Question

the scatterplot shows the relationship between two variables, x and y. which equation is the most appropriate quadratic model for the data shown in the scatterplot? a $y=-0.3x^{2}+1.7x + 7.1$ b $y=-0.3x^{2}-1.7x + 3.1$ c $y=-3.3x^{2}+1.7x + 7.1$ d $y=-3.3x^{2}-1.7x + 3.1$

Explanation:

Step1: Analyze the general shape of the quadratic function

A quadratic function is of the form \(y = ax^{2}+bx + c\). If \(a<0\), the parabola opens down - ward. Looking at the scatter - plot, the data points first increase and then decrease, so the parabola opens downward (\(a<0\)).

Step2: Estimate the \(y\) - intercept

The \(y\) - intercept of the quadratic function \(y = ax^{2}+bx + c\) is \(c\) (the value of \(y\) when \(x = 0\)). Looking at the scatter - plot, when \(x = 0\), the \(y\) - value should be relatively large.

• For option B: \(y=-0.3x^{2}-1.7x + 3.1\), when \(x = 0\), \(y=3.1\)
• For option D: \(y=-3.3x^{2}-1.7x + 3.1\), when \(x = 0\), \(y = 3.1\)
• For option A: \(y=-0.3x^{2}+1.7x + 7.1\), when \(x = 0\), \(y=7.1\)
• For option C: \(y=-3.3x^{2}+1.7x + 7.1\), when \(x = 0\), \(y=7.1\)

Step3: Analyze the coefficient \(a\)

The coefficient \(a\) affects the "width" of the parabola. A larger \(|a|\) makes the parabola narrower. Looking at the scatter - plot, the data points do not seem to be very "narrowly" clustered. If \(a=-3.3\) (options C and D), the parabola would be relatively narrow. If \(a=-0.3\) (options A and B), the parabola is wider.

Answer:

A. \(y=-0.3x^{2}+1.7x + 7.1\)