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| 14 Theory slides |
| 9 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.
Consider the graph of the parent absolute value function.
Many different absolute value functions can be obtained by shifting the graph of the parent absolute value function. Absolute value functions obtained in this way have the following form.Transformations of y=∣x∣ | |
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Vertical Translations | Translation up k units, k>0y=∣x∣+k
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Translation down k units, k<0y=∣x∣+k
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Horizontal Translations | Translation to the right h units, h>0y=∣x−h∣
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Translation to the left h units, h<0y=∣x−h∣
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Tadeo just learned about translations of absolute value functions. He believes in the motto that practice makes perfect, so he decides to study more. The following graphs are the graphs of the absolute value parent function after a certain translation.
Help Tadeo match each graph with the corresponding function rule.Compare the given graphs to the graph of the absolute value parent function to identify the translation applied to each graph.
Begin by identifying the translation of each graph when compared to the graph of the absolute value parent function f(x)=∣x∣.
Now that the translations have been identified, recall the translation rules.
Transformations of y=f(x) | |
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Horizontal Translations | Translation to the right by h units, h>0y=f(x−h)
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Translation to the left by h units, h<0y=f(x−h)
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Vertical Translations | Translation upwards byk units, k>0y=f(x)+k
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Translation downwards by k units, k<0y=f(x)+k
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The graph of an absolute value function y=∣x∣ can be translated vertically by adding a number to — or subtracting from — the function rule. Likewise, it can be also translated horizontally by adding a number to — or subtracting from — the rule's input.
The following applet shows the graph of an absolute value function in the form of f(x)=∣x−h∣+k, where h and k are integers. Considering the translation rules, determine the values of h and k.
Transformations of f(x) | |
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Vertical Stretch or Shrink | Vertical stretch, a>1 y=af(x) |
Vertical shrink, 0<a<1 y=af(x) |
Transformations of f(x) | |
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Horizontal Stretch or Shrink | Horizontal stretch, 0<b<1 y=f(bx) |
Horizontal shrink, b>1 y=f(bx) |
Transformations of y=∣x∣ | |
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Vertical Stretch or Shrink | Vertical stretch, a>1y=a∣x∣
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Vertical shrink, 0<a<1y=a∣x∣
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Horizontal Stretch or Shrink | Horizontal stretch, 0<b<1y=∣bx∣
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Horizontal shrink, b>1y=∣bx∣
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Emily uses a water tank that contains 10 cubic meters of water to water her plants. She drains the water tank from one faucet and then refills it from another identical faucet.
The following graph models the water level of the tank when it is drained and then refilled after t minutes.
Transformations of f(x) | |
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Reflections | In the x-axis y=-f(x) |
In the y-axis y=f(-x) |
Transformations of y=∣x∣ | |
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Reflections | In the x-axisy=-∣x∣
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In the y-axisy=∣-x∣
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W.Write the equation of the reflected function by determining the type of the reflection.
W,begin by graphing the given function by using a table of values.
x | f(x)=∣2x−5∣ | f(x) |
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-1 | f(x)=∣2(-1)−5∣ | 7 |
0 | f(x)=∣2(0)−5∣ | 5 |
1 | f(x)=∣2(1)−5∣ | 3 |
2.5 | f(x)=∣2(2.5)−5∣ | 0 |
4 | f(x)=∣2(4)−5∣ | 3 |
5 | f(x)=∣2(5)−5∣ | 5 |
6 | f(x)=∣2(6)−5∣ | 7 |
Now plot the ordered pairs and connect them to graph the absolute value function.
Looking at the graph of the function, it can be concluded that f(x) needs to be reflected in the y-axis.x | f(x)=21∣x−4∣−2 | f(x) |
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-2 | f(x)=21∣-2−4∣−2 | 1 |
0 | f(x)=21∣0−4∣−2 | 0 |
2 | f(x)=21∣2−4∣−2 | -1 |
4 | f(x)=21∣4−4∣−2 | -2 |
6 | f(x)=21∣6−4∣−2 | -1 |
8 | f(x)=21∣8−4∣−2 | 0 |
10 | f(x)=21∣10−4∣−2 | 1 |
Plot the ordered pairs and draw the graph.
It can be seen that, to form a quadrilateral, the graph of the function needs to be reflected in the x-axis.
The parent function of an absolute value function is f(x)=∣x∣ and its graph is as follows.
Determine the graph of each of the following functions by translating the absolute value parent function.
Let g(x)=∣x+2∣.
Which of these is the graph of g(x)?Let h(x)=∣x∣−2.
Which of these is the graph of h(x)?Let's recall the given function. g(x)=|x+ 2| We can see that a number is added to the input of f(x) to get g(x). This means that g(x) is the horizontal translation of f(x) to the left 2 units.
Therefore, the graph of g(x) is given in option D.
When we look at the function h(x), we can see that a number is subtracted from the output of f(x).
h(x)=|x|- 2
With this operation, we are translating the graph of f(x) down 2 units.
From here, we can conclude that the graph of h(x) is given in option A.
The table below summarizes the different types of translations that can be performed for shifting an absolute value function.
Transformations of y=|x| | |
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Vertical Translations | Translation up k units, k>0 y=|x|+ k |
Translation down k units, k<0 y=|x|+ k | |
Horizontal Translations | Translation to the right h units, h>0 y=|x- h| |
Translation to the left h units, h<0 y=|x- h| |
We can use this table to identify the translations applied to the absolute value functions.
The graph of the absolute value parent function f(x)=∣x∣ is given along with the graph of h(x)=a∣x∣, where a is a real number other than 0.
Find the value of a by comparing f(x) and h(x).Let's begin by recalling the general form of a horizontal stretch or shrink of an absolute value function.
Transformations of y=|x| | |
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Horizontal Stretch or Shrink | Horizontal stretch, 0< b<1 y=| bx| |
Horizontal shrink, b>1 y=| bx| |
Since we want a horizontal shrink by a factor of 3,
we can write the required function by substituting 3 for b in the given form.
g(x)=| bx| ⇒ g(x)=| 3x|
Note that we can get the graph of h(x) by multiplying the output of f(x) by a. Therefore, the graph of h(x) is the graph of f(x) vertically stretched or shrunk. Let's choose a point on each graph and compare them vertically.
We can see that the y-coordinate of the point on h(x) is half the y-coordinate of the point on f(x). This means that we can get the y-coordinate of the point on h(x) by multiplying the y-coordinate of the point on f(x) by 12. Let's recall the general form of a vertical stretch or shrink of an absolute value function.
Transformations of y=|x| | |
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Vertical Stretch or Shrink | Vertical stretch, a>1 y= a|x| |
Vertical shrink, 0< a< 1 y= a|x| |
This table tells us that the graph of h(x) is the graph of f(x) vertically shrunk by a factor of 12. Therefore, the value of a is 12.
Let's begin by recalling the general form of a reflection of an absolute value function.
Transformations of y=|x| | |
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Reflections | In the x-axis y=- |x| |
In the y-axis y=|- x| |
This table tells us that if we want to reflect an absolute value function in the y-axis, we should multiply its input by -1. ccc Original Function & & Reflection iny-axis [0.5em] f(x)=3|x-2| & & g(x)=3|- x-2| We can simplify the function g(x) by using the properties of absolute value.
The function whose graph is the reflection of the graph of f(x)=3|x-2| in the y-axis is g(x)=3|x+2|.
Consider the graph of the absolute parent function f(x)=∣x∣.
Let the graph of h(x) be the graph of f(x) translated 3 units to the right, vertically stretched by a factor of 2, and reflected on the x-axis.
Which of these graphs is the graph of h(x)?To determine the graph of h(x), we will apply the transformations to the graph of f(x) one at a time. Let's begin by translating the graph of f(x) to the right 3 units. To do so, we add 3 to the x-coordinate of the points on the graph of f(x).
Next, we will stretch the resulting graph vertically by a factor of 2 by multiplying the y-coordinate of the points on the graph by 2.
Finally, we reflect the graph in the x-axis by multiplying the y-coordinate of the points on the graph by -1.
Now that we have the graph of h(x), we can see that it matches the graph in option B.