Question

a startup company sells watches. its per - item profit based on number produced is given in the table below. use technology to find the quadratic curve of best fit for the data. choose the equation of the curve.

show your work here

Explanation:

Step1: Recall the general form of a quadratic function

The general form of a quadratic function is \(y = ax^{2}+bx + c\).

Step2: Analyze the shape of the parabola

Since the profit first increases and then decreases (from \(-124.87\) at \(x = 1\) to \(3.5\) at \(x=7\) and then to \(-3.26\) at \(x = 10\)), the parabola opens downwards. For a quadratic function \(y=ax^{2}+bx + c\), if \(a<0\), the parabola opens downwards.

Step3: Check the coefficient of \(x^{2}\) in each option

• For \(f(x)=2.924x^{2}-42.812x - 145.543\), \(a = 2.924>0\) (parabola opens upwards).
• For \(f(x)=-2.924x^{2}+42.812x - 145.543\), \(a=- 2.924<0\) (parabola opens downwards).
• For \(f(x)=-2.924x^{2}-42.812x - 145.543\), when \(x = 1\), \(f(1)=-2.924\times1^{2}-42.812\times1-145.543=-2.924 - 42.812-145.543=-191.279

eq-124.87\)

Answer:

\(f(x)=-2.924x^{2}+42.812x - 145.543\)