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It might help to start by looking at the parent function: f(x)=|x| For this function, what is the domain? That is, what numbers can you put in for x?
For example, can you take the absolute value of a negative number? Of a positive number? Of zero? Are there any numbers for which you cannot find the absolute value?
If you think about it, you will see that you can put any number in for x, and you will be able to find its absolute value. So, the domain of the function is (- infinity, + infinity) , since all numbers can be used. Now, for the range, think about the y-coordinates of the points on your graph. The point (0,0) is the lowest point, or minimum of your graph. So, the lowest y coordinate any of the points on your graph has is 0. So, since all of the y-coordinates of the points on your graph are greater than or equal to 0, the range of the function is: [0, + infinity) .
The graph touches the x-axis at the point (0,0), so that is your x-intercept. You will see that the graph also touches the y-axis at (0,0), so (0,0) is also your y-intercept. The graph is decreasing (slope going downhill) on the interval from (- infinity, 0) and increasing (positive slope) on the interval from (0, + infinity). So, those are the characteristics of the parent function. The characteristics of the translated function can be determined by using the characteristics of the parent function and then performing the translations given in the equation of your translated function.
What your translated function tells you is this: the " - B", inside the absolute value sign tells you that you are shifting your function right B units. The "+ C" at the end tells you that you will also shift the graph up vertically C units (assuming C is positive). So the vertex of your new graph is at (B,C), rather than at (0,0). (B,C) is the minimum of your graph, so that means that the y-coordinates of the points on your graph will all be greater than or equal to C. Therefore, your range is [C, + infinity). There will be no x-intercept, since the lowest point of the graph is now C units above the x-axis. The y-intercept will be (0, C). Since the graph has been shifted right B units, it will be decreasing on the interval from (- infinity, B). It will be increasing on the interval from (B, + infinity). I am assuming that A is positive. The effect the "A" has on the graph is that it makes the slope steeper (I am assuming that A > 1; if 0 < A < 1, then the slope would be less steep.
OHHHHHH that makes so much sense THANK YOU SO MUCH YOU'RE AWESOME!
I hope that was helpful :)
It Really Was THANK YOU THANK YOU THANK YOU
Oops ... I made an error on the y-intercept!
Let me correct that!
You get your y-intercept when x = 0 So, A |0 - B| + C = A|-B| + C = AB + C ; So, your y-intercept is at (0, AB +C)
I apologize for that! Your minimum would still be at your vertex, which is (B, C)
And it would still decrease from (- infinity, B) and increase from (B, + infinity)
ohhh ok i seee its cool Thanks Again!
You are welcome! If you would like, I could also draw a graph of the translated function?
oh really ? That would be great
Sure! Just a moment, please. :)
I didn't do a very good drawing, since I meant to make the graph look steeper. :)
And narrower, since that would be the case if A > 1
More like this:
That's better :)
So basically the parent function shifted right B units, up C units, and made A times as steep.
Yay ok i understand it now thank you so much for showing me how to do this and for graphing it cant thank you enough!!
You are very welcome! :D Have a good night!!!