Question

describe how the graph of the equation relates to the graph of $y = x^2$.
$f(x) = (x - 2)^2 + 7$
\bigcirc a translation 2 units to the left and 7 units up
\bigcirc a translation 7 units to the right and 2 units up
\bigcirc a translation 2 units to the left and 7 units down
\bigcirc a translation 2 units to the right and 7 units up

question 36
0.1 pts
solve the problem.
you have 184 feet of fencing to enclose a rectangular region. find the dimensions of the rectangle that maximize the enclosed area.
\bigcirc 92 ft by 92 ft
\bigcirc 46 ft by 46 ft
\bigcirc 92 ft by 23 ft
\bigcirc 48 ft by 44 ft

Explanation:

First Question (Graph Transformation)

Step1: Recall transformation rules

For a quadratic function in the form \( f(x)=(x - h)^2 + k \), compared to \( y = x^2 \), the graph is translated \( h \) units horizontally and \( k \) units vertically. If \( h>0 \), it's a translation to the right; if \( h < 0 \), to the left. For \( k \), if \( k>0 \), translation up; if \( k < 0 \), translation down.

Step2: Analyze the given function

In \( f(x)=(x - 2)^2+7 \), we have \( h = 2 \) (so 2 units to the right) and \( k = 7 \) (so 7 units up). So the graph of \( f(x)=(x - 2)^2 + 7 \) is a translation of \( y=x^2 \) 2 units to the right and 7 units up.

Second Question (Maximizing Area of Rectangle)

Step1: Define variables

Let the length of the rectangle be \( l \) and the width be \( w \). The perimeter of a rectangle is given by \( P=2(l + w) \). We know that the perimeter \( P = 184 \) feet. So, \( 2(l + w)=184 \), which simplifies to \( l + w=\frac{184}{2}=92 \), so \( l=92 - w \).

Step2: Area formula

The area \( A \) of a rectangle is \( A=l\times w \). Substitute \( l = 92 - w \) into the area formula: \( A=(92 - w)w=92w - w^{2} \).

Step3: Maximize the quadratic function

The area function \( A(w)=-w^{2}+92w \) is a quadratic function in the form \( y = ax^{2}+bx + c \) with \( a=- 1 \), \( b = 92 \), \( c = 0 \). The vertex of a quadratic function \( y=ax^{2}+bx + c \) occurs at \( x=-\frac{b}{2a} \). For maximum area, the width \( w=-\frac{92}{2\times(-1)}=\frac{92}{2} = 46 \) feet. Then the length \( l=92 - 46=46 \) feet. Wait, but a rectangle with length = width is a square. But among the given options, 46 ft by 46 ft is an option (the second option: 46 ft by 46 ft). Wait, maybe I made a mistake earlier. Let's check the perimeter: \( 2(46 + 46)=2\times92 = 184 \) feet, which matches the given perimeter. And the area \( A = 46\times46=2116 \) square feet. Let's check other options:

• Option 1: 92 ft by 92 ft. Perimeter \( 2(92 + 92)=2\times184 = 368

eq184 \). So wrong.

• Option 2: 46 ft by 46 ft. Perimeter \( 2(46+46) = 184 \), area \( 46\times46 = 2116 \).
• Option 3: 92 ft by 23 ft. Perimeter \( 2(92 + 23)=2\times115 = 230

eq184 \). Wrong.

• Option 4: 48 ft by 44 ft. Perimeter \( 2(48 + 44)=2\times92 = 184 \), area \( 48\times44 = 2112 \) square feet. Which is less than \( 46\times46 = 2116 \) square feet. So the maximum area occurs when the rectangle is a square with side length 46 feet. So the correct option is the second option: 46 ft by 46 ft.

Answer:

D. a translation 2 units to the right and 7 units up