Question

find the minimum value of the function $f(x) = 1.7x^2 + 25.8x + 80.7$ to the nearest hundredth.

Explanation:

Step1: Identify the vertex formula

For a quadratic function \( f(x) = ax^2 + bx + c \), the x - coordinate of the vertex (which gives the minimum or maximum value, here \( a = 1.7>0 \), so it's a minimum) is \( x=-\frac{b}{2a} \). Here, \( a = 1.7 \), \( b = 25.8 \), \( c = 80.7 \).

Step2: Calculate the x - coordinate of the vertex

Substitute \( a = 1.7 \) and \( b = 25.8 \) into the formula \( x =-\frac{b}{2a} \).
\( x=-\frac{25.8}{2\times1.7}=-\frac{25.8}{3.4}\approx - 7.5882 \)

Step3: Calculate the minimum value of the function

Substitute \( x\approx - 7.5882 \) into the function \( f(x)=1.7x^{2}+25.8x + 80.7 \).
\( f(-7.5882)=1.7\times(-7.5882)^{2}+25.8\times(-7.5882)+80.7 \)
First, calculate \( (-7.5882)^{2}\approx57.58 \)
Then, \( 1.7\times57.58\approx97.886 \)
\( 25.8\times(-7.5882)\approx - 195.7756 \)
Now, \( f(-7.5882)=97.886-195.7756 + 80.7 \)
\( f(-7.5882)=97.886 + 80.7-195.7756=178.586 - 195.7756=-17.1896\approx - 17.19 \)

Answer:

\( - 17.19 \)