This article I am going to completely splurge what comes to my head. Maybe letting you see my thought processes will help you in super detective detectivity work.
Here are some basic notes to have in your mind when you look at these types of questions.
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1. x + 3 will always be positive inside of absolute brackets. regardless of the value of x itself
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2. we can see, however, the - sign sitting outside the absolute brackets so we know that regardless of any of that, the value of x + 3 will result in a negative value.
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3. Then we have the easy peasy, all by itself, subtraction of value positive 2.
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From looking into the equation, and daydreaming about math powers...
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we can envision among ourselves the revelation
back to real life me, jotting things as I think and then ... nothing .... where was I going ...
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WHICH ART IS HUMAN GENERATED?!?!?!
Ok, back to track. Now, personally, I really do sit and imagine what the graph is going to look like.
THEN, I grab a pencil and paper.
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THEN, I remember the Internet!
BUT I IMAGINE
Google and ChatGPT are not there for you
When you are alone at your desk doing
tests and other work
Regardless. Keep this in your back pocket if you are presented the opportunity. For now, we can verify that I am telling you the truth.
FROM CLOSING EYES AND EXPERIENCING VISUALIZATIONS WE CAN CLEARLY SEE AND UNDERSTAND
The value of this function never sees the chance to be a positive value along the y -axis. We have absolute brackets, but with a negative in front of them we still result in a negative value.
The largest value you will get for g(x), or y-value, is going to be when the value of x + 3 is equal to 0.
Which puts us at x = -3
Then, if we plug in x = 0 for ease we see the y-value at
g(x) = - | 0 +3 | - 2
= -3 - 2
y = -5
Remember g(x) and y are the same in this graph and equation.
Let us make a nice shape and find x when y = -10!
If we plug in
-10 = - | x +3 | - 2
-8 = - | x + 3 |
But wait ....
There are two x values to satisfy that equation!
If I have one x value, and there are two y values as a result, the line must be a dome shape. But there is one location of x for which there is only one value of y, and remember, that is also the greatest value of y.
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As we stated earlier, the greatest value that y can behold is when the value of x + 3 is zero. From that point in either positive or negative x values, y value never rises above -2.
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Rinse with some more of
-8 = - | x + 3 |
The two x-values that satisfy this as true are x=5 and x=-11
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So when x = -3, y = -2. That will be the tip we notch onto our coordinate system map that is really just two lines.
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We also now know that, when x = 5, y = -10
Another dot to draw!
And we know that, when x = -11, y = -10
Another dot to draw!
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Now draw two lines. One starting at (5, -10) and progressing towards (-3, -2).
Then the second line is drawn from (-11, -10) with progression towards (-3, -2)
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Now you can see the shape!
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So, the transformative properties we have gathered.
1. This graph reflects down across the x-axis. The negation of the absolute value. So, we know how it points.
2. The dome is going to shift to the left 3 units, because we are subtracting 3 from x.
3. The dome is going to be shifted down 2 units on the y-axis because of the whole number subtraction at the end.
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What is a vertex??
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A vertex is the point where our graphed line changes direction.
We saw this to be true at x = -3.
Because prior to reaching this x-value, the value of y, though negative, was increasing. One the line surpassed the x-value of -3,
the y-value then began to decrease.
(-3, -2)
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The domain, when working with absolute value functions, will be.
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any number that keeps the denominator from totaling 0
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In our example, the denominator will never reach the value of 0.
This our domain is all real numbers.
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This is the range I have. However, I have to make sure this is correct.
So, to be continued.
( -∞ , -2]
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