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 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 is
the slope or gradient of the function and 𝑏 is the 𝑦-intercept. We can calculate the value of
the slope 𝑚 using the following formula, 𝑦 two minus 𝑦 one over 𝑥 two minus
𝑥 one. This is the change in
𝑦-coordinates over the change in 𝑥-coordinates, where any two points 𝐴 and 𝐵
have coordinates 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two, respectively.
In our table, we have three
coordinates, firstly, zero, four. Our second coordinate has an
𝑥-value of two and a 𝑦-value of 10. Our third coordinate, which we
will call 𝐶, is four, 16. We can select any two of these
three coordinates. In this question, we will begin
by considering point 𝐴 and point 𝐵. The 𝑦-coordinates of these two
points are 10 and four. The corresponding
𝑥-coordinates are two and zero. The slope 𝑚 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 𝐴 and point 𝐶. 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 𝐵 and 𝐶. 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 𝑥-values between
the first and second point is plus two. The change in the 𝑦-values
between the first two points is plus six. As the slope is equal to the
change in 𝑦-values divided by the change in 𝑥-values, this also gives us an
answer of three. For each single unit the
𝑥-value increases, the 𝑦-value will increase by three units.