Comparing features of quadratic functions
Sal compares the y-intercepts, the zeros, and the concavity of quadratic functions given graphically and algebraically.
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- I can't understand how the coeficient of x square terme tell's us wether the vertex is minimum or maximum!(9 votes)
- If the coefficient is a positive number (2x²) the vertex will be a minimum point (the parabola will be in the shape of a "U"). The vertex is a low as your parabola will go.
if the coefficient is a negative number (-2x²), the vertex will be a maximum point (the parabola will be in a shape like "∩"). Which means the vertex is as high as your parabola will go.(11 votes)
- At0:38, how did you determine that the y intercept for y=g(x) was 3. I understand how you found it for the first equation because the y intercept was 4, but how did you find it for g(x)?(5 votes)
- You look for the point on the y-axis that is (0, #). for that one it happened to be 3(3 votes)
- So concavity is either something two functions do or don't have in common, not something they can have more or less of?
Sort of like how two lines either are or aren't parallel?(2 votes)
- This is correct (to a degree), concavity describes the open up/down nature of the graph of a function. Later on in mathematics (Algebra 2 and beyond), concavity can identify if there's a minimum or maximum.(2 votes)
- How do you write a quadratic function in standard form?(0 votes)
- Why is it we can simplify the equation by dividing by two to make it easier to find its x's but we cannot leave it at the simplified version to find our y intercept? I thought manipulation keeps the formula the same(0 votes)
- I could not find where in the video you are talking about, but your final statement is true if you manipulate the equation correctly, it is just a different form, but same equation. I do not know what you consider the simplified version, so could you provide an example where you cannot find the y intercept?
If you have f(x) = 2x^2 + 6x - 8 you see y int of - 8, if you take out 2 and factor, you end up with f(x) = 2(x^2 +3x - 4) = 2(x + 4)(x - 1) so if x=0, y int is 2(4)(-1) which still gives -8 and x intercepts are -4 and 1.(5 votes)
- There is another way to it also that is discriminant also determines the number of solutions a Quadratic Equation have, Right?(1 vote)
- Yes, the discriminant can be used to determine the number of roots and what type they are. If you need their actual values, then you will need to do the complete solution.(3 votes)
- 3:38what does Sal mean when when he says “as x gets further and further away from the vertex”?(1 vote)
- As an example, let's say that the vertex is at (3,4) The xs on both sides (2,)(4,) will be relatively close to the vertex, two values away (1,)(5,) will be a little further away, but as x gets 10 or 15 units from the vertex (farther and farther away) the y values really zoom up.(2 votes)
- at2:04, why is x -3 or 2 instead of 2 or -3 which also works?(1 vote)
- The order of the numbers does not matter.
Saying x=-3 or x=2 is the same as saying x=2 or x=-3.(2 votes)
I have a question:
Lets say you have a rope with its ends attached to 2 ropes, such that the distance between the poles is greater than the length of the rope. This would form a parabolic shape. Then, we graph the parabola on a coordinate plane, such that the zeroes of the parabola correspond with the points where the rope meets the 2 poles (the endpoints of the rope). If the length of the rope is 10 meters, what would be the distance between the 2 poles (or graphically, the distance between the zeroes)?
I don't get how to answer this problem without using calculus. It would be nice if someone could show if, and how, you could answer this problem using algebra.
Thanks for your help!(1 vote)
- I don't think thats possible for the length of the rope to be less than the distance of between the poles.(1 vote)
- Why did Sal refer to the x-solutions as "roots" @2:31?(0 votes)
- A "root" is the solution to an equation. When a quadratic function is set equal to zero, it is simply a quadratic equation that can be solved.
A "zero" is the value of a variable that makes a function equal to zero. For polynomials like quadratics, and many other types of functions (especially in two dimensional space), the "root" and "zero" are basically synonymous - they are both the x-values where the graph would intercept the x-axis (where the y is zero).(4 votes)
- [Voiceover] So we're asked which function has the greater y-intercept? So, the y-intercept is the y-coordinate when x is equal to zero. So, f of zero, when x is equal to zero, the function is equal to, let's see, f of zero is going to be equal to zero minus zero, plus four, is going to be equal to four. So this function right over here, it has a y-intercept of four. So it would intersect the y-axis right over there. While the function that we're comparing it to, g of x, we're looking at its graph, y is equal to g of x, its y-intercept is right over here, at y is equal to three. So which function has a greater y-intercept? Well, it's going to be f of x. F of x has a greater y-intercept than g of x does. Let's do a few more of these where we're comparing different functions. One of them that has a visual depiction, and one of them where we're just given the equation. How many roots do the functions have in common? Well, g of x, we can see their roots. The roots are, x equals negative one and x is equal to two. So these two functions, at most, are going to have two roots in common, because this g of x only has two roots. There's a couple of ways we could tackle it. We could just try to find f's roots, or we could plug in either one of these values and see if it makes the function equal to a zero. I'll do the first way, I'll try to factor this. So let's see, what two numbers, if I add them I get one, 'cause that's the coefficient here, or implicitly there. And if I take the product, I get negative six. Well, their gonna have to have different signs since their product is negative. So, let's see, negative three and positive two. No, actually, the other way around 'cause it's positive one. So positive three, and negative two. So this is equal to x plus three, times x minus two. So f of x is going to have zeros when x is equal to negative three. X is equal to negative three. Or, x is equal to two. These are the two zeros. If x is equal to negative three, this expression becomes zero. Zero times anything is zero. If x equals two, this expression becomes zero, and zero times anything is zero. So f of negative three is zero, and f of two is zero. These are the zeros of that function. So let's see, which of these are in common? Well, negative three is out here, that's not in common. X equals two is in common, so they only have one common zero right over there. So how many roots do the functions have in common? One. All right. Let's do one more of these. And they ask us, "Do the functions have the same concavity?" And one way to think about concavity is whether it's opening upwards or opening downwards. So this is often viewed as concave upwards, and this is viewed as concave downwards. Concave downwards. And the key realization is, well, if you just look at this blue, if you look at g of x right over here, it is concave downwards. So the question is, "Would this be concave "downwards or upwards?" And the key here is the coefficient on the second degree term, on the x squared term. If the coefficient is positive, you're going to be concave upwards, because as x gets suitably far away from zero, this term is going to overpower everything else, and it's going to become positive. So as x gets further and further away, we're not even further away from zero, as x gets further and further away from the vertex, as x gets further and further away from the vertex, this term dominates everything else, and we get more and more positive values. And so that's why if your coefficient is positive, you're going to have concave upwards, a concave upwards graph. And so if this is concave upwards, this one is clearly concave downwards. They do not have the same concavity, so no. If this was negative four x squared minus 108, then it would be concave downwards and we would say yes. Anyway, hopefully you found that interesting.