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Lesson 12: Quadratics- DEPRECATED Compare properties of quadratic functions
- DEPRECATED Compare properties of quadratic functions
- IDEAS Algebra 1: Quadratics
- GRAVEYARD Algebra 1: Quadratics
- Parabolas intro
- Graphing parabolas intro
- Quadratic word problem: mosquitoes
- Quadratics: unit review
- Graphing quadratic functions intro
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Graphing parabolas intro
Graphs of quadratic functions have a U shape called "parabola." Here, Sal graphs f(x)=-3x²+8 by creating a table of values and plotting the resulting points. Created by Sal Khan and Monterey Institute for Technology and Education.
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- in my class we just say Y=5x^2 not F(x). Why does he use F(x)?(6 votes)
- they are just two different notations for an equation. When F(x) is used, it indicates that the equation that follows the 'equal' sign is a function. In this case, F(x) = Y. You can learn about what defines a function in one of Khan's videos. But if Y is used instead, it simply means that it is just an equation but does not necessarily have to be a function. In other words, F(x) = Y does not necessarily hold true in this regard(14 votes)
- In this graph, the vertex of the parabola is reached when x=0. Is that always the case , or can it be different ?? Thanks !! :)(10 votes)
- No, it is not always the case, the parabola can be shifted along the x axis just like it can be sifted along the f(x) axis.
Example: f(x)= x^2 is your parabola, to shift the parabola to the left by 1 unit, the equation becomes to f(x)=(x+1)^2.(6 votes)
- How do you pick a number to substitute "x" at2:05?(5 votes)
- You can use any numbers you want. However, you usually want them near the vertex (peak), and it's often easier trying with numbers like -10, -2, -1, 0, 1, 2, 10 etc.(8 votes)
- What is f(x) and what does it mean?(2 votes)
- It means it's a function. For every input there is only 1 output. Or, for every number you input into the function there is ONLY 1 output. You subsitute whatever number is supposed to represent x. If the function is f (x)=x +2.
Your told x=2. Then plug 2 into x and you will get 2+2=4. So the function is 4. 4=2+2. The input is 2 (the number you plugged into x) and the output is 4, the result of plugging in 2 in for x and adding another 2 to that. Input=2 Output=4 Or you can say 4 is a function of 2.
To be considered a function, every input can have only 1 output. If it has more than 1 output (or more than one answer or result) it's not a function but instead a relation. The output is dependent on the input. What you input determines what your output will be. Functions are confusing, Sal has videos about functions.
https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-relationships-functions/cc-8th-function-intro/v/what-is-a-function(6 votes)
- The roots are +/- 2 sqrt 2 over 3 which equals approx 0.94, so less than one (and more then -1 respectively). The parabola crosses the x-axis around approx 1.8, roughly double the correct value. The graph seems to be fine, the method makes sense - so how come a correct approach leads to an incorrect representation?(3 votes)
- Okay - I see what happened now - the roots of this function are
2*sqrt(2/3) as opposed to [2*sqrt(2)]/3
when the three is under the square root sign, as it should be, the value is 1.63(3 votes)
- How do you choose what values to plug in for x when making a table?(3 votes)
- Generally to start, try 0, 1, 2, 3 and -1, -2, -3 and any x value that makes the function equal to zero.
With time and practice, the values to choose will become obvious to you.(3 votes)
- What if i'm trying to find the vertex of a function with a formula without graphing it? My teacher said i can turn it into an equation (instead of f(x)=x^2-6x+9 --> y=x^2 -6x+9) and use a formula but i didn't quite get it.(1 vote)
- You don't really need to find the axis of symmetry to find the vertex... Instead, you can put that into vertex form which is y = a(x-h)^2 + k.
(h,k) would be your vertex in (x,y) form, respectively.
Here's the work:
y = x^2 - 6x + 9
(Factor)
y = (x - 3) (x - 3)
or
y = (x - 3)^2
Because there is no k value in this specific function then we can assume it is zero so your vertex is (3,0).
Hope that helped!(6 votes)
- What makes Parabola's smooth? When drawn, they have nice curves, even though there's only a handful of points.
Am I correct in saying the more points you use, the smoother the lines would be? For example, when interpreted/drawn by a computer?(3 votes)- Parabolas are smooth curves. That is simply their nature in this universe we live in. You are not supposed to plot a few points and then connect the segments because that is just careless. A parabola contains an infinite amount of points but for simplicity, we just draw its approximate shape. A computer could obviously draw it more accurately.(3 votes)
- how do you graph y=(x+3)^2(3 votes)
- Option 1: expand (x+3)^2 and follow the steps Sal set down.
Option 2: Create a table of values for y and (x+3)^2.(3 votes)
- Can't we technically view a parabola as half of an infinitely wide and long ellipse? As the quadratic goes up (assuming it has a positive
a
value), it gains in width as well as height, and of course, like all functions, goes infinitely in that direction. So, if we were to say, take a segment of the parabola, cut it off, and then close it with a reflection of it, would it make an ellipse?(3 votes)
Video transcript
We are asked to
graph the function f of x is equal to
negative 3x squared plus 8. So we'll do this by essentially
trying out different points for x, and seeing what
we get for f of x, and then graphing it. But the first question
I have for you is, just looking at
this function definition for f of x, what type
of graph will this be? Will just be a line? Will this be a parabola? Will this be something else,
a circle, something else, maybe something else
bizarre or strange? Well, this is pretty clearly
going to be a parabola here. You have the
function is defined, it's negative 3x squared, so
you have this second degree term here. You don't have any x thirds or
x to the fourths or anything else bizarre, so this is
going to be a parabola. Now, the other thing
that we could think about is whether the parabola is
going to open up like that or whether it's going
to open down like that. And just looking at this
function definition, do you have any intuition of
whether it's going to open up or it's going to open down? Well, if you look
at the coefficient on the x squared
term, the negative 3, that tells you that this
parabola is going to open down. It's going to open down. So with that
intuition now that we know it's going
to be a parabola, we know it's going to
open down, let's actually try to graph the thing. And let me draw some axes here. So let's say that this is my
x-axis, so that's my x-axis. And then let's make this right
over here, this is my y-axis. And let me make
a table of values and see what values
f of x takes on. So on one column, I'm
going to do my values for x and over on
the right I'm going to do my values for f of x. And then we can
plot these things. And actually I want to
take all of these values before I draw the
scale on these axes, so I know what might be
an appropriate scale. So I'm just going to
try a bunch of values. So let's try first what happens
when x is equal to negative 2. So when x is equal
to negative 2-- and I'm just
picking numbers that will be relatively
easy to compute. When x is equal to
negative 2 what's f of x? Well, f of x is going
to be negative 3, this negative 3,
times negative 2 squared plus 8, which is going
to be equal to, let's see. Negative 2 squared
is 4, positive 4, then we multiply that times
a negative 3, which gives us negative 12 plus 8,
gives us negative 4. Let's try another point. Let's see what happens when
x is equal to negative 1. What do we get for f of x then? Well, f of x is going to be
negative 3 times negative 1 squared plus 8. So that's going to be-- see
negative 1 squared is just 1, and then that times
negative 3 is negative 3. Negative 3 plus 8 is 5. Now, what does f of x
equal when x is equal to 0? Well, this is pretty
easy to compute. When x is equal to
0, you get negative 3 times 0 squared,
which is equal to-- and we could write that
either way-- negative 3 times 0 squared plus 8. Well, this just simplifies to 0,
and so you're just left with 8. Now, let's see what happens
when x is equal to 1. What do we get for f of x? Well, it's going to be negative
3 times 1 squared plus 8. So 1 squared is just 1,
negative 3 plus 8 is equal to 5. And then finally, what do we get
when x is equal to positive 2? What does f of x equal,
or another way of thinking about it, what is f of 2? Well, let's think about it. You get negative 3
times 2 squared plus 8. 2 squared is 4, times
negative 3 is negative 12, plus 8 is equal to negative 4. So let's see if
we can plot this. So the x values that I picked go
from negative 2 to positive 2. So let's make this
negative 2, negative 1. This is 0. This is positive 1, and
that could be positive 2. And then our f of x values,
or we are essentially graphing y is equal to f of x,
so I can even say this is going to be the graph of y
is equal to f of x. Our f of x values take on
things between negative 4 and positive 8. Let me try to draw that. So if this is positive 8, that's
positive 8, that is positive 4, and this is negative 4. This is negative 4. And if that's positive 4,
then this is positive 6, and then that right
there is 5, that is 7, this would be 2 that would be
3, and then that would be 1. Now, let's graph the points. When x is negative 2,
f of x is negative 4. And actually I
could say, this is the y is equal to f of x-axis. I'm going to plot f of x. I'm graphing, and
this is going to be the graph of y is
equal to this function. So let's graph
negative 2, negative 4. So that gets us,
when x is negative 2, f of x is negative 4. It's right over there. When x is equal to negative
1, f of x is equal to 5. And we're saying that y is
equal to f of x in this context. When x is 0, f of x or y-- I
could even write over here, I could say, y is
equal to f of x. When x is equal to
0, our f of x is 8. x is 0, f of x is 8. When x is 1, f of x is 5. When x is 1, y
equals f of x is 5. And then finally, when
x is equal to 2, f of x is equal to negative 4. So 2, negative 4,
gets us right there. And now we can connect the dots. We know this is going
to be a parabola. And I will do it in blue. So my best attempt-- I like
to draw it as a dotted line, just because it's
easier to not mess up-- so it would look
something like that. And it keeps on
going just like that, and then I can actually make the
line a little bit more solid. So we see that we
definitely got a parabola, and just as our
intuition told us, our ability to inspect
the coefficient on the x-squared term told us,
that our parabola is indeed opening downwards.