Part a) Given that 𝑦 equals a half 𝑥 squared plus three, complete the table of values by filling in the remaining 𝑦-values.
So we have five values that we need to fill in our table using the relationship between 𝑥 and 𝑦, which is that 𝑦 is equal to a half 𝑥 squared plus three. Now, we need to be careful because of this first term. If we recall BIDMAS or BODMAS, then we remember that powers or indices come before multiplication and division in the order of operations, which means we need to square each 𝑥-value first and then multiply it by a half.
A common mistake would be to half each 𝑥-value first and then square the result. Now, we could work each of these values out by hand. For example, when 𝑥 is equal to four, 𝑦 would be equal to a half multiplied by four squared plus three. Four squared remember means four multiplied by four. So this is equal to 16. And multiplying by a half is the same as dividing by two. So a half multiplied by 16 or a half of 16 is eight and eight plus three is 11.
However, we do have access to a calculator in this question. So perhaps, the easier route — certainly for the odd numbers — is going to be to use our calculator. You may find it easier to think of a half as 0.5. So we can type this into our calculator as 0.5 multiplied by 𝑥 squared plus three.
Before we start calculating any new values, we could check it for one of the existing values. So if I substitute 𝑥 equals one and on my calculator type 0.5 multiplied by one squared plus three, exactly as it’s written on the screen here, then this gives 3.5, which is the correct value when 𝑥 is one.
So we can find all of the other values in the same way. For zero, we have 0.5 multiplied by zero squared plus three, which is three. And then for the values of two, five, and six, we have five, 15.5, and 21, respectively. So we’ve completed the table of values.
Part b of the question says, “Plot the graph of 𝑦 equals a half 𝑥 squared plus three for zero is less than or equal to 𝑥 which is less than or equal to six on the grid.”
Now, this inequality zero is less than or equal to 𝑥 which is less than or equal to six is just telling us the interval of 𝑥-values that we’re plotting our graph for. So this means that we’re plotting it for 𝑥 between zero and six. Notice that this is the same interval of 𝑥-values that we already have the 𝑦-values worked out for in our table.
Now, before we start plotting, let’s just look at the scale on the 𝑦-axis of the grid we’ve been given. We notice that five small squares corresponds to 2.5 on the 𝑦-axis. Dividing by five, we, therefore, find that one square is equal to 0.5. So this is the scale of the small squares on the 𝑦-axis.
So next, we just need to plot each of these points accurately. The first point has an 𝑥-coordinate of zero and a 𝑦-coordinate of three. So three will be one small square that 0.5 above 2.5. So the point is here.
The next point has an 𝑥-coordinate of one and a 𝑦-coordinate of 3.5. So this will be two small squares above the 2.5 or you can think of it as one small square above the point we just plotted. So the point when 𝑥 equals one goes here.
The next point has an 𝑥-coordinate of two and a 𝑦-coordinate of five. So this point is perhaps more easy to plot as five is one of the large-scale markers on our 𝑦-axis. Next, we have an 𝑥-coordinate of three and a 𝑦-coordinate of 7.5, which again is quite straightforward to plot as 7.5 is one of the large-scale markers on our 𝑦-axis.
Next, we have the point four, 11. And remember each small square represents 0.5. So this will be two small squares above 10 on the 𝑦-axis, then the point five, 15.5, which will be one small square above 15 on the 𝑦-axis, and then finally the point six, 21, which will be two small squares above 20 on the 𝑦-axis.
You need to make sure you take the time to plot these points accurately. They usually need to be within one small square of the exact location. So don’t rush them. Next, we need to connect all of the points together.
Now, this is a quadratic curve as its 𝑦 equals something 𝑥 squared. So this means we need to use a smooth curve through all of the points. Don’t try to do, for example, a dot to dot using straight lines; it must be a smooth curve.
So now, we’ve answered part b of the question and we’ve plotted the graph of 𝑦 equals a half 𝑥 squared plus three for zero is less than or equal to 𝑥 which is less than or equal to six.