Video Transcript
In this video, we will learn how to
complete a function table for a linear function. We will begin by defining what we
mean by some of the key terms.
A function table displays the
inputs and corresponding outputs of a function. If we consider the function machine
shown, we need to add seven to the input to calculate the output. Creating a function table as shown
allows us to substitute different values for the input and then calculate the
corresponding output. If we select the inputs three, six,
and 10, we can calculate their corresponding outputs by adding seven to each of the
numbers. Three plus seven is equal to 10,
six plus seven is equal to 13, and 10 plus seven is equal to 17. The inputs three, six, and 10 in
this function give us outputs of 10, 13, and 17, respectively.
A linear function is a function
whose graph is a straight line. It is usually written in the form
𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 is the slope or gradient and 𝑏 is the
𝑦-intercept. A linear function has one
independent variable 𝑥 and one dependent variable 𝑦. Our 𝑥-values are the inputs, and
our corresponding 𝑦-values are the outputs. If we consider the function machine
and table above, we can replace input with 𝑥 and the output with 𝑦. As our rule was adding seven, our
linear function is 𝑦 equals 𝑥 plus seven.
We will now look at some questions
where we need to complete function tables given their linear function or rule.
Complete the given function
table.
We’re given in the table that our
input is 𝑥 and our output is two plus 𝑥. If we let the output be the letter
𝑦, we can see that we have the linear function 𝑦 is equal to two plus 𝑥. We know this is linear as the
highest power of 𝑥 is one. We can now calculate our output
values by substituting the input for 𝑥. When 𝑥 is equal to nine, 𝑦 is
equal to two plus nine. This is equal to 11. So when the input is nine, the
output is 11.
We can repeat this process when 𝑥
is equal to five. Two plus five is equal to
seven. This means that when the input is
five, the output is seven. Finally, we have an input of
16. 𝑦 is therefore equal to two plus
16, which is 18. When our input is 16, our output is
18. The linear function 𝑦 equals two
plus 𝑥 means that each value of 𝑦 or the output is two greater than the value of
𝑥 or the input. Our correct answers are 11, seven,
and 18.
Our next question involves a more
complicated linear function.
Use the rule to find the missing
number in this input–output table. Rule equals three 𝑛 minus
seven.
We are told that when the input is
four, the output is five. To check this, we can substitute
the value 𝑛 equals four into the equation. Three 𝑛 means three times 𝑛, so
we are left with three multiplied by four minus seven. Three multiplied by four is 12. And taking away seven does give us
five. Our second set of values are an
input of five and an output of eight. Substituting five into the rule
does give us eight, as three multiplied by five is 15 and subtracting seven from
this is eight. We can do the same for an input of
seven, giving us an output of 14.
We need to calculate the missing
value in the table. This is the output when the input
is six. We can calculate the output by
substituting 𝑛 equals six into our rule. Three multiplied by six is equal to
18. Subtracting seven from this gives
us an answer of 11. When our input is six, our output
is 11.
An alternative method to solve this
question would be to notice that our rule, three 𝑛 minus seven, is a linear
function. This means that for every increase
of one to the input, the output will increase or decrease by the same value. This value is the coefficient of
𝑛, the number we’re multiplying by 𝑛, as this would be the slope or gradient of
the linear function. We can see this in our function
table as five plus three is equal to eight, eight plus three equals 11, and 11 plus
three is 14. As the input increases by one, the
corresponding output increases by three.
In our next question, we need to
identify which table corresponds to a linear function.
Which of the following is a table
of values for the function 𝑦 equals seven 𝑥 minus three?
There are a couple of ways of
approaching this question. The quickest way would be to
recognize that we have a linear function of the form 𝑦 equals 𝑚𝑥 plus 𝑏. We know that a linear function is a
straight-line graph, and the value of 𝑚 is the gradient or slope of the line. This is the increase in 𝑦-values
for each increase of one in the 𝑥-values. In this question, our slope or
gradient is seven. The value of 𝑏 is the 𝑦-intercept
of the graph. This is the point at which the
graph crosses the 𝑦-axis. When 𝑥 is equal to zero, this is
the 𝑦-value. In the equation 𝑦 equals seven 𝑥
minus three, the 𝑦-intercept is negative three. This means that our table must
contain the corresponding values 𝑥 equals zero and 𝑦 equals negative three.
Options (A), (B), (C), and (E) all
contain this value. Option (D), on the other hand,
contains the value zero, three. When 𝑥 equals zero, 𝑦 is equal to
three. Therefore, this is not the correct
table. As the 𝑥-value in each of the
tables increases by one, the 𝑦-value needs to increase by seven. In table (A), negative three plus
seven is equal to four, four plus seven is equal to 11, and 11 plus seven is equal
to 18. This means that table (A) does
satisfy the function 𝑦 equals seven 𝑥 minus three. Option (B) is not correct as the
difference between four and 10 is six and between 10 and 18 is eight. This means that table (B) does not
represent a linear function and is therefore not equal to 𝑦 equals seven 𝑥 minus
three.
We have a similar problem with
table (C). Negative three plus six is equal to
three. And three plus eight is equal to
11. Therefore, this function is not
linear. In option (E), our first two steps
are correct. Negative three plus seven is equal
to four, and four plus seven is equal to 11. However, the final number, 19,
should be 18 as 11 plus eight is equal to 19. The correct answer is option
(A). The function 𝑦 equals seven 𝑥
minus three has coordinates zero, negative three; one, four; two, 11; and three, 18
that lie on the straight line.
An alternative method here would be
to substitute our 𝑥-values, zero, one, two, and three, into the function 𝑦 equals
seven 𝑥 minus three. When 𝑥 is equal to zero, 𝑦 is
equal to seven multiplied by zero minus three. This is equal to negative
three. Therefore, when 𝑥 is zero, 𝑦 is
negative three. When 𝑥 is equal to one, 𝑦 is
equal to four as seven multiplied by one minus three is four. Substituting in 𝑥 equals two gives
us a value of 𝑦 equal to 11. Finally, when 𝑥 is equal to three,
𝑦 is equal to 18. This confirms that the four values
of 𝑥 and 𝑦 need to be zero, negative three; one, four; two, 11; and three, 18.
Our next two questions require us
to find missing 𝑥- and 𝑦-values in a function table.
Given that 𝑦 is equal to 40
divided by 𝑥, complete the following table.
There’re three missing values in
the table. We need to calculate 𝑦 when 𝑥 is
eight, 𝑥 when 𝑦 is two, and 𝑦 when 𝑥 is 40. The first and third missing numbers
are relatively straightforward. Substituting 𝑥 equals eight into
our equation or function gives us 𝑦 is equal to 40 divided by eight. This is equal to five. So our first missing value is
five. We can repeat this process for the
third missing value. Substituting 𝑥 equals 40 gives us
𝑦 is equal to 40 divided by 40. Dividing any number by itself gives
us an answer of one. When 𝑥 is equal to 40, 𝑦 is equal
to one.
For the second missing value in our
table, we are given the value of 𝑦 and need to calculate the value of 𝑥. This means that two is equal to 40
divided by 𝑥. We can rewrite this as 40 over 𝑥
and then solve the equation to calculate 𝑥. Multiplying both sides of the
equation by 𝑥 gives us two 𝑥 is equal to 40. Dividing both sides of this
equation by two gives us 𝑥 is equal to 20. When 𝑦 is equal to two, 𝑥 is
equal to 20. We can check this answer by
dividing 40 by 20. As this is equal to two, our answer
is correct. The missing numbers are five, 20,
and one.
Given that 𝑦 is equal to two 𝑥
minus five, complete the following table.
We need to calculate the 𝑥-values
that correspond to the 𝑦-values of 11, 19, and 30 in the linear function 𝑦 equals
two 𝑥 minus five. The easiest way to do this is to
substitute each of the 𝑦-values in turn into the function. We will begin with 𝑦 equals
11. This gives us 11 is equal to two 𝑥
minus five. We can solve this equation using
the balancing method, firstly, by adding five to both sides. 11 plus five is equal to 16, so we
have 16 is equal to two 𝑥. Our final step is to divide both
sides of this equation by two. This gives us a value of 𝑥 equal
to eight. When 𝑦 is equal to 11, 𝑥 is equal
to eight.
We can repeat this process when 𝑦
is equal to 19. Adding five to both sides gives us
24 is equal to two 𝑥. Dividing by two gives us a value of
𝑥 equal to 12. When 𝑦 is equal to 19, 𝑥 equals
12. The same method can then be applied
for our final missing value when 𝑦 is equal to 30. Once again, we follow a two-step
process. We add five and then divide by
two.
Adding five to both sides of the
equation gives us 35 is equal to two 𝑥. Two does not divide exactly into 35
as 35 is an odd number. We could leave this answer as a
top-heavy or improper fraction or change it to the decimal 17.5 as this is half of
35. When 𝑦 is equal to 30, 𝑥 is equal
to 17.5. The three missing values in the
table are eight, 12, and 17.5. We could substitute each of these
values back into the function 𝑦 equals two 𝑥 minus five and ensure we get answers
of 11, 19, and 30.
An alternative method here would be
to consider function machines. Our value of 𝑥 is our input and
our value of 𝑦 is our output. In the function 𝑦 equals two 𝑥
minus five, there are two steps. We multiply 𝑥 by two and then
subtract five. As we need to calculate the
𝑥-values, we need to reverse these functions. The inverse or opposite of
subtracting five is adding five, and the inverse of multiplying by two is dividing
by two. In order to calculate our
𝑥-values, we need to add five to our 𝑦-value and then divide the answer by
two. This is the same two steps that we
completed when solving the equation. We added five to 11 and then
divided by two, giving us an answer of eight.
We will now summarize the key
points from this video. A function is a rule that assigns a
set of inputs to a set of outputs. We looked at linear functions which
are written in the form 𝑦 equals 𝑚𝑥 plus 𝑏. These are straight-line graphs
where 𝑚 is the slope or gradient and 𝑏 is the 𝑦-intercept. Our 𝑥-values correspond to the
inputs, and our 𝑦-values correspond to the outputs. We completed a variety of function
tables by using rules and also by substitution and then solving equations. When we were given the output and
needed to calculate the input, we used inverse functions. This involved using our knowledge
of inverse or opposite operations.
