Graph each quadratic function. Give the vertex, axis, x-intercepts, y-intercept, domain, range, and largest open intervals of the domain over which each function is increasing or decreasing. ƒ(x)=-3x^2-12x-1

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Welcome back. I am so glad you're here. We are asked to consider the quadratic function F of X equals a negative two X squared plus 16 X minus 14. And graph it with the eight of the graph identify and we have a lot of things to identify the largest open intervals within its domain at which it is increasing and decreasing. Also determine its X and Y intercepts the axis of symmetry, the vertex, the domain and the range. And then for our answer choices, we have various answer choices describing each of those concepts coupled with a graph. And we will look at all of those graphs more closely and compare our answer choices in a little bit. But now let's focus on graphing our quadratic function. So we know from previous lessons that the standard form of a quadratic function is given in the format of F of X equals a multiplied by the quantity of X minus H squared plus K where A is not equal to zero. And we have it in the standard quadratic format of with an X squared and an X and then a constant term. So we need to get it into that format for a quadratic. How do we do that? Well, we recall from previous lessons that we are going to complete the square so we can bring this negative two X squared down plus the X minus 14. We need to have nothing in front of our X squared in order to complete the square. So we are going to factor out this negative two but only from the terms with the XS. So we'll have a negative two and then open parentheses X squared plus while we take a positive X divided by negative two. So now that's going to be minus eight X, then we leave a space for completing the square, close parentheses and bring down that minus 14. And now to complete the square, we recall from previous lessons that we're going to take the term in front of our X, we're going to divide it by two. So negative eight, divide it by two and then we'll square it and negative eight divided by two is negative four. Keeping in mind that number before we square it, we'll need that in a moment. Negative four square, it is 16. So we will be adding 16. And now we're working on one side, we're not working on both sides of the equal sign. So because we're adding here, we're adding 16, but are we, this is actually a plus 16 multiplied by a negative two? Well, 16 multiplied by negative two. Is a negative 32. So actually on this side, we're subtracting 32. And in order to not technically change what we're doing, we need to add 32. So after this minus 14, we're going to be adding 32 there. And now we can simplify this, we bring down this negative two and then we can complete the square here. We have this negative two multiplied by open parentheses. We know that this is going to be X and here's this middle number before we squared it X minus four, closed parentheses squared and then negative 14 plus 32 is equal to a positive 18. And there, now we have this in the proper format. We can see that our, a term is negative two, that our H term is equal to a positive four. And our K term is equal to a positive 18. And so we've identified those parts of it. How are they helpful? All right, we know that the vertex is H K. So our H is four, our K is 18. There's the vertex at 4 18. We know that the axis of symmetry is at our X equals H, our H value is four. So our axis of symmetry is X equals four. And our A term that's very helpful too because we know that when our, a term is less than zero, our verte or excuse me, our parabola is going to open downward. And this is enough information. To begin to graph and then we can identify all of those other things that we've been asked to identify for this problem. So let's get to where there's a little more space on the screen and we'll draw a rough sketch of a graph. We've got a vertical Y axis, a horizontal X axis. They come together at the origin in the middle, we first find our vertex from the origin. We're going to the 0.4 18. So we go to the right four units and up 18 units. This is very rough, but that's approximately here. So here's our vertex at 4 18. This opens downward. Now to help us graph this, we can determine our X and Y intercepts. That's another thing we need to identify. So for our X intercepts, we are going to set this function equal to zero. So we'll have zero equals a negative two X squared plus 16 X minus 14. We can factor this negative to or we can, we can divide everything by negative two, excuse me. So we'll have still zero on the left zero equals we've got X squared minus eight X plus seven. Now, we can factor this two open sets of parentheses. This will be an X minus seven multiplied by X minus one. We set both of those factors equal to zero X minus seven equals zero. Solving for X adding seven to both sides, we'll have X equals positive seven. And for X minus one setting that equal to zero, adding one to both sides, we get X equals a positive one. So we have two X intercepts, one is going to be at 10 and the other at from the origin, we go to the right one unit for our X intercept of 10. And then from the origin, we go to the right seven units for our X intercept of 70. OK. Now let's work on our why intercept for the Y intercept, we plug zero in for X. So this one's going to be a negative two, multiplied by zero squared minus 16, multiplied by zero minus 14. Well, the first two terms are just zero. So the Y intercept is at negative 14 when X equals zero, Y equals negative 14. So from the origin, we're going down 14 units and R Y intercept is at zero negative 14. And now we can draw the line connecting all of these points. So it's opening downward. That means it's heading toward negative infinity for our Y values. We can draw that coming up through our points zero, negative 14 and then 10 hitting our vertex at 4 18, then turning around and going down through our X intercept at 70 and heading toward negative infinity. OK. What do we still have left? OK. We need to know where this is increasing the domain where it's increasing. Those are X values it's increasing, it goes toward negative infinity even though it doesn't look like it, it's heading toward negative infinity for the X values. And it's increasing all the way up to where X is equal to four. And at that point, it turns around and it'll be decreasing from four all the way to positive infinity. What else do we need to identify? We need to identify the domain and the range. So the domain is all the possible X values and the X values they can go all the way from negative infinity to positive infinity for a Parabola. Very cool. How about the range? Well, it can go all the way down to negative infinity for the Y values but it's only going up as high as a positive 18. That's including positive 18. But there it stops and that's the range. All right. Now, we need to find the matching answer choice and we want to do a quick comparison of all of these different graphs. So let's begin by looking for a vertex of 4 18. And let's start down here with answer choice D. We look at answer choice D and that graph has a vertex negative 18. So that is not what we're looking for. We can eliminate that answer. Choice C has a vertex of positive 5 18. That's close. But it doesn't work. We have 4 18. So not answer choice C answer choice B that has a vertex of negative 4 18. Not gonna work. We're in the positive range here and answer choice A that one does have the vertex at 4 18. So let's compare the rest. So we're looking for increasing from negative infinity to four. Yes, decreasing from four to infinity. Yes, X intercepts of 1070. Correct Y intercept at zero negative 14. Yes axis of symmetry at X equals four. Yes, we have that vertex at 4 18. Yes domain at negative infinity to infinity. Correct and range from negative infinity to positive 18. And the graph looks good answer. Choice. A it is well done. We'll catch you on the next one.

Graph each quadratic function. Give the vertex, axis, x-intercepts, y-intercept, domain, range, and largest open intervals of the domain over which each function is increasing or decreasing. ƒ(x)=-3x^2-12x-1

Verified Solution

Key Concepts

Quadratic Functions

Vertex and Axis of Symmetry

Intercepts and Intervals of Increase/Decrease