Question

graph the equation.
$y = 6|x - 7|$

Explanation:

Step1: Identify the vertex of the absolute - value function

The general form of an absolute - value function is \(y = a|x - h|+k\). For the function \(y = 6|x - 7|\), we can compare it with the general form. Here, \(h = 7\), \(k = 0\) and \(a=6\). The vertex of the absolute - value function \(y=a|x - h|+k\) is at the point \((h,k)\). So, the vertex of \(y = 6|x - 7|\) is at \((7,0)\).

Step2: Find another point on the graph

We can choose a value of \(x\) and find the corresponding value of \(y\). Let's choose \(x=8\). Substitute \(x = 8\) into the equation \(y=6|x - 7|\). Then \(y=6|8 - 7|=6\times|1| = 6\). So, the point \((8,6)\) lies on the graph. Also, since the absolute - value function is symmetric about the line \(x = h\) (here \(x = 7\)), if we take \(x = 6\), then \(y=6|6 - 7|=6\times|- 1|=6\), so the point \((6,6)\) also lies on the graph.

Step3: Determine the shape of the graph

Since \(a = 6>0\), the graph of the absolute - value function opens upwards. The graph is a V - shaped graph with the vertex at \((7,0)\), passing through points like \((6,6)\) and \((8,6)\), and as \(x\) moves away from \(x = 7\) (either to the left or to the right), \(y\) increases with a slope of \(a = 6\) for \(x>7\) and a slope of \(-a=- 6\) for \(x<7\). To graph it, we first plot the vertex \((7,0)\). Then we plot the points \((6,6)\) and \((8,6)\) (or other points obtained by substituting values of \(x\) into the equation). Then we draw a V - shaped graph with the vertex at \((7,0)\) and passing through the other plotted points, opening upwards.

(Note: Since this is a graphing problem, the final answer is the graph of the function \(y = 6|x - 7|\) with vertex at \((7,0)\), opening upwards, and passing through points like \((6,6)\), \((8,6)\) etc. But if we are to describe the key steps to graph it, the above steps are used. If we were to draw it on the given coordinate plane, we would mark the vertex at \((7,0)\), then mark a point to the right of \(x = 7\) (e.g., \(x = 8,y = 6\)) and a point to the left of \(x = 7\) (e.g., \(x = 6,y = 6\)) and draw the two arms of the V - shape.)

Answer:

The graph of \(y = 6|x - 7|\) is a V - shaped graph with vertex at \((7,0)\), opening upwards, passing through points such as \((6,6)\) and \((8,6)\) (and other points) and symmetric about the line \(x = 7\). (To graph it, plot the vertex \((7,0)\), then plot points like \((6,6)\) and \((8,6)\) and draw the V - shaped graph opening upwards.)