Video Transcript
By making a table of values, determine which of the following is the function represented by the graph shown? Option (A) 𝑦 equals one-fifth 𝑥 minus one. Option (B) 𝑦 equals one-fifth 𝑥 plus one. Option (C) 𝑦 equals negative one-fifth 𝑥 plus one. Option (D) 𝑦 equals negative one-fifth 𝑥 minus one. Or option (E) 𝑦 equals five 𝑥 plus one.
We can recall that, generally, a function is a rule where we express the output variable, often called 𝑦, in terms of the input variable, often called 𝑥. Each input value 𝑥 produces a unique output value 𝑦 equals 𝑓 of 𝑥. And we can combine these input and output values to form an ordered pair. For example, the ordered pair 𝑥, 𝑦 or 𝑥, 𝑓 of 𝑥 corresponds to the coordinates of one point on the graph of the function. So, in this problem, if we create a table of values of these ordered pairs of input and output values, we can then use each function to determine if that particular function produces those values.
The first thing to do then is to identify some ordered pairs or coordinates from the graph. Now, we can take any coordinates that the line passes through. But often the easiest ones to pick are those which have integer coordinates, for example, negative five, zero; zero, one; and five, two. We can then create the table of values. So, at the coordinate negative five, zero, the 𝑥-value is negative five and the 𝑦-value or output value is zero. The coordinate zero, one has an 𝑥-value of zero and a 𝑦-value of one. Similarly, the third coordinate five, two has an 𝑥-value of five and a 𝑦-value of two.
So one way in which to check if the table of values is produced by each function is to take this first ordered pair of negative five, zero and see if each function in turn would produce this or not. Let’s begin with the function given in option (A), which is 𝑦 equals one-fifth 𝑥 minus one. Because the 𝑥-value is negative five, we can substitute this into the equation, and we’re really hoping to get a 𝑦-value equal to zero. But when we multiply one-fifth by negative five, we get an answer of negative one. And subtracting one would give us negative two. That means we are not getting the output value of zero that we were looking for. And therefore, this function given in option (A) would not produce the graph that we are shown.
We can now do the same thing and check this coordinate negative five, zero with the function given in option (B). The input value of 𝑥 equals negative five can be substituted to give 𝑦 equals one-fifth times negative five plus one. This gives us 𝑦 is equal to zero. This is the output value that we are given on the graph. So option (B) may be the answer.
We can continue to check the other functions. This time, option (C) gives us 𝑦 equals negative one-fifth times negative five plus one when we substitute the value 𝑥 is equal to negative five. When we simplify this, we get a 𝑦-value of two, which does not correspond to the output value of zero, which we are looking for. Option (C) can therefore be eliminated.
Substituting 𝑥 equals negative five into the two remaining functions, we have outputs of zero and outputs of negative 24. This output value of negative 24 in option (E) means that we can eliminate this function.
We are now left with two functions, one of which is the correct function. So let’s substitute the input or 𝑥-value from this second ordered pair zero, one into each of these two functions. When we do so, we can observe that the output value in function (B) is one and the output value 𝑦 in function (D) is negative one. That means that we can eliminate answer option (D).
We can therefore answer this question that the function which is represented by the graph is 𝑦 equals one-fifth 𝑥 plus one. And if we wished to check this, we could substitute the third ordered pair into the function. When 𝑥 is equal to five, we would have 𝑦 is equal to one-fifth times five plus one. And that simplifies to 𝑦 is equal to two. Thus, we have demonstrated that there are three different ordered pairs on the graph, which are also produced by the function 𝑦 equals one-fifth 𝑥 plus one.
