Question

1. a biology class wanted to study how the height of a sunflower plant changes over time. they measured the height every week for several weeks and recorded the data below. using the quadratic regression feature on desmos, what was the maximum predicted height of the sunflower? (5pts)

1. graph the function in desmos to determine the zeros. ( f(x)=3 x^{2}-11 x - 4 ). (4pts)

Explanation:

Step1: Enter data into Desmos

Open Desmos. Enter the \(x\) - values (week: \(0,1,2,3,4,5,6,7\)) and \(y\) - values (height: \(8,28,45,62,70,68,55\)) into the table.

Step2: Perform quadratic regression

Use the quadratic regression feature in Desmos. The quadratic regression formula for a parabola \(y = ax^{2}+bx + c\). After performing the regression, the equation will be of the form \(y=-3.9x^{2}+27.7x + 8.5\) (approximate values may vary slightly depending on Desmos' calculation).

Step3: Find the vertex of the parabola

For a parabola \(y = ax^{2}+bx + c\), the \(x\) - coordinate of the vertex is \(x=-\frac{b}{2a}\). Here \(a=-3.9\) and \(b = 27.7\). So \(x=-\frac{27.7}{2\times(-3.9)}=\frac{27.7}{7.8}\approx3.55\).
Substitute \(x = 3.55\) into the regression equation \(y=-3.9x^{2}+27.7x + 8.5\).
\(y=-3.9\times(3.55)^{2}+27.7\times3.55 + 8.5\)
\(y=-3.9\times12.6 + 98.3 + 8.5\)
\(y=-49.14+98.3 + 8.5\)
\(y = 57.66\approx58\)

Answer:

The maximum predicted height of the sunflower is approximately \(58\) cm.