Hey, everyone. Let's work through this example together. So here we want to graph the given quadratic function identifying all of the possible information about it. Now here we have the function f of x is equal to negative one half times (x plus one)2 plus 2. So let's get right into graphing.
Now looking at the very first thing I want to do, I want to identify my vertex, which when written in vertex form is h comma k. Now just a reminder of vertex form, remember it's a times (x minus h)2 plus k. So identifying our vertex, we need to identify both h and k. Now here in my function, this is x+1, and I know in vertex form it's x minus h. So whenever we have a plus in that, we want to make sure and be careful and identify this as x minus negative one so that we know our h is not positive one it is negative one here. So here my x value, my vertex is negative 1, and then k we have this positive 2. So the vertex point is negative 1, 2. Now, is this a minimum or a maximum point? Well, looking back at my function, it has this negative at the front, which tells me that my parabola is going to be opening downward, telling me that I have a maximum point as my vertex. So my vertex is negative 1, 2 and it is at a maximum.
Now let's move on to step number 2 and identify our axis of symmetry, which is simply x is equal to h, which we know that h is negative one. So this is simply the line x is equal to negative one.
Now moving on to step 3, finding our x intercepts, steps 3 and 4 are going to be a little bit more involved because we need to calculate some stuff, so I'm going to be doing my work for these 2 right at the bottom here. Let's go ahead and set up our equation for step 3 and set up f of x is equal to 0. So my function here, negative one half times (x + 1)2 + 2 is equal to 0.
So looking down here at my function, since I have something squared and a constant, I know that I'm going to go ahead and use the square root property. Whenever we have our function in vertex form, we can always solve using the square root property. So I'm going to go ahead and move this 2 over to the other side. So subtracting 2 from both sides leaves me with negative one half times (x + 1)2 is equal to negative 2. Now from here, I want to go ahead and cancel out this negative one half, which I can do by simply multiplying both sides by negative 2, canceling that one negative one half out and leaving me on this side with (x + 1)2 is equal to 4. Now from here, I can go ahead and apply the square root property and simply take the square root of both sides, leaving me with x + 1 is equal to plus or minus 2. Now from here, I want to go ahead and move that positive one over to the other side by simply subtracting it, so minus 1 on both sides, and I'm left with x is equal to negative one plus or minus 2, which I know I can split into my 2 possible answers. Negative one plus 2 gives me a positive one, and then negative one minus 2 is going to give me a negative 3. So here I have my 2 zeros or my 2 x intercepts, one and negative 3. So filling that in on my table up here, I have 1 and negative 3 as my x intercepts, and we can go ahead and move on to step 4 and find our y intercept by calculating f of 0, plugging 0 into my function for x.
So my function here, f of 0 negative one half times (0+1)2 plus 2. Now I can simply compute this by simplifying negative one half. 0 + 1 is just 1, so this is 12 plus 2. Simplifying this further, ...