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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.
Write the function rule $g(t)$ that models Emily's jog, taking into account the advantage that she will give to Heichi.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.
Distribute $m$
Commutative Property of Addition