
report
code
security
report
code
We are asked to write the vertical translation of the given function by 2 units up. Vertical translations are of the form g(x)=f(x)+k. Since in this case we want to translate the function 2 units up, k= 2. We can now write the translation. Let's do it!
We will now take a look at the graph of both functions to identify the translation.
We will now translate the given function by 2 units down. This time the translation is 2 units down, so k= -2. Let's write the translation!
We will now take a look at the graph of both functions to identify the translation.
Finally, we will translate the given function by 3 units down. This time the translation is 3 units down, so k= -3. Let's write the translation!
We will now take a look at the graph of both functions to identify the translation.
security
report
code
E
Hint & Answer
S
Solution
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"g(x)=","formTextAfter":null,"answer":{"text":["-3x+11","11-3x"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"m(x)=","formTextAfter":null,"answer":{"text":["-3x-7","-7-3x"]}}
security
report
code
security
report
code
A horizontal translation of a function is a shift along the x-axis. This means that we have to manipulate the inputs — the x-values — of the original function. To move a function f(x) to the right by h units, we have to subtract h from the function's input before evaluating. f(x-h) We know that the function g is a horizontal translation 3 units to the right of f. This means we have to subtract 3 from the input of f. g(x)=f(x- 3) Next, we evaluate f(x) at x-3. f(x)=-3x+2 [0.5em] ⇓ [0.5em] f( x-3)=-3( x-3)+2 Finally, we simplify the right-hand side expression.
We are now given function m, which is is a horizontal translation 3 units to the left of f. This means we have to subtract -3 from the input of f. This is the equivalent to adding 3 to the input of f. m(x)=f(x-( -3)) [0.5em] ⇕ [0.5em] m(x)=f(x+3) Next, we evaluate f(x) at x+3. f(x)=-3x+2 [0.5em] ⇓ [0.5em] f( x+3)=-3( x+3)+2 Finally, we simplify the right-hand side expression.
security
report
code
E
Hint & Answer
S
Solution
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["f","x"],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"g(x)=","formTextAfter":null,"answer":{"text":["f(x-4)"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["f","x"],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"g(x)=","formTextAfter":null,"answer":{"text":["f(x)+4)"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["f","x"],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"g(x)=","formTextAfter":null,"answer":{"text":["f(x+4)"]}}
security
report
code
security
report
code
A horizontal translation is a shift of a graph along the x-axis. This means that we have to manipulate the inputs — the x-values — of the original function. To move a function f(x) to the right by h units, we have to subtract h from the function's input before evaluating. f(x-h) We are told that the graph of g is a horizontal translation 4 units to the right of the graph of f. This means we have to subtract 4 from the input of f. g(x)=f(x- 4)
A vertical translation is a shift of a graph along the y-axis. This means that we have to manipulate the outputs — the y-values — of the original function. To move a function f(x) up by k units, we have to add k to the function's output after evaluating.
f(x)+k
The graph of g is a vertical translation 4 units up of the graph of f. This means we have to add 4 to the output of f.
g(x)=f(x)+ 4
This time g is a horizontal translation 4 units to the left of the graph of f. This means we have to subtract -4 from the input of f.
g(x)=f(x-( -4)) ⇕
g(x)=f(x+4)
security
report
code
E
Hint & Answer
S
Solution
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["f","x"],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"g(x)=","formTextAfter":null,"answer":{"text":["f(x)+9"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["f","x"],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"g(x)=","formTextAfter":null,"answer":{"text":["f(x-9)"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["f","x"],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"g(x)=","formTextAfter":null,"answer":{"text":["f(x)-9"]}}
security
report
code
security
report
code
A vertical translation is a shift of a graph along the y-axis. This means that we have to manipulate the outputs — the y-values — of the original function. To move a function f(x) up by k units, we have to add k to the function's output after evaluating. f(x)+k The graph of g is a vertical translation 9 units up of the graph of f. This means we have to add 9 to the output of f. g(x)=f(x)+ 9
A horizontal translation is a shift of a graph along the x-axis. This means that we have to manipulate the inputs — the x-values — of the original function. To move a function f(x) to the right by h units, we have to subtract h from the function's input before evaluating.
f(x-h)
We are told that the graph of g is a horizontal translation 9 units to the right of the graph of f. This means we have to subtract 9 from the input of f.
g(x)=f(x- 9)
This time g is a vertical translation 9 units down of the graph of f. This means we have to subtract 9 from the output of f.
g(x)=f(x)- 9
security
report
code
E
Hint & Answer
S
Solution
Transformations of Linear Functions
Recommended exercises
To get personalized recommended exercises, please login first.
Loading content
Edit Lesson
≤
>
■2
■▢
e▢
7
8
9
×
÷■1
=
=
√▢
▢√
∫▢▢
4
5
6
+
−
≤
<
log
ln
log▢
1
2
3
()
∣
sin
cos
tan
0
.
π
x
y
Loading content