Transformaciones de funciones

Desplazamientos verticales de gráficas.

Suponga que c>0.

Para gráficas:

  • y=f(x)+c, desplace c unidades hacia arriba la gráfica de y=f(x)
  • y=f(x)-c, desplace c unidades hacia abajo la gráfica de y=f(x)

Desplazamientos horizontales de gráficas.

Suponga que c>0.

Para graficar:

  • y=f(x-c), desplace la gráfica y=f(x) a la derecha c unidades.
  • y=f(x+c), desplace la gráfica y=f(x) a la izquierda c unidades.

Estiramientos y compresiones verticales.

Se usa la gráfica de f(x) = x² para trazar.

  • g(x) = 3x². La gráfica de g se obtiene al multiplicar la coordenada y de cada punto sobre la gráfica de f por 3. Es decir, para obtener la gráfica de g se alarga verticalmente la gráfica de f por un factor de 3. El resultado es la parábola más estrecha.
  • h(x) = x²/3. La gráfica de h se obtiene al multiplicar la coordenada y de cada punto sobre la gráfica de f por 1/3. Es decir, para obtener la gráfica de h, se acorta verticalmente la gráfica de f por un factor de 1/3. El resultado es la parábola más amplia.

Estiramientos y compresiones horizontales.

Si f(2x) se acorta horizontalmente la gráfica de f(x) por un factor de 1/2.
Si f(x/2) se alarga horizontalmente la gráfica de f(x) por un factor de 2.

Reflexión de gráficas.

¿Cómo se hace para obtener las gráficas de y = -f(x) y y = f(-x) cuando se conoce la gráfica de y = f(x)?

  • La coordenada de cada punto sobre la gráfica y = -f(x) es simplemente el negativo de la coordenada de y = f(x). Por lo tanto, la gráfica deseada es la reflexión de la gráfica de y = f(x), en el eje x.
  • Por otro lado, el valor de f(-x) es el mismo valor de y = f(x) en -x. Por lo tanto, la gráfica deseada es la reflexión de y = f(x) en el eje y.

3 ways to let your audience know you published something new

There are two big mistakes every blogger has made in the beginning. 1) Start the email list later. 2) Never announce they’ve published something new to the people who already want to know about new posts.

That’s crazy! We often are so obsessed with getting more eyes to read our stuff and we’re tracking new visits every day but we forget to actually tell the people who already said they want to hear from us that we published something new.

That madness has to stop. You don’t need more eyes to read you, you need to pay attention to the ones that are already doing it.

I’ve been making this mistake for so long. I started my email list this year (english list, I mean, the spanish list started on 2012) and I always thought that the next post would give me more people and the next one, too, oh, but wait to publish the next one… and besides being a tiring task, it’s greedy, in its purest form.

I focus my last few years in getting more people but my content suffered a little bit. I was always trying to write the next great thing but the people who were already my audience got a lot of inconsistent posts about a lot of topics. And, let’s say I wrote something about Social Media today and they signed up but the next post they’d get was about this restaurant and the next one was about this event and the next one was about this prize and they’d had to be really patient to get another Social Media post, the main topic they signed up in the first place.


When I figured out that was my mistake, I knew exactly how to fix it. By focusing on the people who were already on my list, to write for them, to serve them, to build blog series for them.

But here’s the deal, there’s a lot of people out there who hasn’t realise they’re chasing the idea of getting more readers instead of focusing in their own audiences. And I thought it was a great opportunity to talk about how we almost always forget there are people who are waiting to know about us, already.

Your audience followed you somewhere for some reason. Your audience signed up your newsletter for some reason. Your audience added your blog on a syndication tool for some reason. Don’t ever forget about them.

Instead, everytime you published something new, try to tell them you wrote a post for them.

Don’t worry if you lost time or if you’re starting to let them know a little bit later. The only thing that matters is that, from here, you’re focusing on the right thing: your existing audience.

The 3 ways to let your audience know you published something new are, basically, 3 features I found on WordPress for this purpose. But it doesn’t mean you can’t find a way to add it to your own site if you don’t use WordPress. Just be creative enough to add them or improve them!

3 ways to let your audience know you published something new

Connect your Social Media channels with Publicize.

Publicize is a built-in WordPress feature to help you spread the word on your Social Media channels every time you publish a new post.

And yes, you can totally share your new posts on Social Media. And there are 2 reasons for doing so (I bet there are more but if you’re doubting of doing so, you only need these 2).

  1. If people followed you on Social Media is because they’re interested in you. This interest could be you’re friendly or you’re smart or you’re creative or you’re, you guessed it, interesting. There’s a big chance they want to know what you have to say. You might won’t get them all to read you but I bet some of them will!
  2. We’re living in the post-Google-Reader era. “Having a blog” is no longer that common so there’s a big chance they don’t know you write one. Publishing your new posts on your Social Media channels will give them a heads up!

You can connect your Social Media channels to Publicize right here:

Just click in the “connect” button of the social network you want to publicize your posts in and follow the instructions. 🙂

If you’re using WordPress.ORG, visit your dashboard and go Settings > Sharing to find the Publicize options.

No subscription box? No problem! Add a form instead.

I’ve heard a lot of people claiming that the abscense of subscription boxes is a problem in my platform. I only think the problem is creativity. You always have to do the most you can with the tools you have. If you really need a subscription box and can’t add one, add a form where the only field you’re asking is the email or email and name.

In every page or post, there are 4 buttons above the editor page. Those are Add Media, Add Poll, Add Contact Form and Add Location. Well, click on Add Contact Form and create a form with a single field and you’re done with your new subscription box.

For example, if you leave me your email address in this box, you’ll be the first to know when I publish something new. I bet you won’t want to miss it!

Of course you’ll have to do something extra to contact the emails you get, later. But it’s as easy as eating a pie. You just have to download a .csv archive from the Feedback menu. To do this go to the WordPress Admin Dashboard (both WordPress dot com and org), then Feedback, then Feedback again and you’ll find all your new friends’ email addresses and / or messages if you set up a contact form, instead.

You can download that email list as a .csv (at the end of the page there’s a button to do it). And import it to your favorite email marketing platform and you’re done. ❤

Just remember that they gave you their email address because they want to hear from you. Don’t be lazy and try to contact them every time you publish good stuff or every time you want to say hello (you can write a letter to them, you know, they’re people)!

Or, you know, add a link at the end of every post that goes to your newsletter landing page, too. Like this one 😉

Use all the “follow blog” WordPress features.

The WordPress original features help you to spread the word without any extra effort. People who follow you blog, get every post you publish immediatly. There are 3 features to help you with this task, try to activate them all.

The Follow Button.

The follow button is a widget you can add in any side bar you have on your blog. As I don’t have a side bar on my theme, I didn’t add it but it’s a tiny blue button that can be very useful if your main audience is already a WordPress user. The people who will be able to follow you by clicking that button will be the people who already have a WordPress profile.

If you don’t want to add a widget but want to create your own button and add it wherever you want, create your own Follow Button over here: it will only work with sites powered by WordPress (com and org).

And that’s it. Every person who clicks that follow button on your blog, will get every post you publish from now on.

The Follow Button on the corner for the non-Wordpress-users.

Now, we’re talking! What if your audience doesn’t use WordPress or doesn’t have a WordPress profile or doesn’t read that many blogs but, somehow, want to keep reading yours? Well, give them an opportunity to keep reading you by activating this follow button at the bottom right corner of your blog.

It will appear to every person who is not logged in on WordPress (the only way I can see it is if I visit my own blog in a new incognito window).

To turn this feature on go to the WordPress Dashboard > Settings > Reading > Follower settings > Logged out users and check the “Show follow button to logged out users” option box. 🙂 And that’s it.

What I like about this feature is that it is a button and doesn’t disturb anybody.


But the people who are really interested in follow your blog and get all your new posts, will click that button and see this:


Who said doesn’t offer subscriptions forms!? That’s a form! A more discrete one but a form at the end. I love this option because it offers the people who doesn’t have a WordPress profile a way to follow my blog and keep in touch with me. ❤

And you just have to click a checkbox to get this feature. They only have to click and leave their email address and automatically will be notified every time you publish again. ❤

The follow blog widget.

This is the most popular follow feature WordPress has. It’s a beautiful combination of the two above. You add the widget in your side bar and it will detect if the person reading you is logged in or logged out WordPress. If logged in, they’ll see a follow button. If logged out, they’ll see a subscription form to leave their email.

I’ve put mine at the very end of the page.


Activating these 3 follow blog WordPress features was a permanent solution for me. After activating it, I had to do nothing but publishing to let my audience know I published something new. Which is perfect because you know how much I love permanent solutions.

Now, I can focus on creating content for my existing audience knowing that any new person reading me will find a way to be part of my audience without me tricking them to be so and I can focus on sending emails to my existing audience who subscribed to my newsletter (I need to improve my skills on this, but I’ll tell you that story later).

If you want to be the first one to read my next post, join my newsletter here:

ZEN PENCILS » 186. My spirit is a roaring sea

I’m born to ride the crashing wave
To conquer each challenge anew
Ready to fight, forever brave
Ready to face a foe like you

Out from the shadows you appear
And steal me from my sheltered home
The day has come, the day we fear
Now I must face you on my own

You have me now, you didn’t ask
Lust and greed your only reason
But all that strength is just a mask
Stripped away, I see you demon

Your stench, your breath, your icy touch
Are not enough to close my eyes
And through the pain I know this much
I’ll never be your helpless prize

So we begin our violent dance
You have the rage, but no control
Don’t you realise there is no chance
Against the thunder of my soul?

Beyond the darkness into light
Is a warrior breaking free
Did you think that I would not fight?
I had no doubt, I count on me

Beyond my pain beyond my fear
For no one will I bend a knee
My heart is fire, my mind is clear
My spirit is a roaring sea

Source: ZEN PENCILS » 186. My spirit is a roaring sea

Website analytics for humans. What all those numbers mean?

When you’re looking at your website analytics, there’s more to look at than visits. It might be scary to even think to visit your WordPress stats at first but I assure you behind those numbers are people just like you and me.

I hope I can help you read your own stats in human language so you can understand better the humans behind your stats.

When you’re writing on a WordPress blog hosted at WordPress, you have the stats built in. It means, you get to see your numbers since day 1. If you’re hosting your WordPress blog elsewhere, you can add the Jetpack plugin and get the data as we all do here.

There are 2 places where you can see your data. You can see it at or at My Sites > WP Admin > Stats. But I recommend you highly to go to the first address because is more user friendly, it’s cleaner and easier to understand.

Understanding your data is primordial. There’s a lot that can happen from there. You can see if your ideas are working, if your work is being read and more important, if you didn’t know what to write about for certain and started publishing to see what happens, by learning your stats the right way, you can have your answer.

I’m gonna focus on a blog again, because this is the platform I use. If you use a blog + the Jetpack plugin, please, visit the stats in the right direction. If you’re using Google Analytics, there might be some things that you can’t see on your screen or there might be a lot of things you’re seeing on you’re screen that I won’t mention here. Don’t panic. Stay with the basic and look deeper into your own stats. 

So, visit and see your WordPress stats. Admire the numbers your work has made so far and let’s get into it!

We’re only gonna focus on the right column.


You’ll see a rectangle with 5 words: Insights, Days, Weeks, Months, Years. These are tabs, basically. When you click on one or another, you’ll see different numbers. By default, when you visit your stats through the link I gave you (, you’ll get the days tab. And you’ll be seeing the current day stats.

The second rectangle is a bar chart where you can see your visits/visitors from the last 30 days. The orange color shows your current day (it’s set by default to today but if you want to know the stats of three days ago, click the bar from three days ago and that’s the one that’ll turn orange). The difference between visits and visitors is that visitors are people and visits are the number of times they loaded one of your pages. See? Stats are more than visits. If you’re talking in your blog to people, I think you’ll pay more attention to the visitors number than the visits.


Right down the bar chart, you’ll see four words: visits, visitors, likes, comments. Every time you click one of these words, the bar chart is segmented to the word you chose. If you want to have a clear panoramic of how the visitors had been looking at your work, just clic the visitors word and, since you’re still in the day tab, you’ll see the numbers of your visitors for the last 30 days. One bar represents each day and the orange one is the current day.

Below this main bar chart, you’ll find another sections showing the stats for the day. These sections are: Post & Pages, Countries, Referrers, Clicks, Authors, Search Terms and Video.

Post & Pages show the most popular posts and pages of your site in the defined period. As we haven’t moved from the Day tab, this section will list the most popular posts and pages of the current day with its own views. This section is important because you can find what’s resonating with your audience.

I found out something funny, my popular posts and pages are from different categories, talk about different stuff and there’s no common topic in any of it. But, after study, really study every single one popular post and page found out that most of you really liked my super complete long articles. And that gave me some kind of direction (and also permission to not post every day to work in long depth articles).

Try to study to see what really resonates with your audience in this section or found another way to use it that’s helpful to you. These are YOUR stats, you created these numbers with your work, you can create better conclusions.


Countries show you where your visitors come from. It seems like a vanity stat but it’s not. With this section you could study the language of your platform or, if you’re writing a travel blog, maybe write a guide o special places of the countries that are visiting you.

The countries section helped me a lot of finally making the decision to make this a bilingual blog. I had a lot of spanish readers but every once in a while saw another country taking an yellow color in my map. As I started writing english posts, I saw how those yellow colored countries began to turn orange (the more intense the color, the more visits you have from that specific country). This way I knew turning my blog into a bilingual one was the way to go.

Again, this section can mean another thing to you, just study your stats and get your own conclusions.

Referrers show where your visitors come from. But this is not about countries anymore, this is about sites. How did they find your site? I cannot emphatize this section enough.

If you’re doing a good job, you’ll see here several search engines (like Google, Bing and Yahoo!) and if you’re doing a great job, you’ll find some bloggers out there are sharing your blog and if you’re doing an outstanding job, you’ll see social media sites here, meaning that people who read your stuff are sharing it on their profiles and their audience is clicking to read you. But if you’re doing a beyond the sky amazing job, you’ll see all three and maybe more.

If you can study where your visitors came from, it will give you more ideas of where more potential readers could be. You need to know where your readers come from. And also, where they’re going when they left your site.

Clicks is the section that shows you where your visitors go when they left your site. This is also very important because if you know where they’re going, you know what they’re looking for. I need to be clear here and say this section only shows the clicks that are made to get out of your site. If you linked to another blogpost or if you put a popular posts widget on your sidebar and a visitor clicks it, it will not show here, or anywhere in the WordPress stats. This section only shows the clics that made people go away from your site.

Authors is the section where you can see how many posts per author had been written. This is specially helpful if you have a collaborative blog. As you can see, I’m the only one writing here, so, this section is indifferent for me.

Search Terms is another section that is very very important. This could help you to actually know what was the urgency of the visitor who found you. Maybe, if you wrote about this urgency in a very short way and barely touched the subject and you see that very topic in this section, you might want to reconsider editing that post or write another more deepening post about it.

The search terms are super important because this tells you how somebody who didn’t know about your existence and didn’t have a direct link for your site actually found you and chose to visit you.

Video shows how many reproductions your video hosted at WordPress had, but, as you can see, I host my videos on my Youtube account so I can’t really see how this section works because I have it blank.

If you visit the other tabs, the ones in the first rectangle, you’ll get the same exact sections we just study. Obviously, the numbers will change, you’ll get to see more numbers and more patterns when you’re numbers accumulate in larger periods.

Do you want me to be honest about this? Don’t loose so much time on the days or weeks tabs, unless you’re looking for something specific (like one day of promotions or what happened to that lovely christmas post or so), study your numbers with the months and years tab. You’ll get to have more clear messages from your audience when you take the time to study a larger period.

And if you are patient enough to keep blogging for a full year, assuming you started on January because we, humans, like to start things on January, WordPress will send you a really pretty infographic at the end of every year so you can admire your work and give yourself a little pat on the back. Seriously, this summary is so pretty, you should keep blogging for more than a year.

WOWOWOW! Wait a minute, Katherine! What about the Insights tab?

Well, that’s my favorite. The insights page has other type of data but it’s shown in an human way, so you just can look at it and take action. You can see your activity, your best hour to post, the activity your last post had, etc.

The insights page is the best way to get a full picture of your current work. And the resume at the end of the year is the best way to get the big picture of your whole year work. Which is amazing to remind ourselves that we, in fact, did work the last 365 days.

The insight pages has different sections: Posting activity, most popular day and hour, all time posts, time and visitors, today’s stats, latest post summary, comments, followers, tags & categories, publicize.


Posting activity shows in a very graphic way the amount of published posts in the last year! Every square is a day and every column is a week. Every bunch is a month and the more intense the color, the more posts were published that day! Are you aiming for consistency? This is the right section for you. If you intended to write 1 post per day the whole year, you’ll be able to see that here. If you have a writing day and published 3 posts every thursday, you’ll be able to see a straight horizontal line of dark blue squares on this section.

As you can see, I started publishing one article per day on January (because I’m a human and I like to start things on January, hahaha) but then analyze my popular posts and realized you’re really into my super long articles, so I step back a little and started publishing weekly, instead.

But, please, don’t stay with my conclusions for your own site. Really try to find the message your own audience is telling you and build a plan based on that!

Most popular day and hour shows the best hours in your blog. But this is a number that you really didn’t work for. You just happen to track that number and have it there. While you can’t do anything to change it, you might try having something new posted one hour before that time and day for the next week. Just to see what happens.

For example, if your best hour happens to be Monday at 2pm, try to publish something new on Monday at 1pm. That way, if stats are correct, you might be waiting new visitors at 2pm and have something new for them.

All time posts, time and visitors tells you the summary of all of your work. This is one of my favorites, too! This will help you realize that, in fact, you’re doing something amazing. If you have the feeling that you don’t do anything and time goes by and you achieve nothing, this section will help you realize you’re wrong. It will tell you how many posts have you published, how many of your pages are being visited, how many people read your work and it also will tell you what days you got the more visits ever!

Maybe you’ll want to check that day specifically on the days tab to see what did you do different on that day and try to replicate it to get more visits some day soon, huh?

Today’s stats is the resume of the bar chart we saw at the Days tab. You can access directly to that chart clicking on the icon on the upper-right corner.

Latest post summary is a cool way to see quickly what’s been happening with your last published post. Basically, it will tell you how many views, likes and comments has had. But, as you know, I highly recommend not to stress about quickly numbers and go for the long run.

Comments don’t show the comments on your blog but shows who has commented the most and also shows a list with the most commented posts. This is very helpful to realize who are your more loyal and active readers. I don’t know about you, but I think it will be a kind gesture to treat the first 10 people on this list as VIP members of your site. Gift them a little something like your last ebook or consider to send a postcard in the holidays. Again, this whole stats conclusions are up to you and only you!

Followers, no surprises here, shows you the last followers your site has got. If you’re a user then, you’ll get to see the WordPress followers and the email followers. If you’re a user, you’ll get to see the email followers only. These people subscribed to your site and are willing to get every single post you publish in the future. Every time you hesitate about doing something, think there are people waiting to hear from you! Keep it going!

Tags & categories, contrary to the belief, don’t show the most used tags and categories. Tags & categories section shows the most viewed categories and tags! This is also very important for analysis purposes. If you took the time to categorize and tag all your posts, you get to see what’s really resonating with your audience.

For example, if you published about salvadoran food and salvadoran recipes and tagged them correctly, you’ll see what’s better for your audience. If salvadoran recipes are the most viewed, maybe you want to keep sharing recipes or start making videos while you cook. Again, the conclusions are yours and you get to decide what to do when you find out what your data means.

Publicize shows the social media sites that automatically update with a link to your more recent post when you click the publish button.

I bet you have a clearer vision on what’s happening in your site right now. And maybe, you already have a few ideas to pursue your goals thanks to the analysis you did on your own site.

But let me remind you something, you still can crossover some sections to have even more data. For example, publicize and referrers, likes and visits, followers and visitors, categories and search engines. If these grow at the same levels, you have found more patterns to study on your site!

I just want you to know something before I go for this day. None of these conclusions, data, patterns or numbers happen without work. If you don’t put something out first, you’ll have no data to analyze. You need to start writing and publishing so you can be able to analyze and repeat to grow. So don’t hesitate about it and start publishing!

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