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Introduction to Parabolas

Introduction to Parabolas

Author: Anthony Varela

Determine the vertex and direction of opening of the parabola for a quadratic equation.

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Video Transcription

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Hi, my name is Anthony [INAUDIBLE]. And today, I'm going to introduce parabolas. So we'll talk about the shape of parabolas, the vertex of parabolas and then also how we can find the vertex just from an equation.

So first, I'd like to talk about the shape of parabolas. So here, we have a sketch of a parabola that could be y equals 2x squared minus 6x plus 1.

And looking at the ending behavior of this parabola-- so it's a U-shaped parabola. And as x approaches negative infinity-- so that would be on this side of the graph-- we can see that y continues to approach positive infinity. Well, as x approaches positive infinity, y also approaches positive infinity as well. So on both ends, y is traveling upwards.

Let's contrast this with a parabola that can open downwards. So this might be the graph of y equals negative 2x squared minus 5x minus 7. So we can see here then, that as x approaches negative infinity, so does y. y approaches negative infinity, as well.

As x approaches positive infinity-- so the other direction on the x-axis-- y still continues to travel towards negative infinity. So this is a downward-opening parabola. And on both ends, y travels towards negative infinity.

And this actually depends on one variable in our equation, y equals ax squared plus bx plus c. And that is this leading coefficient, this a value, the coefficient in front of the x squared term.

If a is a positive number, so greater than 0, this will be a parabola that opens up, so something that looks like this, a U-shaped curve. If this a value is a negative number, so less than zero, then we're going to have a parabola that opens down. So we're going to have that upside down U-shape to it.

So next, let's talk about the vertex of a parabola and symmetry in parabolas. So the vertex of a parabola is either the maximum or minimum point of a parabola. And it's located on what's called the axis of symmetry. So we'll get to that in a moment. But I'd like to point out on the graph what vertices look like.

So it's either a maximum or a minimum point. So we can see here, in our parabola that opens downward, it is a maximum point. In our parabolas that open up, it is a minimum point.

Now it is located on the axis of symmetry. So the axis of symmetry is a line of reflection passing through the vertex of a parabola. In up and down parabolas, it's a vertical line. So here, we see this vertical line that acts as a line of reflection. So parabolas are symmetrical at the vertex at this axis of symmetry.

And what this means then is we can plot a point on our parabola and reflect it across the line. And we'll get another point on the parabola. So we can see that in both of these pictures. Plotting a point, it gets reflected about this axis of symmetry. And it lands on the other side of our parabola.

So the vertex is either a minimum or a maximum point. It is on this axis of symmetry or a line of reflection for parabolas. So how can we find the vertex of a parabola? So we're going to be using an equation to find the vertex. And we'll go through two different examples.

One, if our equation is given to us in standard form-- so y equals ax squared plus bx plus c-- we can find the vertex using this formula, x equals negative b over 2a. Now, this happens to also give us the equation of that vertical axis of symmetry. So this gives us the x-coordinates of our vertex. And b and a come from our standard form. There is a, b and c.

So if we have this equation, y equals 2x squared minus 2x plus 3, what we can do is use this equation, x equals negative b over 2a, find the x-coordinate of the vertex, plug that x value into our equation, solve for y. That would give us the y-coordinate of the vertex.

So a is 2. B is negative 2. And c is 3. We don't really need c. We just need b and a. So x equals negative b over 2a. That would mean that x equals negative negative 2-- so that'd be a positive 2-- over 2 times 2. That would be 4.

So our x-coordinate to the vertex is 2/4 or 1/2. So now we can substitute 1/2 in for x and solve for y. So when x equals 1/2, 2x squared is 1/2 minus 2x, would be minus 1, and then plus 3. So y equals 2 and 1/2. So that's the y-coordinate to our vertex. So this vertex is at 1/2 comma 2 and 1/2.

Now we can also find the vertex from another form of an equation. And this is the vertex form. And it's called vertex form for a reason. We'll get to this in a minute. And this is y equals a times x minus h quantity squared plus k.

Now we call it vertex form because our variables h and k represent the x and y-coordinates, respectively, of the vertex. So this h value corresponds to the x-coordinate of the vertex. k corresponds to the y-coordinate of the vertex.

So if we're given this equation, y equals x minus 3 quantity squared minus 4, we can line up h and k in the equation and easily identify the vertex. Now be careful, because this is generally x minus h. So we have x minus 3. So I know that 3 is going to be h, not negative 3.

If this were x plus 3, then h would be negative 3. And here, we can see, generally, we have plus k. So this is minus 4. So k equals negative 4. So keep the signs in mind. Our vertex is at 3 comma negative 4.

So we can locate then that on our graph of this equation. So here's our vertex at 3 negative 4.

So let's review our introduction to parabolas. If we have our equation, y equals ax squared plus bx plus c, our leading coefficient, a, will determine if our parabola opens up. That would be if a is a positive number or down if a is a negative number.

We talked about the vertex being either a minimum or maximum point of a parabola. And it's on the axis of symmetry, which can be thought of as a line of reflection.

We can find the x-coordinate of the vertex if we have our equation given in standard form, using negative b over 2a. If we have our equation in vertex form, we can find the vertex by looking at the variables h and k.

So thanks for watching this tutorial and an introduction to parabolas. Hope to see you next time.

Terms to Know
Axis of Symmetry

a line of reflection passing through the vertex of a parabola; in up and down parabolas, it is a vertical line.

Vertex (of a parabola)

the maximum or minimum point of a parabola located on the axis of symmetry

Formulas to Know
Vertex Form of a Quadratic Equation

y equals a left parenthesis x minus h right parenthesis squared plus k

X-Coordinate of Vertex

x equals fraction numerator negative b over denominator 2 a end fraction