Example: Roots and vertex of a parabola

Algebra

Quadratics

Graphing quadratics

Ex 3: Graphing a quadratic function
Example: Graphing a quadratic
Example: Roots and vertex of a parabola
Example: Parabola vertex and axis of symmetry
Graphs of Quadratic Functions
Examples: Graphing and interpreting quadratics
Applying Quadratic Functions 1
Applying Quadratic Functions 2
Applying Quadratic Functions 3
Graphing parabolas in standard form
Parabola Focus and Directrix 1
Focus and Directrix of a Parabola 2
Vertex of a parabola
Graphing parabolas in vertex form
Graphing parabolas in all forms
Parabola intuition 3

Example: Roots and vertex of a parabola Quadratic Functions 3

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Example: Roots and vertex of a parabola

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We're asked to graph this function, finding the roots
and the vertex of this parabola.
So to find the roots, when people talk about roots,
they're talking about the points of intersection of this
function and the x-axis.
So this is the x-axis here, and this is the y-axis.
And you're going to intersect the x-axis when
y is equal to 0.
So to find the roots, we said y is equal to 0, so we solve
the equation, x squared plus 4x minus 12 is equal to 0.
All I did is I said y is equal to 0, and I swapped the left-
and right-hand sides.
Now, to factor this, we have a 1 as a leading coefficient, so
we don't have to do the all out grouping.
We can just think, are there two numbers whose products are
negative 12, and if I add them I get 4?
So let's think about it.
It looks like 2, positive 6, and negative 2 would work, if
I go through all of negative 12's factors.
So this can be factored as x plus 6, times x minus 2.
6 minus 2 is 4.
6 times negative 2 is negative 12, so that's going to be
equal to 0.
And if we have two numbers, and if, when you multiply
them, they equal 0, that means that either x plus 6 is equal
to 0, or x minus 2 is equal to 0.
Subtract 6 from both sides of this equation, you get x is
equal to negative 6.
Add 2 to both sides of this equation, and you get x is
equal to 2.
So y is equal to 0 when x is equal to negative 6, or x is
equal to 2.
So let's graph both of those points.
x is equal to negative 6, y will be equal to 0.
So x is 1, 2, 3, 4, 5, 6.
So negative 6 comma 0.
That will be on the graph of this function, or the graph of
this parabola.
And also the point, x is equal to 2.
So 1, 2.
Just like that. x is equal to 2.
So this is the point 2, 0.
Now to find the x value of the vertex, there's actually
several ways of doing it.
There is a formula that gives it to you.
But actually, the easiest way to figure it out is it's
actually going to be the midpoint between the two 0's.
If you already figured out the 0's, the x value where the
vertex was going to lie-- because the two 0's are always
symmetric around the axis of symmetry, which is the same x
value as the vertex.
You can just find the midpoint of these two points, and that
will be the x value of the vertex.
So let's just do that.
So if we take the midpoint, we can literally just average
these two numbers.
So negative 6 plus 2, over 2 is negative 4 over 2, which is
equal to negative 2.
So the vertex is x is equal to negative 2.
And when x is equal to negative 2, what is y?
So x is the vertex, and then y will be equal to negative 2
squared, is positive 4.
4 times negative 2 is negative 8.
And then we have a minus 12 there.
So 4 minus 8 gets us to negative 4, minus another 12
is negative 16.
So the point negative 2, negative 16 is
going to be our vertex.
So let me graph that.
I think it's going to fall off of this graph a little bit.
So negative 2, and then we're going to go 1, 2, 3, 4, 5, 6,
7, 8, 9, 10 11, 12, 13, 14, 15, 16.
So it's going to be right around here.
We went off the graph paper.
But negative 2 all the way down to negative 16.
So that right there is the vertex.
Negative 2, negative 16.
All the way down here.
We can even lower the y-axis.
And so if I were going to graph the parabola, it would
look something like this.
It would look something like that.
Just like that and obviously it would keep going.
Now, I figured out the vertex in this problem just by
averaging the x values of the two routes.
That gave me the x value of the axis of symmetry, which is
this right here. x is equal to negative 2.
That's the axis around which the parabola is symmetric.
The other way to figure out the x value of the vertex, is
to use the formula, negative b over 2a.
So the x value of the vertex is equal to negative b--
negative 4-- over 2 times a.
a is 1, so over 2, which is also going to give you x is
equal to negative 2.
Now, the last way to solve for the vertex is to complete the
square on this parabola right here.
So what we can do is we can rewrite it.
We can say this is y is equal to x squared plus 4x.
I'm going to put the minus 12 out here.
And what we want to do is add and subtract the same number,
so I can express this as a perfect square.
So if I take 1/2 of 4x, or if I take 1/2 of 4, I get 2 if I
square that.
I could add a 4 here, and then subtract a 4.
And then what that does is it turns this into x plus 2
squared, right? x plus 2 squared is x squared
plus 4x, plus 4.
We've seen that multiple times.
So you have x plus 2 squared.
And then you have the minus 4 minus 12, minus 16.
So this is just another way of writing
this exact same function.
But when you write it this way you see that the lowest point,
this part of the function right here is always positive.
It's a quantity squared.
The lowest value that it can be is 0.
But then it can only go up from 0.
So the lowest point, the lowest y value we can take on,
is when this expression right here is 0.
And when that happens, the y value is negative 16.
Now, what x makes this value 0?
Well, x equals negative 2 will make this 0.
If you wanted to say, x plus 2 is equal to 0, this expression
right here is going to be 0 when x plus 2 is equal to 0.
Subtract 2 from both sides, x is equal to negative 2.
So that's another way to find the vertex, all of them
equally valid.

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