We want to estimate
the slope of each tangent line at the points on the graph.
Let's begin by estimating the slope of the tangent line
and X equals negative three.
Notice that X equals negative three,
this should be the point in the function.
So notice how when X or the input is negative three,
the output or Y value
is approximately, let's say, 3.7.
So, the coordinates of this point of tangency
are negative three comma 3.7.
And now let's sketch the tangent line at this point.
The tangent line at this point
would look something like this.
That will help us find the slope of this line.
Let's find two points on the line.
So we'll use the point of tangency, this point here,
as well as, let's say, this point here.
Notice how the coordinates of this point
would be negative four comma one.
And now there are two ways we can find
the slope of this line.
In general, the slope is equal to the change in output
divided by the change in input.
Well, if our function is in terms of X and Y,
we can say the slope is equal to the change of Y
divided by the change of X,
which graphically represents the vertical change
divided by the horizontal change,
plus first approximate the slope using the graph.
Notice how if you want to move
form this point to this point in the line,
you would have to go up
approximately 2.7 units
for the change of Y is positive 2.7,
and then we'd have to go right one unit,
which means the change of X is positive one.
So now we know our approximation of the slope,
M, is going to be equal to 2.7 divided by one,
which equals 2.7.
Again, the 2.7 is the change of Y,
and the positive one is the change of X.
So X equals negative three.
The slope of the tangent line is M, is approximately 2.7.
The other option would be to use the slope formula here,
the quantity of Y sub two minus Y sub one,
divided by the quantity X sub two minus X sub one.
Let's also show this.
So we'll call these the twos and these the ones.
So using the formula, the slope is equal to
getting Y sub two minus Y sub one would be 3.7
minus one divided by X sub two minus X sub one
would be negative three minus negative four.
Simplifying, we get 2.7 divided by...
This simplifies the negative three plus four,
which is positive one,
giving us the same approximation of 2.7
for the slope of this tangent line.
Now let's approximate the slope of the tangent line
at X equals zero.
At X equals zero, this should be the point in the function,
we notice how the operator Y value is positive three.
So the coordinates of this point are zero comma three,
and now let's sketch the tangent line at this point.
The tangent line would look like this.
And I will select two points on this line
to help us approximate the slope.
So again, we'll use that point of tangency.
Let's also use this point here.
Notice how the coordinates at this point
would be negative one comma five.
Determine the slop graphically,
notice how to move from this point to this point,
we have to go down two units,
which means the change of Y is negative two,
and then we'd have to go right one unit,
which means the change of X is positive one.
And therefore, the approximation of the slope M
is equal to negative two divided by one,
which equals negative two.
So X equals zero.
The slope of the tangent line as M
is approximately negative two.
Let's also show it using the slope formula.
We already have the coordinates of the two points,
so we'll call these the ones and these the twos.
So the slope M would be equal to three minus five
divided by zero minus negative one.
So we have negative two divided by one,
which once again does gives us negative two.
For now, let's approximate the slope at X equals 1.8.
At X equals 1.8,
this should be the point on the function.
We notice how the output or Y value
is approximately 0.5.
So the coordinates of this point would be 1.8 comma 0.5.
Now let's go ahead
and sketch the tangent line at this point.
Notice how this point,
the tangent line is going to be horizontal,
which we should recognize would have a slope of zero.
If we choose two points on this line
to help us approximate the slope.
Let's go ahead and use the point of tangency
as well as this point here.
And notice how the second point would have coordinates,
four comma 0.5.
Notice how it move from this point to this point.
The vertical change, and the change in Y would be zero,
then the horizontal change would be 2.2.
Zero divided 2.2 is zero,
which is the reason why at X equals 1.8,
the slope of the tangent line as M is approximately zero.
If we want to shop that it's using the slope formula
we'd have the slope M is equal to
Y sub two minus Y sub one As 0.5 minus 0.5,
and then X sub two minus X sub one would be four minus 1.8.
Simplify, we get a zero divided by 2.2 which is zero.
I hope you found this helpful.