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the first problem: max and min of h(x)=-16x^2 +32x +4

h(x)= -16x^2 +32x +4 creates a parabola... plot some points to see for yourself to see this. A parabola is a function that only has a maximum or a minimum value, but not both. All you have to figure out which way the parabola is pointing and calculate the vertex and the vertex will be your max or min depending on the problem. Since the 1st coefficient (-16) is negative this graph points downwards. (this is similar to the fact that y=-x is a graph that is negative). That means there is no minimum points because the parabola goes down for forever. This means you only have a maximum value, which is the vertex.

So now we just use the equation for the vertex of the parabola. In order to so this it is helpful to think of this graph as y=ax^2+bx+c where -16=a, 32=b and 4=c. The equation for the vertex is x=-b/2a so x=-32/(2*(-16)) which comes out to be x=1. Then plug it back into the equation to figure out what y is... -16(1)^2+32(1)+4=20=y

so the max is the point (1,20)

The second problem: roots of x^2 +8x = -16
1st of all "find the roots of" is a fancy way of saying find the places where y=0. So let solve this: x^2 +8x = -16 that's the same as x^2 +8x +16 = 0. So we have to find the x's that make this true. Again this function is a parabola, which means that it either crosses the x axis at one place (at the vertex), at 2 places most of the time, or the rare case where the parabola never crosses the x-axis (in other words the case where the vertex is above the x-axis, and it is a positive parabola or the case where the vertex is below the x-axis, and it is a negative parabola).

So, how do we figure this out? Well the answer comes from remembering foiling. remember when you were given something like this (x+3)(x+2) and you had to solve by foiling? Well now we want to figure out how to go backwards. The reason is because in this form we can figure out easily where (x+3)(x+2)=0. This is a multiplication problem so we know that if x+3=0 then (0)(x+2)= 0 and if x+2=0 (x+3)(0)=0. And then once we have x+3=0 we can solve that x=-3 and similarly x+2=0 implies x=-2 so in this case x=-3 and x=-2 would be your answer.

So lets learn how to "unfoil" or factor. Let's start with looking at what foiling does using our example (x+3)(x+2)= x^2+2x+3x+6 if we look at what's happening carefully we have an x^2 so that means we started with 2 separate x's multiplied together during the process. So the factored form x^2 +8x +16 has to look like (x + ?)(x+ ?). In order to figure out what those question marks are lets look again at our example (x+3)(x+2)= x^2+2x+3x+6 = x^2+5x+6. We have 2 numbers that multiply together to get 6 but also add together to get 5. 2+3=5 and 2*3=6. So when unfoiling that is exactly what we want to look for. Looking at our problem we have x^2 +8x +16 so we want two numbers that multiply together to get 16 and add together to get 8. So lets look at the factors of 16: they are 1x16, 2x8, or 4x4. Do any of these numbers add up to 8? The answer is yes! 4+4=8. our question marks from before are 4 and 4. x^2 +8x +16= (x+4)(x+4) foil this and check that it's the right answer.

Ok so setting (x+4)(x+4)=0 we get that either the 1st x+4=0 which means x=-4 or the 2nd x+4=0 which again means x=-4. Since both of the x's are equal in this case this parabola only crosses the x axis at one point (-4,0) which is the vertex.

Last problem xp Write a quadratic function in standard form for each given set of zeros. 5 and -1.
Ok so we know that the parabola crosses the x-axis at the point (5,0) and at the point (-1,0) so we kinda just do the last problem backwards (which is the easier direction). We know that for the function x=5 and x=-1 so just solving you can get x-5=0 and x+1=0. These are your factor groups (x-5)(x+1)... now just foil. (x-5)(x+1)=x^2+x-5x-5

=x^2-4x+5

and we're done.