Video Transcript
The graph in the 𝑥𝑦-plane of the
function 𝑦 equals 𝑓 of 𝑥 is shown. If the graph of 𝑦 equals negative
seven 𝑥 plus two were drawn on the same axes, which of the following would not be a
point of intersection of the two graphs? Is it A) zero, zero; B) one,
negative five; C) negative two, 16; or D) negative one, nine?
The equation 𝑦 equals negative
seven 𝑥 plus two is written in the form 𝑦 equals 𝑚𝑥 plus 𝑏. This means that it will have a
slope equivalent to 𝑚 and a 𝑦-intercept equivalent to 𝑏. The straight-line graph 𝑦 equals
negative seven 𝑥 plus two has a slope of negative seven and a 𝑦-intercept of two
or positive two.
As the 𝑦-intercept is equal to
positive two, the graph will pass through the point or coordinate zero, two. As the slope of the graph is
negative seven, for every one unit we move in the 𝑥-direction, we need to move
negative seven in the 𝑦-direction. This means that the graph will also
pass through the point one, negative five, as two minus seven is equal to negative
five. We can draw a straight line through
these points as shown on the graph.
As our graph is quite small, it is
not clear exactly which points the graph passes through. We will now go through a method to
show that it passes through negative one, nine and negative two, 16. If we consider the 𝑥-coordinates
negative two, negative one, zero, and one, we can work out their corresponding
𝑦-values by substituting the 𝑥-values into the equation 𝑦 equals negative seven
𝑥 plus two.
Substituting in 𝑥 equals negative
two gives us 𝑦 is equal to negative seven multiplied by negative two plus two. Negative seven multiplied by
negative two is equal to 14, as multiplying two negative numbers gives us a
positive. Adding two to this gives us an
answer of 16. This means that our first
coordinate or point that lies on the straight line 𝑦 equals negative seven 𝑥 plus
two is negative two, 16.
Substituting in 𝑥 equals negative
one gives us 𝑦 is equal to negative seven multiplied by negative one plus two. Negative seven multiplied by
negative one is equal to seven. Adding two to this gives us an
answer of 𝑦 equals nine. This means that our second
coordinate that lies on the graph 𝑦 equals negative seven 𝑥 plus two is negative
one, nine. We can repeat this process with 𝑥
equals zero, which gives us a value of 𝑦 equal to two. Therefore, the third coordinate
that lies on the line is zero, two.
Finally, substituting 𝑥 equals one
into the equation 𝑦 equals negative seven 𝑥 plus two gives us a 𝑦-value equal to
negative five. Our fourth coordinate is one,
negative five.
We were asked in the question to
work out which of the four points given would not be a point of intersection of the
two graphs. This means that it would not lie on
the function 𝑓 of 𝑥 and on the graph 𝑦 equals negative seven 𝑥 plus two. We have shown from our table and
the graph that the three points — negative two, 16; negative one, nine; and one,
negative five — all lie on the graph 𝑦 equals negative seven 𝑥 plus two. As the point zero, two lies on the
graph 𝑦 equals negative seven 𝑥 plus two, then the point zero, zero is not a point
of intersection of the two graphs.
Whilst the function 𝑦 equals 𝑓 of
𝑥 passes through the point zero, zero, it is clear from our table and the straight
line on our graph that the line 𝑦 equals negative seven 𝑥 plus two does not pass
through this point.
