### Video Transcript

In this video, we will learn how to
find the slope of a line using graphs and tables. We will begin by recalling some key
facts of linear functions. The graph of any linear function is
a straight line. And the equation of any linear
function is written in the form π¦ equals ππ₯ plus π. The letters π and π are
constants, where π represents the slope or gradient of the line. π represents the π¦-intercept, the
point at which our line crosses the π¦-axis. This is sometimes written as π¦
equals ππ₯ plus π instead of π.

The value of π will be positive if
our straight line slopes up from left to right. π will be negative if our line
slopes down from left to right. The absolute value of π determines
how steep the slope is and its sign gives the direction of the slope. For example, the equation π¦ equals
three π₯ plus four will be steeper than the equation π¦ equals two π₯ minus
seven. This is because the value of π is
greater. As π represents the slope, it
follows that the value of π is the rate of vertical change in the π¦-coordinates to
the horizontal change in the π₯-coordinates between any two points.

This can be written using the
following formula. π is equal to π¦ two minus π¦ one
divided by π₯ two minus π₯ one, where two points on the line π΄ and π΅ have
coordinates π₯ one, π¦ one and π₯ two, π¦ two. This is often referred to as the
change in π¦ over the change in π₯ or the rise over the run. We will now look at how we can
apply this to find the slope of a linear function given its graph.

What is the slope of the
function represented by the given figure?

We know that any straight line
graph must be a linear function written in the form π¦ equals ππ₯ plus π,
where the value of π is the slope or gradient of the function. The value of the slope π can
be calculated using the following formula. π¦ two minus π¦ one over π₯ two
minus π₯ one, where π΄ and π΅ are two coordinates on the line with coordinates
π₯ one, π¦ one and π₯ two, π¦ two. We begin by selecting any two
points on our straight line. If possible, it is often useful
to choose points where the line crosses the π₯- and π¦-axis. In this case, π΄ has
coordinates zero, 10 and π΅ has coordinates five, zero.

At this stage, it often helps
to create a right-angled triangle on our graph. This will help us calculate the
change in π¦ and the change in π₯, otherwise known as the rise and the run. Substituting our π¦-coordinates
into the formula gives us zero minus 10. Substituting in our
π₯-coordinates gives us five minus zero. It doesnβt matter which point
is π₯ one, π¦ one and which is π₯ two, π¦ two. But we must be consistent in
our order. Zero minus 10 is equal to
negative 10. Five minus zero is equal to
five. Negative 10 divided by five is
equal to negative two. This means that the slope of
the function represented on the graph is negative two.

We can check this on the graph
by considering the rise and the run. The rise is negative 10, as the
π¦-coordinate drops from 10 to zero. The run is five, as the
π₯-coordinate goes from zero to five. Once again, we have negative 10
divided by five. An important check is that any
line that goes up from left to right will have a positive slope. Any line that goes down from
left to right will have a negative slope. As our line goes downwards from
left to right, a negative answer is sensible.

We will now look at a second graph
question.

Calculate the slope of the line
in the graph.

We know that any straight line
is a linear function that can be written in the form π¦ equals ππ₯ plus π,
where π is the slope or gradient of the line. The value of π can be
calculated using the formula π¦ two minus π¦ one over π₯ two minus π₯ one. This is the change in
π¦-coordinates over the change in π₯-coordinates, otherwise known as the rise
over the run. We begin by selecting any two
points on the line π΄ and π΅ with coordinates π₯ one, π¦ one and π₯ two, π¦
two. Whilst it doesnβt matter which
two points we choose, it is sensible to pick those with integer coordinates
where possible.

In this question, we will
choose the two points shown on the graph. Point π΄ has coordinates zero,
one and point π΅ has coordinates two, seven. At this point, it is worth
drawing a right-angled triangle on the graph to show the rise and the run. The rise in this case is equal
to six, as the change in π¦-coordinates is six. The run is equal to two. This means that we would expect
the slope to be six divided by two, which is three.

We can check this by
substituting our coordinates into the formula. The two π¦-coordinates were
seven and one. And the corresponding
π₯-coordinates were two and zero. This simplifies to six over
two, which once again gives us an answer of three. The slope of the line in the
graph is three. It is worth recalling that any
line that slopes upwards from left to right will have a positive slope. As three is positive, this
suggests that our answer is correct.

We will now look at a question that
involves finding the slope of a linear function from a table.

What is the slope of the linear
function represented by the given table?

We know that the equation of
any linear function is written in the form π¦ equals ππ₯ plus π, where π is
the slope or gradient of the function and π is the π¦-intercept. We can calculate the value of
the slope π using the following formula, π¦ two minus π¦ one over π₯ two minus
π₯ one. This is the change in
π¦-coordinates over the change in π₯-coordinates, where any two points π΄ and π΅
have coordinates π₯ one, π¦ one and π₯ two, π¦ two, respectively.

In our table, we have three
coordinates, firstly, zero, four. Our second coordinate has an
π₯-value of two and a π¦-value of 10. Our third coordinate, which we
will call πΆ, is four, 16. We can select any two of these
three coordinates. In this question, we will begin
by considering point π΄ and point π΅. The π¦-coordinates of these two
points are 10 and four. The corresponding
π₯-coordinates are two and zero. The slope π is equal to 10
minus four over two minus zero. This simplifies to six over
two, giving us a final answer of a slope of three.

We can check this answer by
selecting a different two points, in this case point π΄ and point πΆ. This time the slope is equal to
16 minus four over four minus zero. 12 divided by four is also
equal to three. We would get the same answer if
we use the points π΅ and πΆ. The slope of the linear
function represented by the table is three.

We could also calculate this
answer just by looking at the table. The change in π₯-values between
the first and second point is plus two. The change in the π¦-values
between the first two points is plus six. As the slope is equal to the
change in π¦-values divided by the change in π₯-values, this also gives us an
answer of three. For each single unit the
π₯-value increases, the π¦-value will increase by three units.

Our next question will include a
graph in a real-world context.

The graph shows the distance
Amelia traveled over her two-hour bike ride. Which of the following is
true? A) She traveled at a constant
speed of four miles per hour for the last hour. B) She traveled at a constant
speed of 10 miles per hour for the entire ride. C) She traveled at a constant
speed of eight miles per hour for the last hour. Or D) she traveled at a
constant speed of seven miles per hour for the entire ride.

We can see from the graph that
the π₯-axis represents the time in hours and the π¦-axis represents the distance
in miles. The speed or velocity in any
distance-time graph can be calculated by dividing the change in distance between
any two points by the change in time. If the graph is a straight line
for the entire journey, then they will be traveling at a constant speed. We can see from the graph that
three parts of the journey have different slopes or gradients. This means that, during these
three parts, Amelia will be traveling at different speeds.

We can therefore rule out
options B and D, as these stated that she traveled at a constant speed for the
entire ride. This is not the case as she
will have traveled at three different speeds. Both of the other statements
relate to the last hour of Ameliaβs journey. This occurs between the two
points π΄ and π΅ on the graph. We can calculate the slope
between any two points on a graph by using the following formula, π¦ two minus
π¦ one over π₯ two minus π₯ one. This is the change in
π¦-coordinates over the change in π₯-coordinates, in this case the change in the
distance over the change in the time.

Point π΄ has coordinates one,
10 and point π΅ has coordinates two, 14. The π¦-coordinates or distances
here are 14 and 10. The corresponding
π₯-coordinates are two and one. 14 minus 10 is equal to four
and two minus one is equal to one. This means that the slope of
the line between points π΄ and π΅ is four. We could also have worked this
out by drawing a right-angled triangle on the graph. We can see here that the
distance has risen from 10 to 14. And the time has gone from one
hour to two hours. Four divided by one is equal to
four. So once again, the slope equals
four.

As the slope in a distance-time
graph is equal to the speed, we can conclude that the speed in the last hour was
four miles per hour. This rules out option C and
therefore option A is correct. Amelia traveled at a constant
speed of four miles per hour for the last hour.

We will now recap some of the key
points from this video. The graph of any linear function is
a straight line. A linear function has a constant
rate of change, which means that the difference in the π¦-coordinates of any two
points on the straight line is proportional to the difference in their
π₯-coordinates. This rate of change is the slope of
the line. The equation of a line is generally
written in the form π¦ equals ππ₯ plus π, where π is the slope or gradient of the
line and π is the π¦-intercept. This is the point where the line
crosses the π¦-axis.

Finally, the slope of a line π is
the rate of the vertical change to the horizontal change between two points. For two points π΄ π₯ one, π¦ one
and π΅ π₯ two, π¦ two lying on a line, the slope is π, which is equal to π¦ two
minus π¦ one over π₯ two minus π₯ one. If this number is positive, our
line will slope upwards from left to right. And if it is negative, it will
slope downwards from left to right.