Question

the function models the number of accidents, ( f(x) ), per 50 million miles driven as a function of a drivers age, ( x ), in years, where ( x ) includes drivers from ages 16 through 68, inclusive. the graph of ( f ) is shown. use the equation for ( f ) to solve the problem below.

for what value of ( x ) does the graph reach its lowest point? use the equation for ( f ) to find the minimum value of ( y )

( x=square, y=square )

describe the practical significance of this minimum value

the minimum number of accidents is ( square ) per 50 million miles driven and is attributed to ( square )-year-old drivers

Explanation:

Step1: Find the \(x\) - value of the vertex

For a quadratic function \(y = ax^{2}+bx + c\), the \(x\) - coordinate of the vertex is given by \(x=-\frac{b}{2a}\).
In the function \(f(x)=0.4x^{2}-33.6x + 885.6\), \(a = 0.4\) and \(b=-33.6\).

$$x=-\frac{-33.6}{2\times0.4}=\frac{33.6}{0.8}=42$$

Step2: Find the \(y\) - value of the vertex

Substitute \(x = 42\) into the function \(f(x)=0.4x^{2}-33.6x + 885.6\).

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Answer:

\(x = 42,y = 180\)
The minimum number of accidents is \(180\) per 50 million miles driven and is attributed to \(42\) - year - old drivers.