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Course: Staging content lifeboat > Unit 19

Lesson 2: Article explorations

Parabolas and their quadratic equations

What you should be familiar with before taking this lesson

What you will learn in this lesson

So far you worked with graphs of linear equations, which have the shape of a straight line. In this lesson, you will learn about the graphs of quadratic equations, which have a different shape that we call a parabola.

You will learn about the main features of parabolas and the way they relate to the quadratic equations that define them.

The graph of $y = x^{2}$ ‍

Here is the graph of

y = x^{2}

with a few of its points highlighted.

Let's note a few of the graph's prominent features:

1. It is U-shaped

This curvy shape is shared by all graphs of quadratic functions, and it is what we call a parabola.

2. It has a minimum point at $(0, 0)$ ‍

This point, the tip of the U-shape, is what we call the vertex of the parabola.

This minimum point is at

(0, 0)

because for any non-zero

x

-value (both positive and negative),

x^{2} > 0

. Only for

x = 0

, we have

y = 0^{2} = 0

3. It is symmetrical about the $y$ ‍-axis

In other words, we can fold the graph across the

y

-axis and both halves would match exactly. We call this line the axis of symmetry of the parabola.

y = x^{2}

is symmetrical about the

y

-axis because the square of any number is equal to the square of its opposite number.

The graph being symmetrical means that for any point on the graph on one side of the

y

-axis, there's a point on the graph on the other side of the

y

-axis that is at the same height, and has the same distance from the

y

-axis.

The fact that the two points have the same height means they have the same

y

-value. The fact that they are the same distance from the

y

-axis means that their

x

-values are exact opposites.

This makes sense, because we know that exact opposites have the same square.

try floating the image

Parabolas with vertical shifts

Here is the parabola

y = x^{2} - 4

with a few of its points highlighted.

This graph has the exact same shape as the graph of

y = x^{2}

, but it is located

4

units below it. For example, the

x

-coordinate of the vertex is still

0

, but the

y

-coordinate is now

- 4

instead of

0

This is because for every

x

-value,

y = x^{2} - 4

assigns a

y

-value that is exactly

4

units less than

x^{2}

, which is what

y = x^{2}

assigns.

The graph is still symmetrical about the

y

-axis. Specifically, note that both

x = 2

and

x = - 2

correspond to

y = 0

. In other words, both

(2, 0)

and

(- 2, 0)

are

x

-intercepts of the graph. Unlike linear graphs, parabolas can have two

x

-intercepts!

Here is the parabola

y = x^{2} + 4

with a few of its points highlighted.

This parabola is the same as

y = x^{2}

, only

4

units above it. Specifically, its vertex is at

(0, 4)

Another interesting thing to note is that this parabola has no $x$ ‍-intercepts at all.

Putting things together

For any number $k$ ‍, the parabola $y = x^{2} + k$ ‍ is located $| k |$ ‍ units above the parabola $y = x^{2}$ ‍ $($ ‍if $k > 0)$ ‍ or $| k |$ ‍ units below it $($ ‍if $k < 0)$ ‍.
Specifically, the vertex of the parabola is at $(0, k)$ ‍.
No matter what $k$ ‍ is, the axis of symmetry is always the $y$ ‍-axis.

Now check your understanding with some practice questions.

Which of the following graphs can be the graph of $y = x^{2} - 5$ ‍ ?

Choose the equation that corresponds to the graph.

What is the vertex of $y = x^{2} + 7$ ‍?

Parabolas with horizontal shifts

Here is the parabola

y = (x - 3)^{2}

with a few of its points highlighted.

This graph is also a parabola, but it is located

3

units to the right from

y = x^{2}

. For example, the

y

-coordinate of the vertex is still

0

, but the

x

-coordinate is now

3

instead of

0

This is because for every

y

-value,

y = (x - 3)^{2}

assigns

x

-values that are

3

units greater than what

y = x^{2}

would assign to that same

y

-value.

Let's take the

y

-value

y = 4

as an example.

y = x^{2}

, in order for the

y

-value to be

4

, the

x

-value should be either

x = - 2

x = 2

y = (x - 3)^{2}

, in order for the

y

-value to be

4

, the

x

-value should be $3$ ‍ units greater than either

- 2

2

, so when we subtract

3

from it we will either get

- 2

2

This is why the graph of

y = x^{2}

has the points

(- 2, 4)

and

(2, 4)

and the graph of

y = (x - 3)^{2}

has the points

(- 2 + 3, 4)

and

(2 + 3, 4)

, which are

(1, 4)

and

(5, 4)

Another thing that was shifted

3

units to the right is the parabola's axis of symmetry, which is now the line

x = 3

Here is the parabola

y = (x + 5)^{2}

with a few of its points highlighted.

In this parabola, everything is shifted

5

units to the left.

Putting things together

For any number $h$ ‍, the parabola $y = (x - h)^{2}$ ‍ is located $| h |$ ‍ units to the right of the parabola $y = x^{2}$ ‍ $($ ‍if $k > 0)$ ‍ or $| k |$ ‍ units to the left of it it $($ ‍if $k < 0)$ ‍.
Specifically, the vertex of the parabola is at $(h, 0)$ ‍ and the axis of symmetry is $x = h$ ‍.

Now check your understanding with some practice questions.

Which of the following graphs can be the graph of $y = (x - 6)^{2}$ ‍ ?

Choose the equation that corresponds to the graph.

What is the vertex of $y = (x + 7)^{2}$ ‍?

Parabolas with vertical and horizontal shifts

Let's put everything we learned so far together, by considering the general parabola

y = (x - h)^{2} + k

it is U-shaped,
its vertex is at $(h, k)$ ‍, and
its axis of symmetry is the line $x = h$ ‍.

Let's have a little practice with that before moving on.

Which of the following graphs can be the graph of $y = (x - 4)^{2} - 4$ ‍ ?

Parabolas with vertical reflections

Here is the parabola

y = - x^{2}

with a few of its points highlighted.

Note that the graph is still U-shaped, but now it's turned upside down (you can say it's

\cap

-shaped). It is the exact reflection of

y = x^{2}

over the

x

-axis.

The vertex of this parabola is

(0, 0)

, like the parabola

y = x^{2}

, only now this vertex is the maximum point of the graph. The axis of symmetry remains unchanged as well, the line

x = 0

Here is the parabola

y = - (x - 3)^{2} + 5

with a few of its points highlighted.

We can see this graph as a result of the following transformations on

y = x^{2}

a shift to the right by $3$ ‍ units, resulting ni the equation $y = (x - 3)^{2}$ ‍,

a vertical reflection across the $x$ ‍-axis, resulting in the equation $y = - (x - 3)^{2}$ ‍, and

a shift up by $5$ ‍ units, resulting in the equation $y = - (x - 3)^{2} + 5$ ‍.

Because of the above transformations, the graph of

y = - (x - 3)^{2} + 5

has the following properties:

it opens down $(\cap)$ ‍,
its vertex is at $(3, 5)$ ‍, and
its axis of symmetry is the line $x = 3$ ‍.

Stretched and compressed parabolas

Here are the parabolas

y = x^{2}

y = 2 x^{2}

, and

y = \frac{1}{2} x^{2}

The parabola

y = 2 x^{2}

seems like

y = x^{2}

was stretched vertically. This is because for any

x

-value,

y = 2 x^{2}

assigns a

y

-value which is twice the

y

-value assigned by

y = x^{2}

For similar reasons, the parabola

y = \frac{1}{2} x^{2}

seems like a vertical compression of

y = x^{2}

This can be made general by thinking about the parabola

y = a x^{2}

If $| a | > 1$ ‍, the parabola is a vertical stretch of $y = x^{2}$ ‍, and the greater $| a |$ ‍ is, the more the parabola is stretched.
If $| a | < 1$ ‍, the parabola is a vertical compression of $y = x^{2}$ ‍, and the smaller $| a |$ ‍ is, the more the parabola is compressed.

Conclusion

Everything we learned here can be summarized by considering the most general parabola of them all,

y = a (x - h)^{2} + k

If $a > 0$ ‍ the parabola opens up $(\cup)$ ‍, and if $a < 0$ ‍ the parabola opens down $(\cap)$ ‍.
If $| a | > 1$ ‍ the parabola is stretched vertically, and if $| a | < 1$ ‍ the parabola is compressed vertically.
The vertex is at $(h, k)$ ‍.
The axis of symmetry is the line $x = h$ ‍.

Now let's practice everything together.

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Staging content lifeboat

Course: Staging content lifeboat > Unit 19

What you should be familiar with before taking this lesson

What you will learn in this lesson

The graph of y=x2‍

1. It is U-shaped

2. It has a minimum point at (0,0)‍

3. It is symmetrical about the y‍ -axis

Parabolas with vertical shifts

Putting things together

Parabolas with horizontal shifts

Putting things together

Parabolas with vertical and horizontal shifts

Parabolas with vertical reflections

Stretched and compressed parabolas

Conclusion

Want to join the conversation?

The graph of $y = x^{2}$ ‍

2. It has a minimum point at $(0, 0)$ ‍

3. It is symmetrical about the $y$ ‍-axis