Welcome to the first lesson
on connecting different representations of functions.
In the first example,
we're asked to identify the pattern from the table,
use that information to construct the graph,
and determine the formula for the function g(x).
Then, use words to describe the relationship
between the input and output variables.
Or, in this case,
the inputs are the x values,
and the outputs are the function values g(x).
Looking at the given table,
notice how we have several inputs,
as well as the corresponding outputs.
So, the next step will be to write the ordered pair
for each input and corresponding output.
So, the first ordered pair is going to be -3 comma -6.
Again, every ordered pair
is in the form of the input comma the output,
or in our case, x comma g(x).
So, the second ordered pair
is going to be -2 comma -4.
Followed by -1 comma -2.
0 comma 0.
1 comma 2.
2 comma 4,
and 3 comma 6.
Next, we'll plot the points given by the ordered pairs
on the Cartesian plane
where we always find the inputs along the horizontal axis,
and the outputs along the vertical axis.
So, because the input variable is x,
we'll label the horizontal axis x,
and because the outputs are g(x),
we'll label the vertical axis g(x).
Now, let's work on scaling the axes.
Notice how the least input is -3,
and the greatest input is positive 3.
We'll scale the horizontal axis by ones,
but for the horizontal axis,
let's let every two good marks
be equal to one unit.
So, we'll put a tick mark here,
here, here, and here in the positive direction
where we have one, two, three, four
in the positive direction.
Therefore, to the left
we would have -1, -2,
3, and -4,
but now for the vertical axis
because the least output is -6,
and the greatest output is positive 6.
We'll scale the vertical axis by ones.
Bullet every single grid mark equal to one unit.
So, in the positive direction,
we'd have 1, 2, 3,
4, and so on.
Notice how this vertical axis does accommodate
the maximum output of positive 6,
and the native direction are going down.
We'd have -1, -2, and so on.
Now, we'll plot the points given by the ordered pairs.
The first ordered pair is -3 comma -6.
So, because the input is -3,
and the output is -6.
From the origin, we move left three units
to where the input is -3,
and then, down six units
to where the output is -6.
So, at this point,
the input is -3,
and the output is -6.
The ordered pair -2 comma -4.
Input is -2,
output is -4.
The next ordered pair is -1 comma -2.
Input is -1,
output is -2.
Next ordered pair 0 comma 0,
which is here at the origin.
Next ordered pair is 1 comma 2.
Input is 1,
output is 2.
Next ordered pair is 2 comma 4.
Input is 2,
output is 4.
And, the last ordered pair is 3 comma 6,
where the input is 3,
and the output is 6.
So, our function passes through these seven points,
and because the graph is a line,
g(x) is a linear function.
Notice how for this line,
both the vertical and horizontal intercept
are the origin with coordinates 0 comma 0.
We want to write the symbolic rule,
the formula for g(x),
and then, express the rule in words.
So, we're going to determine the formula or equation or rule
that when the input is -3,
the output is -6.
When the input is -2,
the output is -4.
When the input is -1,
the output is -2,
and so on.
Notice how the output g(x)
is twice the value of the input x,
which means we can express the symbolic rule for g(x)
as g(x) equals two times the input x,
or g(x) equals 2x.
To express this in words,
we can say the function g(x)
doubles the input x.
We could also express the function rule in words
as the function g(x) multiplies the input by two.
Technically though, when we say
g(x) multiplies the input by two,
that would be the function rule
g(x) equals x times 2,
but because the commutative property of multiplication
x times two is equivalent to two times x
or just 2x.
I hope you found this helpful.