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Graphing quadratics: standard form

Learn how to graph any quadratic function that is given in standard form. Here, Sal graphs y=5x²-20x+15. Created by Sal Khan.

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  • orange juice squid orange style avatar for user Brent Carter
    Is it just me or is the Vertex Formula left out of this video? Maybe it is left out to avoid relying on memorization but it seems like some of the "Graphing quadratics: Standard Form" Practice/Mastery Challenges seem to rely on it for equations that are difficult to factor. I didn't see it in the "Graphing Quadratics: Vertex Form" section either but I did not miss it there like I did in the Standard Form challenges.
    (44 votes)
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  • leaf green style avatar for user tk12
    Why does -b/2a (given y = ax^2 + bx + c) give the x-coordinate of the vertex?
    (19 votes)
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    • primosaur ultimate style avatar for user asmodeus
      It's derived from a key concept in calculus, called the derivative. A derivative is a measure of how quickly a function changes; for example, velocity (or speed) would be a derivative of position. The derivative of a function ax^2 + bx + c ends up being 2ax + b, for reasons you might learn later. At the vertex of a parabola, the derivative is 0, so we can set up the equation 2ax + b = 0 and solve for x to get 2ax = - b, then x = -b/2a.
      (22 votes)
  • leaf green style avatar for user Lomani O'Hagan
    At when Sal factors out the five from the equation, why does he not put the newly factored equation in brackets with a five on the outside?
    (9 votes)
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    • male robot hal style avatar for user Ben Willetts
      In order to get rid of the 5, Sal divided both sides of the equation by 5, giving (5x^2 - 20x + 15) / 5 = 0/5, which simplifies to x^2 - 4x + 3 = 0. You don't need to include the 5 as a factor in the left-hand side because you divided both sides by it.
      (19 votes)
  • leaf green style avatar for user julien roger
    this is helpful, but what if you can't factor the equation?
    (7 votes)
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    • stelly blue style avatar for user Kim Seidel
      There are multiple ways that you can graph a quadratic.
      1) You can create a table of values: pick a value of "x" and calculate "y" to get points and graph the parabola.

      2) If the quadratic is factorable, you can use the techniques shown in this video.

      3) If the quadratic is not factorable, you can use the quadratic formula or complete the square to find the roots of the quadratic (the x-intercepts) and then find the vertex as shown in this video.

      4) You can convert the equation into vertex form by completing the square.

      All these techniques are covered in various videos in the lessons on quadratics.
      (17 votes)
  • male robot hal style avatar for user Jeremy
    Are you always supposed to switch your x values to positive? In the video, he switched -3 and -1 to 3 and 1. If not please tell me why he did it there. Thanks
    (4 votes)
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    • primosaur ultimate style avatar for user Lott N
      hey, Jeremy
      He didn't exactly switch his x values to positive. the -3 and the -1 that he got were not his x values. the -3 and -1 were numbers that he got when he factored the equation into a binomial. In order to find the x values he used that binomial and made it equal 0, and his x values that he found were 3 and 1. The -3 and -1 were not x values, they were just what he used to factor in a way that he teaches in previous factoring videos. you can check them out if you are still confused.

      I think that was the question you were asking, I hope that helped, if not, hopefully this will:

      (this basically explains how to get the 3 and 1 for the x values and it explains why -3 and -1 do not work for x values.)

      the reason that Sal made the x values positive 3 and 1 is because they are the only two x values that would make his equation equal zero.
      When you have:

      (x-3)(x-1) = 0

      x cannot equal -3 or -1 because if x was -3 then this would happen:

      (-3 -3) (-3 -1) =0
      -6 * -4 = 0
      24 DOES NOT EQUAL 0

      so -3 doesn't work as an x, and the same thing would happen with -1:

      (-1 -3) (-1 -1) = 0
      -4 * -2 = 0
      8 DOES NOT EQUAL 0

      Both of these solved binomial equations show that -3 and -1 cannot be x values for the parabola. Therefore, 3 and 1 are the only possible x values.
      (7 votes)
  • blobby green style avatar for user milo
    How do you find the x-intercept of the vertex when it can't be as easily estimated like he did in this specific equation?
    (5 votes)
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  • blobby green style avatar for user Neimo Mohamed
    what if it doesnt intervene into the x axis?
    (5 votes)
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  • blobby green style avatar for user lizete g
    whats the process of graphing polynomial functions
    (3 votes)
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  • duskpin ultimate style avatar for user ™ ¤♏◭khï¤ ™
    Is there a more simple way because this is pretty time consuming and I have to take a timed test on this and i don't want to waste time if i don't have to
    (3 votes)
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    • leaf blue style avatar for user Stefen
      For the detailed explanation, Sal takes over 4 minutes, but once you understand the process, the question Sal demonstrated can be done in less than 60 seconds (and less than 30 if you really understand the process):
      y=5x² - 20x + 15
      5x² - 20x + 15 = 0
      5(x² - 4x + 3) = 0
      5(x - 3)(x - 1) = 0,
      giving x intercepts of x=1 and x=3.
      The x value of the axis of symmetry is 2 since 2 is equidistant from (that is, in the middle of) x=1 and x=3.
      The value of f when x=2 is 5(2 - 3)(2 - 1) = 5(-1)(1) = -5.
      Now you have all you need to graph.

      Keep studying!
      Keep Asking Questions!
      (4 votes)
  • blobby green style avatar for user Ryan Dufresne
    I only have two things where Sal has three here. My problem is C squared + 3= 0 . How would i go about doing this?
    (3 votes)
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Video transcript

We're asked to graph the following equation y equals 5x squared minus 20x plus 15. So let me get my little scratch pad out. So it's y is equal to 5x squared minus 20x plus 15. Now there's many ways to graph this. You can just take three values for x and figure out what the corresponding values for y are and just graph those three points. And three points actually will determine a parabola. But I want to do something a little bit more interesting. I want to find the places. So if we imagine our axes. This is my x-axis. That's my y-axis. And this is our curve. So the parabola might look something like this. I want to first figure out where does this parabola intersect the x-axis. And as we have already seen, intersecting the x-axis is the same thing as saying when it does this when does y equal 0 for this problem? Or another way of saying it, when does this 5x squared minus 20x plus 15, when does this equal 0? So I want to figure out those points. And then I also want to figure out the point exactly in between, which is the vertex. And if I can graph those three points then I should be all set with graphing this parabola. So as I just said, we're going to try to solve the equation 5x squared minus 20x plus 15 is equal to 0. Now the first thing I like to do whenever I see a coefficient out here on the x squared term that's not a 1, is to see if I can divide everything by that term to try to simplify this a little bit. And maybe this will get us into a factor-able form. And it does look like every term here is divisible by 5. So I will divide by 5. So I'll divide both sides of this equation by 5. And so that will give me-- these cancel out and I'm left with x squared minus 20 over 5 is 4x. Plus 15 over 5 is 3 is equal to 0 over 5 is just 0. And now we can attempt to factor this left-hand side. We say are there two numbers whose product is positive 3? The fact that their product is positive tells you they both must be positive. And whose sum is negative 4, which tells you well they both must be negative. If we're getting a negative sum here. And the one that probably jumps out of your mind-- and you might want to review the videos on factoring quadratics if this is not so fresh-- is a negative 3 and negative 1 seem to work. Negative 3 times negative 1. Negative 3 times negative 1 is 3. Negative 3 plus negative 1 is negative 4. So this will factor out as x minus 3 times x minus 1. And on the right-hand side, we still have that being equal to 0. And now we can think about what x's will make this expression 0, and if they make this expression 0, well they're going to make this expression 0. Which is going to make this expression equal to 0. And so this will be true if either one of these is 0. So x minus 3 is equal to 0. Or x minus 1 is equal to 0. This is true, and you can add 3 to both sides of this. This is true when x is equal to 3. This is true when x is equal to 1. So we were able to figure out these two points right over here. This is x is equal to 1. This is x is equal to 3. So this is the point 1 comma 0. This is the point 3 comma 0. And so the last one I want to figure out, is this point right over here, the vertex. Now the vertex always sits exactly smack dab between the roots, when you do have roots. Sometimes you might not intersect the x-axis. So we already know what its x-coordinate is going to be. It's going to be 2. And now we just have to substitute back in to figure out its y-coordinate. When x equals 2, y is going to be equal to 5 times 2 squared minus 20 times 2 plus 15, which is equal to-- let's see, this is equal to 2 squared is 4. This is 20 minus 40 plus 15. So this is going to be negative 20 plus 15, which is equal to negative 5. So this is the point 2 comma negative 5. And so now we can go back to the exercise and actually plot these three points. 1 comma 0, 2 comma negative 5, 3 comma 0. So let's do that. So first I'll do the vertex at 2 comma negative 5, which is right there. And now we also know one of the times it intersects the x-axis is at 1 comma 0. And the other time is at 3 comma 0. And now we can check our answer. And we got it right.