Ok we obviously need some review work on transformations. Please post "rules" for function transformations. I'll get us started off.
Let's say we have the the function f(x). If I define g(x) = f(x) + 2, it will shift the graph up 2. If I define g(x) = f(x) -2 it will shift the graph of f(x) down 2. So in general if you add or subtract a number outside of the original function, it will shift it up or down that amount. If you add a number it will move up, if you subtract a number it will move down.
Now you guys finish the rules.
If f(x)=(x+2), the graph will translate to the left 2 units.
ReplyDeleteIf f(x)=(x-2), the graph will translate to the right 2 units.
If g(x)=-f(x), the graph will reflect over the x-axis.
If g(x)=f(-x), the graph will reflect over the y-axis.
If the number on the outside is greater than 0, then it is a vertical stretch. Ex: f(x)=3|x| would be a vertical stretch.
If the number on the outside is less than 0, then it is a vertical shrink. Ex: f(x)=.5x would be a vertical shrink.
For horizontal shrinks/stretches you use the reciprocal of the number on the inside and if the reciprocal is less than 0 it's a shrink, if it's greater than 0 it's a stretch. Ex: f(x)=(2x), the reciprocal of 2 is 1/2, since it's less than 0 it would be a horizontal shrink by a factor of 1/2.
anything on the inside of the parentheses, where the x is, is automatically the opposite of its normal function.
ReplyDeletefor example, x+2 would normally mean moving the value up positively, but inside the parentheses, (x+2) means to move left, towards the negative side of the x axis.
Actually, I slightly disagree. I could say that y = x+2 moves left 2, or I could say it moves up 2.
ReplyDeleteAnybody care to explain why?
g(x)=f(x)+k (goes up k units)
ReplyDeleteg(x)=f(x)-k (goes down k units)
g(x)=f(x+h) shifts the graph left h units
g(x)=f(x-h) shifts the graph right h units
g(x)=-f(x) reflects over the x axis
g(x)=f(-x) reflects over the y axis
y= af(x) where x>1 makes its a vertical stretch
y= af(x) where x<1 makes it a vertical shrink
y=f(cx) 01 horizontal shrink
about mr frank's question i think it just has to do with parenthesis. the parenthesis are what tell you what happens in thers of translations to the left or to the right...
one thing you have to remember is when you see
ReplyDeletef(x)=g(x2) that means that means that it will affect the x axis but it will horizontal shrink by 1/2. even though the factor you are multiplying the x values by is the opposite of what your thinking, it will stay with the x values! only the factor is the opposite, not the x value it is affected by. when you see f(x)=g2(x) that means that the y values are being affected. this would NOT be the opposite. it would be a vertical stretch by the factor of 2. i always get this confused.
well i think i might know why Mr.Frank, because .. the x+2 is not in parenthesis and if it was(x+2) then the graph would go left, but if its just x+2 then the graph will read the 'x' as if it stayed on the origin ......
ReplyDeletef(x)= g(x)+k (up k units)
f(x)= g(x)-k (down k units)
f(x)= -g(x) (reflects over x-axis)
f(x)= g(-x) (reflects over x-axis)
f(x)= g(x+h) (tranSLates left h units)
f(x)= g(x-h) (tranSLates right h units)
f(x)= g(cx) (Horizontal Shrink) [where c>1]
f(x)= g(cx) (Horizontal Stretch) [where 01]
f(x)= ag(x) (Vertical Shrink) [where a<1]
Functions were one of my favorite things to go over! At first they seem complicated, but in actuallity, they're really easy! One of the parts of functions we focused on the most were the vertical shrinks/stretches, and the horizontal shrinks/stretches.
ReplyDelete*In the equations, a=the number you are putting in.
Vertical Stretch: For a vertical stretch, the number has to be greater than one, and has to be on the outside of g(x). So, the equation is f(x)=ag(x).
Vertical Shrink: For a vertical shrink, the number has to be greater than zero and less than one, and has to be on the outside of g(x) as well. So, the equation is f(x)=ag(x). (a has to be greater than zero and less than 1)
Horizontal Stretch: For a horizontal stretch, the number has to be greater than zero less than one, and be next to x. So the equation is f(x)=g(ax). (a is greater than zero and less than one)
Horizontal Shrink: For a horizontal shrink, the number has to be greater than 1, and next to x. So, the equation is f(x)=g(ax).
I know there's only one Tariro in all your classes, but I forgot to put my initial. So, its Tariro K.
ReplyDeletelets say you have f(x)=-1/2(x+5)+10 what this equation is telling you is that you translate left 5 and up 10. the - 1/2 is a vertical shrink of 1/2 and the negative is a reflection over the y axis.
ReplyDeleteIf the equation is F(x)=4x^2 it's a vertical stretch by a factor of 4 because the 4 is with x^2 and their is no parenthesis but if it says F(x)=(3x)^2 then it's a horizontal shrink by a factor of 1/3. if f(x)=-x^2 then you reflect over the x-axis because the x is not in parenthesis but if theirs parenthesis life F(x)=(-x)^2 then you reflect over the y axis.
ReplyDeletea trick to remember that things inside the perenthisis move horizontaly (negative=right & positive = left) is:
ReplyDeletethings outside move as they should like things with x never would :D
Mr. Franks examples only deals with shifts VERTICALLY, which occurs in the Y-AXIS.
ReplyDeletef(x)= (x)-2
f(x)= (x-2)
see the difference?
The 1st equation is what he explained in the beginning examples--changes in the y-axis. The 2nd equation is a shift two spaces to the right. This will be a HORIZONTAL shift, occuring in the x-axis. The key thing to remember when dealing with a change horizontally on your x-axis, understanding the signs (-,+), is opposite.
f(x)= (x-2) is a shift two spaces to the RIGHT.
f(x)= (x+2) is a shift two spaces to the LEFT.
Also what you must rememeber is adding inside your original equation will STRECTCH your graph horizonatally, and subtracting from your original eqaution will SHRINK the graph horizontally.
The key rule though is adding or subtracting INSIDE your original equation is changes in your horizontal graph and the signs (-,+) are switched. Adding or subtracting OUTSIDE your original equation will change the graph verticallyyyyyy.
Another thing is that a changed original equation can have BOTH a vertical and horizontal change, you just have to make sure which change occurs where.
f(x)= (x+5)-2 <------ This equation shifts 5 units to the left, and 2 units down.
that was destiny wagner by the way ^^^
ReplyDeleteFiguring out if you flip over x or y axis:
ReplyDeletelets say you are given f(x)= -g(x). with this equation you're gonna flip it over the x axis. The reason you flip it over the x axis is because the (-) is with G meaning it is going to effect the y value of the function. the negative symbol tells you to flip the function over either the x/y axis. a trick to figure out whether to flip a function over the x or y axis is to see if the negative symbol is in the parenthesis with x or not. If the (-) is with the x that means it is going to flip over the y axis if the (-) is outside of the parenthesis that means it flips over the x axis.
Figuring out whether to stretch or shrink:
f(x)= 2g(x) with this equation you stretch the graph VERTICALLY by a factor of 2. you strecth it Verticaly because the number is outside of the parenthesis. and you stretch it by a factor of 2 b/c thats common sense.
Now this is when it gets tricky if the equation is f(x)= g(2x) since it is with x in parenthesis that means you do the OPPOSITE meaning instead of stretching the function by a factor of 2. You are going to SHRINK it by a factor of 1/2.
*important rule whenever a number is with x in the parenthesis that means you do the exact opposite of what you would normally do
deciding whether to move horizontally or vertically:
okay say you're given f(x)=g(x-2) for this equation you are going to move HORIZONTALLY because you are in parenthesis with the X. you are going to move the function to the RIGHT(you are going to ADD 2 to the x value of the function) because like i said before if the number is in the parenthesis you do the opposite. so if the function said f(x)=g(x+2) you are going to move to the LEFT 2. simple.
if the equation is f(x)=g(x)+2 you are going to move the function up by a factor of 2. f(x)=g(x)-2 means you translate down by a factor of 2. thats easy enough.
if you are describing a transformation make sure to use the words
"translate" instead of "move up"
"translate by a factor of 2" instead of "move up by 2 points"
ALSO make sure to say whether you shrink/stretch HORIZONTALLY/VERTICALLY
I might not have used that vocab when i explained the rules but i don't feel like going back and fixing it b/c i just spent so much time writing it out. so just makes sure to use this vocabulary in the future when describing translations.