Listed below are all the families of functions you have studied throughout the course. In a **minimum of 250 words** discuss the effects of *a*, *h* and *k* on each function. Describe any similarities or differences between the effects of *a*, *h* and *k*. Include graphs to support your statements.

Families of Functions:

Quadratic: 𝑦=𝑎,(𝑥−ℎ)-2.+𝑘

Exponential: 𝑦=𝑎∗,𝑏-,𝑥−ℎ..+𝑘

Radical: 𝑦=𝑎,-𝑥−ℎ.+𝑘

Rational: 𝑦=,𝑎-𝑥−ℎ.+𝑘

Michael14
Apr 30, 2015

#16**+13 **

A quadratic function, represented by the formula, y = a(x-h)^{2} + k, produces a parabola or a “U-shaped” function. Based off of the equation, the value of a determines the “steepness” of the parabola. In addition, the value of a also tells us whether or not the curve opens downwards or upwards. For example, if a is negative, the parabola will open downwards, while if a is positive, the parabola will open upwards. In addition, the variable h in the equation is the horizontal (x-axis) value, and the variable k is the vertical (y-axis) value. In the function equation, as h increase, the parabola shifts to the right. For example, if h=2, that means the parabola is shifted to the right 2 units along the x-axis. On the other hand, as k increases, the parabola shifts upwards along the y-axis. If the k value was negative, however, the parabola will shift downwards along the y-axis. For example, if k=-4, the parabola will shift upwards 4 units along the y-axis.

An exponential function, represented by the equation, y = ab^{(x – h)} +k, produces a line that experiences a consistent growth. In other words, the value of the mathematical function is proportional to the function’s current value. According to the equation, the value of a determines a couple things. For instance, if a is positive, the graph faces upwards, whereas if a is negative, the graph reflects across the x-axis and faces downwards. In addition, the value of b is generally known as the base of the exponent. If b is greater than 1, it is known as the “growth factor,” and if b is less than 1, it is known as the “decay factor.” The value of b must always stay positive. Also, in the exponential function, h represents the horizontal shift “h”-units. When h is positive, the function shifts to the right, whereas if the value of h is negative, the function shifts to the left. Finally, the value of k in the exponential function vertically affects the graph when k is positive and negative. If the value of k is positive, the graph will shift upwards “k”-units, and if the value of k is negative, the graph will shift downwards “k”-units along the y-axis.

A radical function, represented by the equation, y = a√x + h + k, produces a curved line. In this radical equation, the changes in variables a, h, and k are similar to the effects of changing parameters in the other functions mentioned. For example, the value of a results in a vertical stretch of the graph by a factor of a. In addition, the value of h determines the horizontal translation. If h is positive, the graph is shifted to the right “h”-units. However, if h is negative, the graph is shifted to the left “h”-units. Finally, the value of k, similar to the quadratic and exponential equations, determines whether or not the graph shifts upwards or downwards along the y-axis. For instance, if k=10, the graph will shift upwards 10 units along the y-axis.

Last but not least, a rational function, represented by the formula, y = a/(x-h)+ k, produces a hyperbola. According to the rational equation, the value of a affects the graph by increasing or decreasing the curve of the branches of the hyperbola. In other words, if a is a small value, the branches of the hyperbola will come very close to the asymptote. However, if a is a large value, it will be farther away from the asymptotes. Also, the variable h determines where the vertical asymptote will be. If h is positive, the vertical asymptote will shift to the right, whereas if the value of h is negative, the vertical asymptote will shift to the left. Finally, the variable k determines where the horizontal asymptote will be placed. For instance, if the value of k is negative, the asymptote will shift downwards. However, if the value of k is positive, the asymptote will shift upwards.

BrittanyJ
Apr 30, 2015

#1**+8 **

This is a repost question Michael,

You need to follow these instructions when you do reposts.

Melody
Apr 30, 2015

#2**0 **

hey michael. ill get right to it. this may take a while. ill tell u when im done :)

BrittanyJ
Apr 30, 2015

#3**+3 **

Melody, i dont care if its a repost, there is nothing wrong with that. i reposted it because i really need help with this problem.

Thank u so much Brittany! :D i have been working on this FOREVER. and its only one problem.

Michael14
Apr 30, 2015

#4**0 **

**You are right, there is absolutely nothing wrong with reposting. I encourage it.**

Just please do it the way you are asked otherwise we end up answering the same question several times and we simply do not have time for that.

If we answer your question twice it means that someone else does not get an answer at all and that is definitely not fair!

Melody
Apr 30, 2015

#5**+3 **

And Melody, im not trying to be rude. I reposted it for a reason, and it worked! Brittany decided to help me which was very nice of her. But u were so worried about the repost that u didnt even help me. WOW

Michael14
Apr 30, 2015

#6**0 **

I posted on your original question.

It is you who ignored me.

I Never criticized you for reposting.

It is to be encouraged.

I just pointed out how it is best to be done.

Melody
Apr 30, 2015

#10**+8 **

lol i like helping people EVEN IF they are lazy. Plus, michael is a nice guy so i will help him :)

BrittanyJ
Apr 30, 2015

#11**0 **

Maybe he is nice most of the time but he was rude to Melody.

Just do the graphs. Let him write his own commentary and exemplifications. The teacher will know it's not his own work, anyway.

Guest Apr 30, 2015

#12**+5 **

1.) calm down...

2.) i have already written 4 large paragraphs already and i dont want to just delete it because i spent a lot of work on it

3.) reread over #2

BrittanyJ
Apr 30, 2015

#13**+3 **

Excuse me asshole, but i dont know who u calling a LAZY A*S KID u dont know how much fuckin work i tried to put in this problem. So next time before assuming that im a LAZY A*S KID, try talking to me without being anonymous and maybe we can figure out this problem together, if ur not being a lil b***h.

Michael14
Apr 30, 2015

#15**+8 **

haha :D yah, i hope he reads it. and sorry Melody, this asshole has to learn.

Michael14
Apr 30, 2015

#16**+13 **

Best Answer

A quadratic function, represented by the formula, y = a(x-h)^{2} + k, produces a parabola or a “U-shaped” function. Based off of the equation, the value of a determines the “steepness” of the parabola. In addition, the value of a also tells us whether or not the curve opens downwards or upwards. For example, if a is negative, the parabola will open downwards, while if a is positive, the parabola will open upwards. In addition, the variable h in the equation is the horizontal (x-axis) value, and the variable k is the vertical (y-axis) value. In the function equation, as h increase, the parabola shifts to the right. For example, if h=2, that means the parabola is shifted to the right 2 units along the x-axis. On the other hand, as k increases, the parabola shifts upwards along the y-axis. If the k value was negative, however, the parabola will shift downwards along the y-axis. For example, if k=-4, the parabola will shift upwards 4 units along the y-axis.

An exponential function, represented by the equation, y = ab^{(x – h)} +k, produces a line that experiences a consistent growth. In other words, the value of the mathematical function is proportional to the function’s current value. According to the equation, the value of a determines a couple things. For instance, if a is positive, the graph faces upwards, whereas if a is negative, the graph reflects across the x-axis and faces downwards. In addition, the value of b is generally known as the base of the exponent. If b is greater than 1, it is known as the “growth factor,” and if b is less than 1, it is known as the “decay factor.” The value of b must always stay positive. Also, in the exponential function, h represents the horizontal shift “h”-units. When h is positive, the function shifts to the right, whereas if the value of h is negative, the function shifts to the left. Finally, the value of k in the exponential function vertically affects the graph when k is positive and negative. If the value of k is positive, the graph will shift upwards “k”-units, and if the value of k is negative, the graph will shift downwards “k”-units along the y-axis.

A radical function, represented by the equation, y = a√x + h + k, produces a curved line. In this radical equation, the changes in variables a, h, and k are similar to the effects of changing parameters in the other functions mentioned. For example, the value of a results in a vertical stretch of the graph by a factor of a. In addition, the value of h determines the horizontal translation. If h is positive, the graph is shifted to the right “h”-units. However, if h is negative, the graph is shifted to the left “h”-units. Finally, the value of k, similar to the quadratic and exponential equations, determines whether or not the graph shifts upwards or downwards along the y-axis. For instance, if k=10, the graph will shift upwards 10 units along the y-axis.

Last but not least, a rational function, represented by the formula, y = a/(x-h)+ k, produces a hyperbola. According to the rational equation, the value of a affects the graph by increasing or decreasing the curve of the branches of the hyperbola. In other words, if a is a small value, the branches of the hyperbola will come very close to the asymptote. However, if a is a large value, it will be farther away from the asymptotes. Also, the variable h determines where the vertical asymptote will be. If h is positive, the vertical asymptote will shift to the right, whereas if the value of h is negative, the vertical asymptote will shift to the left. Finally, the variable k determines where the horizontal asymptote will be placed. For instance, if the value of k is negative, the asymptote will shift downwards. However, if the value of k is positive, the asymptote will shift upwards.

BrittanyJ
Apr 30, 2015

#18**0 **

Wow! If you put half as much work into your maths homework as you do in cursing, you wouldn't need as much help.

You are one lucky little, snot-nosed-kid to have Brittany here to help you. No one else who could do the work would do it.

Brittney, just change some of the facts in your work. He’s too stupid to know, anyway. That way your work won’t go to waste, and you will teach the foul-mouthed, lazy-assed, brat a lesion.

hahaha

Guest Apr 30, 2015

#19**+8 **

no really, anon. cut it out. i understand u r bothered that i spent 2 hours on this large problem and it will be "wasted." In my opinion, this will not be a waste AT ALL! This willl help michael with his math grade or such. Besides... this helps my writing skills.... so in the end, we are both benefited. I love helping people who are nice to me. PERIOD.

BrittanyJ
Apr 30, 2015

#20**+3 **

Wow...u like to spend a lot of time to prove a d**n point, u love attention dont u? im sorry ur lonely bro. if u werent such a p***y b***h and had the b***s to create an account, maybe ill give u some respect. in the mean time, keep entertaining me with these HILARIOUS, ATTENTION HOR COMMENTS. Give me some more bruh, ur fuckin funny. and no wonder u decide to say this s**t on a math website where everyone can see it instead of facebook where nobody gives a f**k about what u say cus its all for attention. GOOD JOB FUNNY B***H! :D

Michael14
Apr 30, 2015

#22**+3 **

LOL. Seriously, this kid makes me laugh! i was laughing the whole time i was typing my comment! :D he just doesnt get it, hes like a fuckin 5 year old, HE LIVES FOR ATTENTION

Michael14
Apr 30, 2015

#24**0 **

That is an impressive piece of work, Brittany!

* I love helping people who are nice to me*.

Maybe so. But in this case you paid for his “niceness.” Consider, Melody has helped him too, and he now throws s**t at her. He’ll likely do the same to you, when you don’t have time to help in the future. It’s in his nature.

A scorpion and a frog meet on the bank of a stream and the scorpion asks the frog to carry him across on its back. The frog asks, "How do I know you won't sting me?" The scorpion says, "Because if I do, I will die too."

The frog is satisfied, and they set out, but in midstream, the scorpion stings the frog. The frog feels the onset of paralysis and starts to sink, knowing they both will drown, but has just enough time to gasp "Why?"

Replies the scorpion: "Its my nature..."

Guest Apr 30, 2015

#25**+3 **

1.) thank you so much, anon. that means a lot to me that people appreciate my work. However, michael and i are bonded with a very tight friendship so i highly doubt that he will do that.

2.) I love the story and yes in certain situations, that is true. there are some people out there who do mean things becasue of "their nature" but guess what? In my opinion, just because it is a person's nature to b mean, doesn't mean that they HAVE to be mean.

When I was young, I was in an environment where drugs, alcohol, s*x was just around the corner. But my dad divorced divorced my sick and twisted mother and found a new wife, my step mother. Because of where I was, I was headed towards those paths. At that time, it was in MY NATURE to do this terrible things. But, because of FREE WILL, i was able to separate from those paths and choose the right one. It led me to successful ilife i hae now.

The point in this is that it doesnt matter where u come from, and your decisions, actions, and words aren't really linked to our "nature." Instead, it links to our free will in this world.

This is my opinion... haters, hate all u want. but an opinion should be respected. Thanks :)

BrittanyJ
Apr 30, 2015

#26**+3 **

wow. i dont think u have ever had a girlfriend before. U ARE REALLY GOOD AT THIS. i have never met anyone that loves to seek attention. IM PRAYING FOR U BRAH. and im still laughin. u are using everything u know to try to make a point. GOOD JOB. but i encourage u to f**k off this website. im here for math help only, and not to deal with unfortunate fucktards like u. Btw that poem or some s**t, was irrelevent. and also notice that u have no friends right now to cover ur back and no body agrees with u. but if u insist, KEEP GOING BRAH. UR DOIN GREAT!! :D

Michael14
Apr 30, 2015

#27**0 **

That makes sense! and i am not what the anon thinks i am. and i am no way related to that story. And i know that sometimes i get VERY ANGRY with these kinds of people. And Brittany, i love that u have the strong heart, mind, soul, and courage to share this :)

Michael14
Apr 30, 2015

#28**0 **

And dude, pleeez get a life and do somthing useful with it. HAVE A BLESSED DAY

Michael14
Apr 30, 2015

#30**0 **

"have a blessed day!" lmfao! that's priceless. (i wonder wat anons thinking)

BrittanyJ
Apr 30, 2015

#31**0 **

Since you joined this site, I have been continually amazed by your intelligence, skill and work ethic. My admiration for you always increases, but it made a quantum leap after reading your post. It is a rare thing to find your way to heaven, but to find your way out of h**l first is astounding.

I do not doubt, in the slightest, your excellence will continue into your chosen career. In your wake, and because of your example, many others will excel too -- who otherwise would not.

Happiness to you, always, Brittany.

BTW No sane person would hate the opinion you gave.

Guest Apr 30, 2015