Graphing Techniques

Homework Statement

I am in College Algebra. We are going over Graphing Techniques. I'm pretty sure I understand it, but my teacher is confusing me (and barely speaks English, so it's tough asking questions). Anyways, the problem given is . . .

f(x) = (x-1)3 +2

and we are supposed to show variations of this problem using what we know about shifting, stretching, reflecting, etc.

The directions tell us to start with the basic form of the problem, in this case being x^3

My question is that my teacher tells us (and expects on the test) to show and graph the different variations of this equation. Such as

y1= x3
y2= x3 + 2
y3= (x-1)3 + 2

First off, I don't understand why we are supposed to do this, and my proficiency in math is primarily based on understanding the why.

Secondly, the book does not provide any information as how to do this. In fact, the only answer provided to this problem is the graph of (x-1)3 + 2 and nothing else. My textbook is Sullivan Algebra8 along with MyMathLab.

Homework Equations

Other problems and the answers expected by my teacher. Again, answers in the book only provide graphs of the original question.

f(x) = (square root of)(x-2)
y1= (sq. rt)x
y2= (sq rt)(x-2)

f(x) = (sq rt)(-x) - 2
y1= (sq rt)x
y2= (sq rt)(x) - 2
y3=(sq rt)(-x) - 2

f(x)= -(x+1)3 - 1
y1= x3
y2= (x+1)3
y3= -(x+1)3
y4= -(x+1)3 - 1

The Attempt at a Solution

In pretty much guessing, I got most of these right. But I still don't understand why I got them right. And I even moreso don't understand why the book is not showing any of this . . .

Any help would be great. The test is Friday.

Answers and Replies

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Mark44
Mentor

Homework Statement

I am in College Algebra. We are going over Graphing Techniques. I'm pretty sure I understand it, but my teacher is confusing me (and barely speaks English, so it's tough asking questions). Anyways, the problem given is . . .

f(x) = (x-1)3 +2

and we are supposed to show variations of this problem using what we know about shifting, stretching, reflecting, etc.

The directions tell us to start with the basic form of the problem, in this case being x^3

My question is that my teacher tells us (and expects on the test) to show and graph the different variations of this equation. Such as

y1= x3
y2= x3 + 2
y3= (x-1)3 + 2
y1 is your untransformed function: the one you start with. If you graph y2, you will see that this graph is the translation up by 2 units of the graph of y1.
y2 is the translation by 1 unit to the right of the graph of y2. Relative to y1, the graph of y3 is the translation up by 2 units and to the right by 1 unit.

First off, I don't understand why we are supposed to do this, and my proficiency in math is primarily based on understanding the why.
Why you are supposed to do this is to help you learn to recognize translations, reflections, and stretches/compressions, relative to untransformed functions.
Secondly, the book does not provide any information as how to do this. In fact, the only answer provided to this problem is the graph of (x-1)3 + 2 and nothing else. My textbook is Sullivan Algebra8 along with MyMathLab.
I'm not familiar with this text, but if it's as you describe, I'm not impressed with it. It should at least summarize translations -- y = f(x - h) and y = f(x) + k; reflections -- y = f(-x), y = -f(x), and y = -f(-x); stretch/compressions -- y = cf(x) and y = f(kx), for a group of fairly well-known functions. It should also have examples of functions that use two or more of the preceding transformations.

Homework Equations

Other problems and the answers expected by my teacher. Again, answers in the book only provide graphs of the original question.

f(x) = (square root of)(x-2)
y1= (sq. rt)x
y2= (sq rt)(x-2)
You should recognize this as a lateral translation (a shift).
f(x) = (sq rt)(-x) - 2
y1= (sq rt)x
y2= (sq rt)(x) - 2
y3=(sq rt)(-x) - 2
You should take care of the reflection first, and then the vertical translation. If you do them out of order, you can sometimes get the wrong graph.
f(x)= -(x+1)3 - 1
y1= x3
y2= (x+1)3
y3= -(x+1)3
y4= -(x+1)3 - 1
Again, you should take care of the reflection first. You did things in this order: translate left 1 unit, reflect across the x-axis, translate down 1 unit.

The right way to do things is to first deal with stretches or compressions, then reflections, and finally translations.

The Attempt at a Solution

In pretty much guessing, I got most of these right. But I still don't understand why I got them right. And I even moreso don't understand why the book is not showing any of this . . .

Any help would be great. The test is Friday.