Okay so all these are a lovely vertex form where we can pluck the vertex right out because the transformation to take the opposite when it's grouped in like this so it'll be a 1 and then a negative 3 so it's gonna be 1 negative 3 and I'll just do all the vertices right so there's nothing happening to X so that's 0 and a 4 and then negative 3 & 1 & 4 & 2 & 3 & 0 and so the direction you can tell because of the sign of the eighth term here so since its negative its opening down since it's positive its opening up since it's negative its opening down since it's positive its opening up since it's negative its opening down since it's positive its opening and our axis of symmetry if I start to plot this one my vertex is 1 negative 3 and my parabola goes up and down and so my axis of symmetry goes straight through the vertex so the equation for your axis of symmetry is always x equals whatever the x-coordinate of the vertex is right that line has a lovely equation x equals 1 x equals 1 all about that line so we got x equals 1 and I'll just keep going pregnant for this one so our y-intercept is going to be always for any graph the y-intercept happens when the x is 0 so if I plug in 0 for this I'll have negative 2 times 0 minus 1 squared minus 3 and that'll be what Y is so negative 1 squared is 1 times negative 2 is negative 2 minus 3 is minus 5 so we've got it have a y-intercept of five and so I can plot that one two three four five that's looking good it's gonna open down just like we thought it would and because I have that this axis of symmetry it's nice and symmetric I get points on the other side of my axis of symmetry for free so if this is a point then it's one away from the axis they go one away from the axis on that side and that'll be enough to get a nice oops I can go through dots ah nice parabola and so they asked for the number of x-intercepts here and it never ever crosses the x-axis so there are no to the next so if I plot my vertex it's zero one two three four my axis of symmetry goes right through my vertex so the equation for my axis of symmetry is x equals whatever the X is in the vertex my y-intercept always happens when the x is zero so the X is zero my Y is 1/2 times zero squared plus four so that's just 0 plus 4 is 4 oh ha ha we already have that but anyway so now it's like okay I don't get any points on the side for free so if I want to graph it I might just need to make a little XY table to get a couple extra points so maybe I'll plug in a 1 and a 2 so if I plug in a 1 I get 1 squared is 1 times 1/2 is 1/2 plus 4 is 4 and 1/2 if I plug in a 2 2 squared is 4 times 1/2 is 2 plus 4 is 6 so I get 1 and 4 and 1/2 and 2 and 1 2 3 4 5 6 then I get points on the other side of my axis of symmetry for free so 1 2 1 2 and it's 1 away so it's one way on the other side we have a slightly fatter parabola than our standard parabola if that value of a is a fraction smaller than one then it gets slightly...