# Patterns And Linear Functions

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This video shows how patterns in function charts give you different values. Linear functions are where the function output changes at a constant rate, relative to the input.

An example of this would be the function:
$y = 4x$
$y = 4(1) = 4$
$y = 4(2) = 8$
$y = 4(3) = 12$
This function increases at a constant rate, as the input increases or decreases at a constant rate.

## Video-Lesson Transcript

Let’s go over patterns and linear functions.

Think of a linear function as a shape. When you put something in, it comes out as something else.

For example, when you put $x$ in a linear function, it will come out as $y$.

For example when we have $x = 0$ our $y = 3$. If $x = 2$, $y = 9$ and if $x = 7$, $y = 28$.

So how does this relates to linear functions?

When we graph using $x$ and $y$ axis and we plot the $x$-coordinates and $y$-coordinates, we will form a straight line. That’s something special about the linear function.

Here’s an example of a linear function:

When $x = 0$, $y = 3$,
when $x = 1$, $y = 5$,
if $x = 2$, $y = 7$,
when $x = 3$, $y = 9$,
and when $x = 4$, $y = 11$.

What makes this linear just by looking at this?

It’s because it increases at a constant rate.

When $x$ increase by $1$, $y$ increase by $2$.

Another example:

When $x = 3$, $y = 15$,
when $x = 4$, $y = 19$,
if $x = 5$, $y = 23$,
when $x = 6$, $y = 27$,
and when $x = 7$, $y = 31$.

Here when $x$ increase by $1$, $y$ increase by $4$.

Now, let’s graph this.

You will see that it’ll give us a straight slash line.

Let’s have another example:

When $x = 0$, $y = 0$,
when $x = 1$, $y = 3$,
if $x = 2$, $y = 6$,
when $x = 3$, $y = 9$,
and when $x = 4$, $y = 12$

Here, its $y = 3x$.

Another example is:

When $x = 0$, $y = 0$,
if $x = 1$, $y = 4$,
when $x = 2$, $y = 8$,
and when $x = 3$, $y = 12$

Here, $y = 4x$.

So we have

$y = 4x$

$y = 4(0) = 0$
$y = 4(1) = 4$
$y = 4(2) = 8$
$y = 4(3) = 12$

Same thing with the example above.

But they don’t always work out this way.

Let’s have a more complicated example.

When $x = 0$, $y = 2$,
when $x = 1$, $y = 5$,
if $x = 2$, $y = 8$,
when $x = 3$, $y = 11$,
and when $x = 4$, $y = 14$

It’s not like our previous examples where it’s a simple equation.

Here, we’re going to have a

$y = 3x + 2$

Let’s check using $x = 4$, let’s substitute

$y = 3x + 2$ $y = 3(4) + 2$ $y = 12 + 2$ $y = 14$