 In the previous video we learned how one can graph a quadratic function and you basically get just one or two pictures If you're your function could be looking like this Which is what examples we saw in the previous video or your parabola if you reflected downward would look something like this So you have this concave up parabola and this concave down parabola Based upon the curvature here or here, right now in this video what I want to focus on is the following idea Look at the vertex in these two situations Well, if you're the concave up situation that means that your leading coefficient was actually positive You did not perform a reflection and the vertex in that situation will represent a minimum value of the range So in this situation your your parabola Let's say the vertex was h comma k in this situation your range the range of the function would be k to infinity Because the vertex represents a minimum value now if you're the concave down situation That's because you actually reflected your graph downward And so your a value is negative and in this situation you actually have a maximum value This is the biggest value in the range and so then the range of the function f would then look like negative infinity up to k so the vertex of a parabola will always represent Some limit to how big or small the range can be the range it always be a maximum or minimum value It'll be a minimum when your concave up, which means you actually have a positive leading coefficient And it'll be a maximum value if your leading coefficient is negative And that'll be a very interesting thing for some future story problems We're gonna do in a later later lecture But consider we have the quadratic function f of x equals 5x squared minus 30x plus 49 Now if we want to determine the vertex of this parabola We have to kind of put this thing in vertex form and go from there, which is what we're gonna do right here So in doing so we're gonna get that f of x equals Factor out the coefficient from the axis you take away the 5 that leaves behind x squared minus 6x leave a space plus 49 Then we need to find the b value That's half of the the 6 right here b equals negative 3 which means b squared equals positive 9 We add 9 to both sides. I'm gonna switch the color We're gonna add 9 to both sides, but then we have to subtract 45 45 of course be 9 times 5 right here So we end up with 5 times x minus 3 squared we get 49 minus 45 That's a plus 4 and so what we see is that the vertex The vertex of this parabola is gonna be 3 comma 4 You always have to switch the sign because you're in the horizontal zone for the for h there So you're gonna get 4 comma 3 comma 4. That's the vertex now is the vertex a maximum or minimum value We're looking at the leading coefficient here. You have a 5 5 which is your a value is greater than 0. So this means we're gonna be a concave up Type picture and so this tells us that the vertex here represents a minimum of the graph It is gonna be the smallest point on the graph and that's an important observation to mention here Now another thing I want to mention is that we actually knew we were gonna have a minimum even before we computed the vertex Because of this fact right here notice that the coefficient You know if you have a standard form a x squared plus bx plus c if you have this standard form versus the vertex form right here a times x minus h Square plus k you'll notice I use the exact same symbol whether you have the standard form or the vertex form That's not a coincidence that number is gonna be the exact same So we started off with a 5 and you ended up with a 5 in which it's a positive 5 the entire time We knew that this parabola even in the standard form was gonna have its vertex was gonna be an absolute minimum of the graph We just had to switch the vertex form to figure out where was the vertex actually located but we knew at Because a was five that it was gonna be a minimum of the graph