Question

use the given points to identify the graph that would be made from adding the two graphs together.

Explanation:

Step1: Assume linear - function form

Let the two lines in the given graph be \(y_1 = m_1x + b_1\) and \(y_2=m_2x + b_2\). For the first line with a negative slope passing through \((- 4,4)\) and \((0,0)\), the slope \(m_1=\frac{0 - 4}{0+4}=-1\) and \(b_1 = 0\), so \(y_1=-x\). For the second line with a positive slope passing through \((0, - 2)\) and \((4,2)\), the slope \(m_2=\frac{2 + 2}{4-0}=1\) and \(b_2=-2\), so \(y_2=x - 2\).

Step2: Add the two functions

The sum of the two functions \(y=y_1 + y_2=(-x)+(x - 2)=-2\) for all \(x\) values. This is a horizontal line at \(y = - 2\). Looking at the options, we need to check the \(y\) - values of the combined - graph at key \(x\) - points. Another way is to add the \(y\) - values of the two lines at specific \(x\) - values. For example, at \(x = 0\), \(y_1=0\) and \(y_2=-2\), so \(y=y_1 + y_2=-2\); at \(x = 4\), \(y_1=-4\) and \(y_2=2\), so \(y=-4 + 2=-2\).

Answer:

The graph of the sum of the two given graphs is a horizontal line at \(y=-2\). Without seeing all the options completely, if there is an option with a horizontal line at \(y = - 2\), that is the correct one.