Question

a company finds that if it charges x dollars for a cell phone, it can expect to sell 1000 - 2x phones. the company uses the function r defined by r(x) = x(1000 - 2x) to model the expected revenue, in dollars, from selling the cell phones at x dollars each. use graphing technology to graph the function r. sketch the graph on your own paper. identify the coordinates of the vertex and any intercepts by entering the coordinates in the blanks below. you may use this graph to help answer the next five questions. a. vertex: (□, □) b. left - most intercept: (□, □) c. right - most intercept: (□, □) graphing calculator

Explanation:

Step1: Rewrite the revenue function

The revenue function is \( r(x) = x(1000 - 2x) \). Expand it to standard quadratic form: \( r(x)=-2x^{2}+1000x \). For a quadratic function \( y = ax^{2}+bx + c \), the x - coordinate of the vertex is given by \( x=-\frac{b}{2a} \). Here, \( a=-2 \) and \( b = 1000 \). So, \( x=-\frac{1000}{2\times(-2)}=\frac{1000}{4} = 250 \).

Step2: Find the y - coordinate of the vertex

Substitute \( x = 250 \) into the revenue function \( r(x)=x(1000 - 2x) \). Then \( r(250)=250\times(1000-2\times250)=250\times(1000 - 500)=250\times500 = 125000 \). So the vertex is \( (250,125000) \).

Step3: Find the x - intercepts

To find the x - intercepts, set \( r(x)=0 \), so \( x(1000 - 2x)=0 \). This gives two equations: \( x = 0 \) and \( 1000-2x=0 \). Solving \( 1000 - 2x=0 \) for \( x \), we get \( 2x = 1000\Rightarrow x = 500 \). The left - most intercept (smaller x - value) is \( (0,0) \) and the right - most intercept is \( (500,0) \).

Answer:

A. vertex: \((250, 125000)\)
B. left - most intercept: \((0, 0)\)
C. right - most intercept: \((500, 0)\)