In Exercises 1–4, use the vertex and intercepts to sketch the gra...

Question

In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x) = - (x + 1)^2 + 4

Accepted Answer

Hey everyone in this problem we are asked to sketch the graph of the parabola defined by F of X equals negative X plus four squared plus nine by using its vertex and intercepts. Okay then we're asked to identify the domain and range based on our graph and to identify the equation for the axis of symmetry. Okay. Alright. So let's start by finding the vertex and the intercepts that were told to use to draw our graph. Okay so we have our function F of X equals negative X plus four squared plus nine. Let's recall that we can write general equation of a parabola and vertex form as F of X is equal to a times X minus H squared plus K. Where the vertex is given by H comma K. Alright, well in our case the vertex H K r H is going to be this number in here. Now pay attention in the general form we have negative H here we have plus four. And so our H is actually going to be negative four. Okay. And r K is going to be nine and so we get the vertex negative 49. Alright, great. Okay. We also noticed that the A value is negative. Okay, A is equal to actually let's just say is less than zero. Okay, it's negative one. So A is less than zero. And so we know this Parabola is going to open downwards. Okay now we're told to find the intercepts or to use the intercepts for our graph. Okay let's just go down. We're gonna give ourselves a bit more space to work. Okay? So for the y intercept we want to set F X equal to zero. Sorry? So we get F of zero is equal to minus four squared plus nine, which gives us minus 16 plus nine, which is negative seven. Okay, so why intercept is that negative 7? What about our X intercepts? Well, in order to find our X intercepts, we want a factor, let's go ahead and expand first and then factor this into that standard factored form. So we have F of X is equal to negative X plus four squared plus nine. Oh this is equal to negative X squared Plus eight. x plus 16 Plus nine. Okay. And that negative has to apply to all three of those terms. And so F of X is going to be equal to negative X squared negative eight. X minus seven. Now, in order to find the X intercept, we want to set this equal to zero when it's equal to zero. We can divide by the negative. So now we're just finding the factored form of X squared plus eight. X plus seven. And we can factor this as X-plus seven and X plus one. Okay, The first term gives us a route of X equals negative seven. And the second term gives us a root of x equals negative one. Okay, so we have our X intercept or y intercept negative seven. An x intercept of negative seven and negative one. We know we have a vertex at negative 49. Let's go ahead and draw this graph. Alright. We're gonna draw it in this space here that we're given. Oh drawing our axes And let's just plot all of those important points. So we have our vertex at -4 and nine. This is -4 and nine. We have our y intercept at -7. We have x intercepts at negative one and negative seven. So we can draw our parabola that we know opens down like this. Okay. And you can imagine that that's a nice smooth parabola. Not as sharp as the graph I've drawn. Okay so we've drawn our graph. What else was the question asking? It wants us to identify the domain and the range based on the graph we've drawn and to identify the equation for the axis of symmetry. So let's start with the domain. We can see from our graph that we can have any X. Value. Okay any X value we put into this equation will get a corresponding Y value. We can represent any X. Value. So our domain is just any X. Value. Okay and if we want to write this Not in set notation we can write -18 or negative infinity. To infinity. In terms of the range. What do we see? But we see that our vertex has a Y value of nine. The parable opens downwards. And so our function will never attain a Y value higher than nine. Okay so we have y value any Y value where y is less than or equal to nine. Okay I'm writing this we get negative infinity up to and including nine. So we use that square bracket. The vertex is at a Y. Value of nine. So we do have to include it. Alright so we've done our domain, we've done our range. The last thing we want to do is find the equation for the axis of symmetry. Now the axis of symmetry when we have a parabola is going to be the line that goes through the vertex straight down like this. Okay now the equation of this axis of symmetry, it's going to be what? Well our vertex is at an X. Value of negative four. And so our axis of symmetry is just going to be the line X is equal to negative four. That's that red line we've drawn there and that's our axis of symmetry. Alright. So we found the graph of our function. We found the domain and range and our axis of symmetry. And if we compare this to the possible answers, we see that we have, see the axis of symmetry is X is equal to negative four. The domain is all real numbers and the range is from negative infinity to nine including nine. And the graph drawn here looks like the graph we found with the intercepts X intercepts at negative seven and negative one. The y intercept at negative seven and the vertex at negative four and nine. Thanks everyone for watching. I hope this video helped see you in the next one.

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In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x) = - (x + 1)^2 + 4