In Exercises 5–6, use the function’s equation, and not its graph,...

Question

In Exercises 5–6, use the function’s equation, and not its graph, to find (a) the minimum or maximum value and where it occurs. (b) the function's domain and its range. f(x) = -x^2 + 14x - 106

Accepted Answer

Hello. Today, we're gonna be determining whether the graph of a parabola produces a minimum or maximum value at what point that minimum or maximum value occurs. And were also asked to identify the domain and range. Alright, so the equation given to us is f of X is equal to negative three, X squared plus 12 x minus 56. Now we can determine whether we have a minimum or maximum value by looking at the leading coefficient of this graph. The leading coefficient is negative three. Since the leading coefficient is negative, that tells us that the graph is going to be opening up in the negative direction. And since this graph is gonna open up in the negative direction, the vertex is going to produce a maximum value. So since we know that the vertex produces a maximum value, we can go ahead and cross off A and B as our possible solutions. Okay, now we need to figure out what the maximum value is and at what point that maximum value occurs. Well, in order to figure out what point this maximum value occurs, we have to recall that this is this form of a parabola in standard form. So the vertex or at least the X coordinate of the vertex could be obtained by negative B over two. A. Well, we need to identify what are B. And R. A. Is if we look back at this equation is the term that sits in front of the X square term. So A is equal to negative three And B is gonna be our middle coefficient. So B is going to be 12 in this case. So now we just have to go ahead and plug that in for X. So that way we can get the X coordinate of the vertex doing so is going to give me X is equal to negative 12/2 times negative three. Two times negative three is going to give me negative six and negative 12 divided by negative six is going to give me positive too. So we know that X is equal to positive two. Okay, now we need to go ahead and determine what the y value of this of the vertex is going to be. Well in order to do that, we can just go ahead and take our X and put it as an input to our equation. The reason why is because this is the X coordinate or the exposition of the vertex. So if we plug it in directly to our equation, we're going to get the y position of our vertex. So plugging into to our equation gives us native three times two squared plus times two minus two squared is going to be four and 12 times two is going to be 24 minus 56. 4 times negative three is going to give me negative 12. And so now we have negative 12 plus 24 minus 56. So negative 12 plus 24 minus 56 gives us the value of negative 44 that's gonna be our y value for the vertex. So we know that the vertex is at the 0.2 comma negative 44. But what's the what's the maximum value? Well keep in mind that the maximum value is the highest y value of the parabola. And we just solve for that. So that means that our maximum value is going to be negative 44. Okay, so now we have a maximum value of negative 44 we know that the location of that maximum value is at two comma negative 44. Well, what about the domain and range of the parabola? Let's just go ahead and make a rough sketch of this parabola, we know that the highest value occurs at negative 44. And we know that the Parabola opens downward so the domain is the lowest and highest possible X value of this graph. And as this graph continues to decrease on both the left hand and right hand side, it's gonna approach positive infinity on the right hand side and negative infinity on the right hand side. So the domain is gonna be the set of all real numbers of negative infinity to positive infinity. Now, what about the range? While the range is the lowest possible Y value, to the highest possible y value. Well, since this graph is opening up towards the negative direction, the lowest possible white value is going to be negative infinity. And since we know that the maximum value is negative 40 for the highest possible y value is negative 44. So the range is going to be negative infinity to negative 44. And that's gonna be our domain and range of this parabola. So we know that we have a maximum value of negative 44. It's located at two comma negative 40 for the domain is the set of all real numbers and the range is from negative infinity to negative 44. That means that the solution to this problem is going to be C. So I hope this video helps you in identifying the different pieces of information of a parabola in standard form. And I'll see you all in the next video.

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