A framed picture has a total area \( y \) , in square inches. The thickness of the frame is represented by \( x \) , in inches. The equation \( y = ( 8 + 2 x ) ( 10 + 2 x ) \) relates these two variables.
1. What are the length and width of the picture without the frame?
2. What would a solution to the equation \( 100 = ( 8 + 2 x ) ( 10 + 2 x ) \) mean in this situation?
length=(8+2x) inches
width=(10+2x) inches
\(100= \left(8+ 2x\right)\left(10+ 2x\right)\quad : \quad x= \frac{- 9+ \sqrt{101}}{2},\space x= -\frac{9+ \sqrt{101}}{2}\)
2x+8=-1+\(\sqrt{101}\)
2x+10=1+\(\sqrt{101}\)
9. A board game company models the profit on its latest board game using the quadratic relation
\( P = - 2 x ^ { 2 } + 28 x - 90 , \) where \( x \) is the number of games it produces in hundred thousands and \( P \) is the profit in millions of dollars.
a) What are the break-even points for the company?
b) What is the maximum profit that the company can earn?
The graph of \( y = f ( x ) \) is stretched horizontally by a factor of \( 5 \) , and reflected in the \( x \) -axis. What is the equation of the image graph in terms of \( f \) ?
A. \( y = - f ( 5 x ) \) C. \( y = - f ( \frac { 1 } { 5 } x ) \)
B. \( y = - \frac { 1 } { 5 } f ( x ) \) D. \( y = - 5 f ( x )\)
