The scatter plot shows the revenue in millions of dollars of a car manufacturer over a 10-year period. Which of the following equations best models the data shown in the scatter plot? Is it A) 𝑦 equals 4.339𝑥 squared plus 56.673𝑥 plus 7.685? B) 𝑦 equals negative 4.339𝑥 squared plus 56.673𝑥 plus 7.685? C) 𝑦 equals 4.339𝑥 squared plus 56.673𝑥 minus 7.685? Or D) 𝑦 equals negative 4.339𝑥 squared minus 56.673𝑥 plus 7.685?
We can see that the coefficients of all four options are the same, albeit some are negative and some are positive. The coefficient of 𝑥 squared is 4.339 positive or negative. The coefficient of 𝑥 is 56.673, again, positive or negative. And the constant term is either positive or negative 7.685. In order to decide which equation is correct, we need to consider the shape of the line of best fit and its key points.
Our line of best fit is an n-shaped parabola, as shown on the graph. Any quadratic graph that has a positive coefficient of 𝑥 squared is U-shaped. Whereas any quadratic with a negative coefficient of 𝑥 squared is n-shaped. As the coefficient of option A and option C is positive, we can immediately rule out these options. We can also rule out option C because of its constant term. The value of the constant term in option C is negative. Therefore, this cannot be the correct answer. We can see from the graph that the line of best fit intercepts the 𝑦-axis above the 𝑥-axis. Therefore, the constant must be positive.
We now need to decide which is the correct answer out of option B and option D. Both of these have the same coefficient of 𝑥 squared and the same constant term. This means that we need to consider the coefficient of 𝑥. For option B, this is positive. And for option D, this is negative. This coefficient will help determine where the maximum point on the graph is.
We know that the first term in any n-shaped parabola is negative 𝑎𝑥 squared. We need to work out whether the second term in this graph would be positive 𝑏𝑥 or negative 𝑏𝑥. When the coefficient of 𝑥 is positive, we know that our maximum point will be on the right-hand side of the 𝑦-axis. However, when the coefficient of 𝑥 is negative, the maximum point will be on the left-hand side of the 𝑦-axis.
Our graph is only interested in positive values of time and revenue, as time must be a positive number. This means that the coefficient of 𝑥 must be positive. We can, therefore, conclude that option D cannot be correct. Therefore, the correct equation that models the data is 𝑦 is equal to negative 4.339𝑥 squared plus 56.673𝑥 plus 7.685. The correct one of our four options was Option B.
It is important that we can recognise the shape of any quadratic graph of the form 𝑦 equals 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 based on whether the values of 𝑎, 𝑏, and 𝑐 are positive or negative. All of these are U- or n-shaped parabolas. However, their shape, along with their intercept and maximum or minimum point, will depend on the signs of 𝑎, 𝑏, and 𝑐.