Video Transcript
What is the initial value and the
rate of change for the function represented by the given graph?
As our graph is a straight line, we
know that the function will be linear. This means that it can be written
in the form 𝑦 equals 𝑚𝑥 plus 𝑏. The value of 𝑚 is the slope or
gradient of the line. The value of 𝑏 is the
𝑦-intercept, which is where the graph crosses the 𝑦-axis.
The initial value of a function is
its value when 𝑥 equals zero. This is the same as the
𝑦-intercept. We can see from the graph that our
line crosses the 𝑦-axis at the point zero, eight. Therefore, the initial value is
eight.
The rate of change of any function
is defined by the change in 𝑦 over the change in 𝑥. This is equivalent to the slope or
gradient of the line and can be calculated using the formula 𝑦 one minus 𝑦 two
over 𝑥 one minus 𝑥 two, where two points on the line have coordinates 𝑥 one, 𝑦
one and 𝑥 two, 𝑦 two.
As we already have one point on the
line zero, eight, we need to select any other point that lies on the line. It is useful to select a point with
integer coordinates. In this case, we will select the
point one, two. Point 𝐴 has coordinates zero,
eight. And point 𝐵 has coordinates one,
two. Subtracting the 𝑦-coordinates
gives us eight minus two. Subtracting the 𝑥-coordinates in
the same order gives us zero minus one. This can be simplified to six
divided by negative one. Dividing a positive number by a
negative number gives a negative answer. Therefore, the rate of change is
equal to negative six.
As our straight line is sloping
downwards from left to right, we know that our slope or gradient will be
negative. Therefore, this answer of negative
six seems sensible. As the rate of change is equal to
negative six and the initial value is eight, the equation of this straight line is
𝑦 equals negative six 𝑥 plus eight.
