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| 10 Theory slides |
| 10 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
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.
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
g(x)=f(x±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) |
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.
Graph:
Graph:
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.
Both a vertical and a horizontal translation will be needed.
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.
The linear functions shown in the following diagram are related by a translation. Be careful with the sign of h.
The linear functions shown in the following diagram are related by a translation.
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.
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.
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.
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.
For the following statements, write a function g in terms of f so that the statement is true.
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)
For the following statements, write a function g in terms of f so that the statement is true.
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