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Consider the function: f ( x ) = − x 2 + 16 x − 60The y-intercept is at y =The x-intercepts are at x = (Enter solutions separated by acomma)The vertex i...

Question

Consider the function: f ( x ) = − x 2 + 16 x − 60The y-intercept is at y =The x-intercepts are at x = (Enter solutions separated by acomma)The vertex is at the point (Give your answer as an orderedpair)The direction of the graph is like which of thefollowing:

Consider the function: f ( x ) = − x 2 + 16 x − 60 The y-intercept is at y = The x-intercepts are at x = (Enter solutions separated by a comma) The vertex is at the point (Give your answer as an ordered pair) The direction of the graph is like which of the following:



Answers

Graph each parabola and give its vertex, axis, $x$ -intercepts, and $y$ -intercept.
$f(x)=-x^{2}+6 x-6$

We're going to find the ex Intercepts and Vertex for this function and draw the graph. So the ex intercepts are points that have a white coordinated zero so we can put zero in the equation for why and sell for X. Let's use the zero product property and set each of these factors equal to zero, and then we'll solve each one. So for the first equation will add to to both sides and we get X equals two. And for the second equation will add six to both sides and we get X equal six. So that means the ex intercepts are the points to zero and 60 So what about the Vertex we know based on the symmetry of a parabola that the Vertex has to be halfway in between the two x intercepts? So what number is halfway in between two and six? That would be four. You can also think of that is the average of the two numbers. What about the white cornet of the vortex? Now that we know the X coordinate is for, we can take that and plug it into the equation for X, and we'll solve for y so we end up with y equals two times negative, too. So why equals negative for so the Vertex is the 0.4 negative for So let's go ahead and sketch this graph. We're interested in X values between zero and six. It looks like and we're interested in Why values between zero and negative for we have the X intercept to zero the X intercept 60 and the Vertex for negative four. So here's a rough sketch of the parabola.

This problem we're going to be given Why equals a negative two X. Where My next to Act 96. Using this, we're going to find some important points. Using the calculator. So using a calculator, we can easily find the vertex To be 0.5 -5.5. And then we can also use the calculator to find That are why intercept is going to be 0 -6. We also see that um -1, is also a point on a graph. And then we're going to want one more point because we're asked to find the X intercepts which there aren't any X intercepts. So yeah, I'm looking for another point. We see that we have X equals two, gives us rex equals one, gives us -10. And also we could do another point X equals negative two gives us X. Y equals negative 10.

In this problem, we are going to graph a quadratic equation which will end up in the shape of a parabola by using the vertex and the X and Y intercepts our formula for the vertex of a parabola, or the graph of a quadratic is the X values negative B over two. A. That will give us this value of the vertex, the Y value is F. Of negative B over two. A. That will give us the Y value of our vertex in order to identify the A. And the B. That we need. We're going to use this standard form of a quadratic equation. F of X equals x squared plus bx plus C. This is the number in front of the X square, the B is the number in front of the acts and our constant is R C. In case we are not needing the quadratic formula. Later in this um example we're going to go ahead and identify it now as well. So are A. Is negative for R B. Is negative eight and R. C. Is negative six. So the X. Part of our vertex is X equals negative B. So that will be the opposite of negative age over to A. Which will be two times negative four. So this will be eight over negative eight which is negative one. So that's the X. Part of our vertex. And this f. Of negative B. Over two. A means f. Of negative one. And that means we're going to take cars and everywhere where there's an X. We're going to put a negative one. So or why is going to equal negative four times negative one squared minus eight times negative one minus six. Um And this equals negative four times one plus eight minus six. So this is negative four plus eight minus six, negative four plus eight is four minus six equals negative two. So our vertex happens at negative one negative two. So let's get some chick marks. Yeah. Oh and negative one. Negative two. Is about somewhere here. We know that this is going to open down because our a value is negative. So I'm gonna mark this is our vertex right here knowing that we're going to get a shape going down from there. Okay. The next thing we'll find is our intercepts. And the easiest intercept to find is our Y intercept. Because for to find the Y intercept we let X equal zero. Just like we just let X equal negative one to find the white part of the vertex. We're going to let X equal zero and solve for why? So why equals negative four times zero square minus eight times zero minus six. Which is negative four times zero square 20 minus eight times zero minus six. So that equals negative six. So why intercept is at zero negative six? Let's plot that. Yes, yes here. Okay. We also would typically find our X intercepts and we can tell that this one is not going to have any X intercepts because this is a downward facing crapola and the vertex is here. So our graph is going to look like this. But let's look real quickly how we know for sure that we do not have any X intercepts. Let's use our quadratic formula. May not have room for that one there. Let's just mark it off here. So we have our A. B and C. We're going to use our quadratic formula here to verify and see why we don't have any X intercept. So we have negative B. B is negative eight plus or minus the square root of the square minus four times A. Which is negative for time, see which is negative six. I'm gonna be running into my graph all over two times a. Okay, so this is eight plus or minus. The square root of negative. Eight squared is 64 minus four times negative. Four times negative six. Four times negative four times negative six is 96. So it's 64 minus 96 Over negative 8, 64 minus 96 is negative 32. So we have eight plus or minus the square root of negative 32 over negative eight. Since this is a negative number under a square root, we know that this is an unreal answer will not show up on our graph. So to sketch are parabola, we go from the vertex, we go through the one point that we know right here and then add some symmetry for the other side. And that is what our sketch is going to look like for our quadratic equation negative four x squared minus eight x minus six. Yeah.

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.


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