Welcome to the second part
of the lesson in writing linear functions
in slope-intercept form.
In this part, we'll find our equations
using tables and graphs.
In this example, we're asked to find the equation
of a linear function that would generate
the following table of values.
So we find the equation in a slope-intercept form
or m is equal to the slope and b is equal
to the vertical intercept.
So we'll find the equation in the form,
y equals mx plus b first and because it asks
for a function, we'll substitute f of x for y.
So this table of values gives us several ordered pairs
that would satisfy of a linear equation.
This also would be coordinates of points
that would be on the line.
So what we can do is select two ordered pairs,
let's use these two here.
Find the slope of the line using our slope formula
then performing substitution to find b.
So let's first find the slope, so we use the ordered pair
negative one comma 110 and the ordered pair one comma 46.
For our slope formula, let's call this x sub one
comma y sub one and x sub two comma y sub two.
So the slope is equal to y sub two minus y sub one
that's 46 minus 110 divided by x sub two minus x sub one
that would one minus negative one.
Simplifying, we have negative 64 divided by,
this becomes one plus one or two which equals negative 32.
So now we know the slope is equal to negative 32
which means the equation must be in the form
of y equals negative 32 x plus b or again b
as the vertical intercept.
And now we'll find the vertical intercept b
by selecting one of the ordered pairs
which we know satisfies the equation,
perform substitution and then solve for b.
So we can use any of the ordered pairs here on the table,
let's go ahead and use this ordered pair here.
So using the ordered pair one comma 46,
we'll substitute one for x and 46 for y.
So we'd have 46 equals negative 32 times one plus b
so we have 46 equals negative 32 plus b.
Notice how if we add 32 to both sides,
we have b equals 78.
And now we have all the information we need.
We know the slope is equal to negative 32.
We know the vertical intercept is equal to 78.
So again, we know m, so again if m equals negative 32
and b equals 78, our linear equation
would be y equals negative 32 x plus 78
but it does ask as a function, so we'll write this
as f of x equals negative 32 x plus 78.
This should be our linear function that generates
the following table.
Now we are asked on the graphing calculator,
so let's quickly do that.
We'll press y equals in our linear function.
So we have y one equals 32 x plus 78.
Now we use the table feature to check this
but let's first press second window and make sure
that the independent variable is set to ask
so we want to have the ask highlighted.
Now we'll press second graph for the table,
now I've already done this but I've entered
all of the x values or the inputs and notice how
the function values or outputs do match our table.
And I entered this one by one, I just entered negative five,
enter, negative three enter and as I enter the x values,
it generates the y value.
And now for the next question, we're asked to give
the equation of the linear function
using the line graph here.
So we'll determine two points on this line
and use those coordinates to determine the equation
of our line in the slope-intercept form.
So notice how this point here is the vertical intercept
with coordinates zero comma five
and now we can also determine the horizontal intercept here,
though we do have to be careful.
The horizontal axis is not scaled by one
so notice how this is negative two,
so this would be negative one
and therefore it's scaled by halves.
So the coordinates to this point would be negative 2.5
comma zero, let's go and write this over here.
We have negative 2.5 comma zero and we have zero comma five
but because this point here is a vertical intercept,
we know that b is equal to five.
Again, b is the vertical intercept
and now we can find the slope using our slope formula.
Let's call these the ones, x sub one, y sub one
and these the twos, x sub two, y sub two.
So the slope is equal to y sub two minus y sub one,
that's five minus zero.
X sub two minus x sub one would be zero minus negative 2.5.
Simplifying we have five divided by zero plus 2.5
or 2.5, five divided by 2.5 is two.
So now we have all the information we need
to write our linear function.
We know the slope is equal to two and b equals five.
So our equation is y equals two x plus five
but again, we're asked for a linear function.
So we'll write f of x equals two x plus five.
Now example four, we want to give the equation
of the horizontal line passing through
the point one comma three.
Let's first graph this line.
So here's the point one comma three
and therefore, this should be the horizontal line.
Now there's a quick, easy way to determine
the equation of a horizontal line
as well as a vertical line.
The equation of a horizontal line passing through
the point x sub one and comma y sub one
will always be y equals y sub one.
So notice the given point has a y coordinate
of positive three which means the equation of this line
is y equals three.
Let's try to understand why this is
the equation of the line.
And a couple ways to explain this, first,
we know that the coordinates of this point
are one comma three, we'll take a look at the coordinates
of several other points, we'll notice a pattern.
For example, this point here would be three comma three,
this point would be five comma three,
moving to the left, this would be negative one comma three
and here we'd have negative three comma three.
Notice for every point on this line,
the y coordinate is three and this is the reason why
the equation of this line is y equals three.
Yet another way to determine the equation of this line
is to recognize that a horizontal line
has a slope of zero, so if we know the slope is zero,
using slope-intercept form, we know the equation
of the line must be y equals zero x plus b.
We know zero times x would be zero.
So the equation would be y equals b
and then from here, we normally perform substitution
using a given point.
For example here, using the point one comma three,
we substitute one for x and three for y,
there's no x, there's only y.
So we substitute three for y, we would get three equals b.
So if we know the slope is zero
and the vertical intercept is three.
We could also find the equation as y equals zero
times x plus three which once again gives us y equals three.
So there are several ways to explain why the equation
of the horizontal line.
For number five, we're asked to give the equation
of the vertical line passing through
the point one comma three.
Let's first graph it, so we have the same point,
one comma three but the line is vertical,
this will be the graph of our line.
So notice out here, this is not a function
because it fails the vertical line test
but we can still determine the equation of this line.
So again, this is the point to one comma three,
the quickest way to determine the equation of this line
is to remember that, the equation of the vertical line
passing through the point x sub one comma y sub one
will always be x equals x sub one.
So you notice here, the x coordinate is positive one
and therefore the equation of this vertical line
is x equals one.
And we can't use slope-intercept form for this equation
because remember, the slope of this line is undefined.
But we can analyze the coordinates again,
to recognize the pattern.
Notice how this will be the point, one comma one,
this will be the point one comma negative one,
this will be one comma negative three and so on.
Notice every point on this line would have an x coordinate
of positive one and this is why the equation of this line
is x equals one, I hope you found this helpful.