In this problem, we are going to graph a quadratic equation using the vertex and the X and Y intercepts. So in order to use our formula for the vertex which is the X value of the vertex is negative B over two. A. And the Y value of the vertex is F. Of negative B over two. A. That's going to give us our X and Y values to plot on a graph. We need to identify the A. B. And the C. Here's the standard form of a quadratic equation. The A. Is going to be in front of the X squared, the B is going to be in front of the X. And in case we need it later for the quadratic formula. Um The sea is a one. We're going to go ahead and identify it. So A. Is negative three. He is six and C. In case we need it later is a one. So to find our X. Value X. And our vertex that we're going to put right down here is equal to negative B over two. A. So that's negative six Over two times negative three. Which is negative six over negative six, which is one. So our expert tex is going to be one. Then this says do F of whatever you got for your ex. So we're going to do F. Of one, which means take your F. Of X. And everywhere where there's an X. You're going to put a one. So to find our Y value, we're going to say why equals F. Of one, which means take negative three times one squared plus six times one plus one. So this is negative three times one. Which is negative three plus six plus one, negative three plus six is three plus one is four. So our vertex is at 14 Let's plot that one in the X. Direction for in the Y direction. It's going to be right here. Okay, we know that this proble is going to open down because our A. Is negative. So I'm gonna put A V right here to remind us that that is our vertex of our graph. Okay, now we're going to find the rest of our graph. We are going to find our intercept and the easiest intercept defined is the Y intercept. Because to find the Y intercept, we need to let X be zero and find what Y is. So here we let X be one and we saw what Y is. So now we're instead we are going to let X be zero. So we have y equals -3 times zero sq Plus six times 0 Plus one. That's a negative right here. So we have Y equals negative three times zero square 20 plus six times zero is zero plus one and this whole thing equals one. So the y intercept 01 So that's going to hit right here. There's a Y intercept. And now we need to find our X intercepts. Need a new color. Let's do I never usually yellow. We'll use yellow to find our X intercept. We need to let why be zero and see what X. Is. So we need to solve zero equals negative three X squared plus six X plus one. And we can use the quadratic formula to do that. We've identified our A. B and C. And here's our quadratic formula. It will give us the two zeros for this equation. So we have negative B. Or be a six plus or minus the square root of B squared. So that's six squared minus four times A. Which is negative three times C. Which is one All over two times a which is two times negative three. So we have negative six plus or minus the square root of 36 minus four times negative three times one is negative 12. All over -6. So negative six plus or minus. The square root of 36 -26 plus 12 Which is 48. is not a perfect square root. We could break it down but we need to graph this thing so we're going to need a decimal in the end. Let's jump to a calculator right here and we are going to key in negative six plus the square root of 48. Yeah and that gives us About .928 with a bunch of decimals. And we are going to divide that by negative six And we get about negative .1547. That's a very small number. So that is going to give us Are negative six plus a squared of 48 over negative six gives us that very small number. -15. Okay, so one of our x intercepts is negative point 15 Right? Yeah comma zero. And then our other one will be found by negative six minus. This one should have just been a plus here. Sorry about that. I know now we're doing negative six minus the square to 48 over negative six. So negative six minus the square root of 48 gives us about negative 12.928 We need to divide that answer by negative six and we get about 2.15 and its approximate. Yeah, yeah, So 2.15/0. So we don't need to be very accurate there because we certainly can't graph this very accurately. So our graph is going to touch our X axis at negative 0.15 I want to be very close here to the origin zero and 2.15 is going to be about their little past 20. And now we have everything our need we need to sketch in this um the graph of the quadratic which is a travel. It's going to open up on this side from the vertex. Down through those two intercepts that we know, it's kind of curved at the top and then it's going to open on this side kind of curved at the top through that intercept that we know. So this is the graph of this quadratic equation. Uh huh. Right. Yeah.