Piecewise functions

[The Velociraptors]

This was on a friend’s precalculus packet:
What I initially thought to do was substitute the restriction’s number into the respective equation, but my final graphed piecewise didn’t match the key. For example, for the first equation, I substituted 1 like this: (-1)^3+1 and got the coordinate (1,0). Then I followed the restriction, put an open circle on the point (1,0) and and arrow pointing to everything on the left of (1,0). I did the same thing for other equations, substituted the restriction number and got the coordinate. But since the key did not match this, I had a thought that my process doesn’t work for certain functions. So should I make multiple tables instead? (Even though I made a table, my final product didn’t match the key still, so maybe it’s an error on my part). For example, for the first equation, should I pick x values less than 1 after I get (1,0) and place the open circle there? Then for those values less than 1, do I substitute it back into the equation? For example, if I pick x value -1, do I substitute it like this: -(-1)^3+1 to get the y-value? Does anyone have a better approach to this? And if possible, can you direct me towards a good resource?

 

[Dr.Peterson]

This was on a friend’s precalculus packet:
View attachment 28865What I initially thought to do was substitute the restriction’s number into the respective equation, but my final graphed piecewise didn’t match the key. For example, for the first equation, I substituted 1 like this: (-1)^3+1 and got the coordinate (1,0). Then I followed the restriction, put an open circle on the point (1,0) and and arrow pointing to everything on the left of (1,0). I did the same thing for other equations, substituted the restriction number and got the coordinate. But since the key did not match this, I had a thought that my process doesn’t work for certain functions. So should I make multiple tables instead? (Even though I made a table, my final product didn’t match the key still, so maybe it’s an error on my part). For example, for the first equation, should I pick x values less than 1 after I get (1,0) and place the open circle there? Then for those values less than 1, do I substitute it back into the equation? For example, if I pick x value -1, do I substitute it like this: -(-1)^3+1 to get the y-value? Does anyone have a better approach to this? And if possible, can you direct me towards a good resource?

Please show your work and your graph. I can help a lot more when you do so (as we request in the guidelines).

It's correct to substitute the boundary values into the relevant expressions to find end points of "pieces" of the graph; but you also need to draw the correct graph within each interval, which you don't say much about other than describing the process of plotting points. A mere "arrow pointing to the left" sounds as if you were just trying to show the domain.

 

[Otis]

Hi. This graphing exercise has to do with basic graph shapes of common equation forms (eg: cubic polynomials, quadratic polynomials, absolute-value expressions).

To start, do you know how to graph these equations separately?

y = 1 - x^3

y = (x - 5)^2 - 2

y = 2|x - 1| - 2

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[The Velociraptors]

Please show your work and your graph. I can help a lot more when you do so (as we request in the guidelines).

It's correct to substitute the boundary values into the relevant expressions to find end points of "pieces" of the graph; but you also need to draw the correct graph within each interval, which you don't say much about other than describing the process of plotting points. A mere "arrow pointing to the left" sounds as if you were just trying to show the domain.

I apologize, it was a digital copy, so I did’t know how to write on it. I guess I was thinking that the first equation had to be a horizontal line. Would I only have a horizontal line when there is a singular numerical value?

 

[The Velociraptors]

Hi. This graphing exercise has to do with basic graph shapes of common equation forms (eg: cubic polynomials, quadratic polynomials, absolute-value expressions).

To start, do you know how to graph these equations separately?

y = 1 - x^3

y = (x - 5)^2 - 2

y = 2|x - 1| - 2

?

yes, I can graph them separately. Oh, then I just follow the restrictions after I graph them separately.
(I think the first one is a cubic, second is a quadratic, third is an absolute value function).

 

[Dr.Peterson]

I apologize, it was a digital copy, so I did’t know how to write on it. I guess I was thinking that the first equation had to be a horizontal line. Would I only have a horizontal line when there is a singular numerical value?

You can attach a picture of your work on paper, or use whatever tool you like to draw digitally.

But, yes the graph is a horizontal line only if the equation is y = c, not y = 1 - x^3. Do you know what that looks like? You can plot a few points, but as Otis said, knowing the shape is the main skill you should be exercising.

 

[The Velociraptors]

You can attach a picture of your work on paper, or use whatever tool you like to draw digitally.

But, yes the graph is a horizontal line only if the equation is y = c, not y = 1 - x^3. Do you know what that looks like? You can plot a few points, but as Otis said, knowing the shape is the main skill you should be exercising.

Well, I understand now that it must be a portion of a cubic due to the restriction. Do I just pick x values less than one and substitute them into the equation after I substitute in the boundary point?

 

[Dr.Peterson]

Well, I understand now that it must be a portion of a cubic due to the restriction. Do I just pick x values less than one and substitute them into the equation after I substitute in the boundary point?

Correct. Though, again, you will be using knowledge of the shape to fill in between a few plotted points; for example, if you think about shifting (translation) and reflection, you can see which point looks like the origin of y = x^3, and graph it appropriately. If you just plotted points, you might easily make wrong guesses.

 

[Otis]

yes, I can graph them separately. Oh, then I just follow the restrictions after I graph them separately.

Yes, make a rough plot of each separately. Yes, the given intervals tell you which section (piece) from each of your three graphs that you'll then plot together on a final graph (the 'piecewise' graph).

Then you determine whether an open or closed dot goes at each endpoint.

Were I to do it, I'd wait until I'd finished the graph before stating the domain and range of the given function.

I totally agree with Dr Peterson about first knowing basic shapes. This exercise is exactly what I'd expect to see from a student who had just spent a couple months learning/reviewing basic graph shapes, plotting them, reflecting them across axes, compressing or stretching, shifting vertically, horizontally.

If you haven't done those sorts of things, and you're working by hand, then make a table of values for each interval, to plot your three graph sections. Use a lot of x values, if you're uncertain of the shape and where it lies within each interval.

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[Otis]

For anyone interested in reviewing function graphs at the precalculus level, there are many online resources. Here's one site with lots of video lessons and demonstrations.

mathispower4u - Algebra 2

www.mathispower4u.com

Locate the sub-section 'Function Transformations, Determining Transformations of Functions'. These examples are relevant:

https://youtu.be/hvyH-WJtMpc

https://youtu.be/9wcALZGyce0

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[Deleted member 4993]

@Jomo do you have video/lesson regarding "function graphs"?

 

[pka]

[imath]w(x)=\begin{cases}-x^3+1 & x\le 1 \\2|x-1|-2 & 1<x<3\\ (x-5)^2-2 & 3\le x \end{cases}[/imath]
I reposted the question so that I can have it to see.
Can you answer these limits?
[imath]1)~\mathop {\lim }\limits_{x \to - \infty } w(x) = {\kern 1pt} ? [/imath]
[imath]2)~\mathop {\lim }\limits_{x \to 1^{{\large\bf-}} } w(x) = {\kern 1pt} ?[/imath]
[imath]3)~\mathop {\lim }\limits_{x \to 1^{{\large\bf+}} } w(x) = {\kern 1pt} ? [/imath]
[imath]4)~\mathop {\lim }\limits_{x \to 3^{{\large\bf-}} } w(x) = {\kern 1pt} ? [/imath]
[imath]5)~\mathop {\lim }\limits_{x \to 3^{{\large\bf+}} } w(x) = {\kern 1pt} ? [/imath]
[imath]6)~\mathop {\lim }\limits_{x \to \infty } w(x) = {\kern 1pt} [/imath]?
If you can answer these limits then the questions about domain and range should be answered.
If not, ask about your doubts.

 

[Steven G]

@Jomo do you have video/lesson regarding "function graphs"?

Subhotosh,
I have videos for all topics in Arithmetic/PreAlgebra, Basic Algebra, Intermediate Algebra, Trigonometry, Pre Calculus, Geometry, Statistics & Probability, Calculus 1, 2 & 3, Linear Algebra, Differential Equations, Probability Theory, SAT, ACT and Math Tricks.

Yes, I have videos on graphing various type of functions.
Go here and choose which type of graphs you want to learn about.

 

[The Velociraptors]

[imath]w(x)=\begin{cases}-x^3+1 & x\le 1 \\2|x-1|-2 & 1<x<3\\ (x-5)^2-2 & 3\le x \end{cases}[/imath]
I reposted the question so that I can have it to see.
Can you answer these limits?
[imath]1)~\mathop {\lim }\limits_{x \to - \infty } w(x) = {\kern 1pt} ? [/imath]
[imath]2)~\mathop {\lim }\limits_{x \to 1^{{\large\bf-}} } w(x) = {\kern 1pt} ?[/imath]
[imath]3)~\mathop {\lim }\limits_{x \to 1^{{\large\bf+}} } w(x) = {\kern 1pt} ? [/imath]
[imath]4)~\mathop {\lim }\limits_{x \to 3^{{\large\bf-}} } w(x) = {\kern 1pt} ? [/imath]
[imath]5)~\mathop {\lim }\limits_{x \to 3^{{\large\bf+}} } w(x) = {\kern 1pt} ? [/imath]
[imath]6)~\mathop {\lim }\limits_{x \to \infty } w(x) = {\kern 1pt} [/imath]?
If you can answer these limits then the questions about domain and range should be answered.
If not, ask about your doubts.

I can answer stuff about limits based on a graph, but I don’t understand how to think about it within the domain of a piecewise functions

 

[Otis]

Can you share your piecewise graph for this exercise?

?

 

[Otis]

choose which type of graphs you want to learn about

I searched for 'piecewise' but that word's not on the page. Do you have any examples of graphing such, using open and closed dots?

 

[Steven G]

I can't find it either. It must be under limits and continuity in my Calculus videos.

 

[mmm4444bot]

S'ok, Jomo -- it's been a week. I'll post some graphs.

The graph of -x^3+1 is the graph of x^3 reflected across the x-axis and shifted up one unit. We can see the x-intercept by inspection, then by symmetry we know (-1,2) also.

The graph of 2|x-1|-2 is the graph of |x| compressed horizontally by a factor of 2 and shifted right one unit and down two units.

Also, y=a|x-h|+k is vertex form, and that shows our vertex point (h,k) is at (1,-2). The intercepts are easy to calculate, so plotting the vertex with points (0,0) and (2,0) is another way to graph the absolute-value function.

The graph of (x-5)^2-2 is the graph of x^2 shifted right five units and down two units.

y=a(x-h)^2+k is also a vertex form, for parabolas. It shows our vertex (h,k) is at (5,-2). Calculating (4,y) and (3,y) and then using symmetry to also plot (6,y) and (7,y) is another way to graph the parabola.



Consider the given domain intervals, for the piecewise graph. Use an open dot for endpoints not included in the interval and a closed dot for those that are.



Some people use arrow heads to indicate that the graph continues forever upward at each end, some people don't.

Now we can write the piecewise function's domain and range, by examining the graph.

By the way, I hope the OP's friend isn't expected to use the given grid.

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