Question

graph the equation.
$y = 5\left|x - 2\
ight|$

Explanation:

Step1: Identify the vertex of the absolute - value function

The general form of an absolute - value function is \(y = a|x - h|+k\), where the vertex is \((h,k)\). For the function \(y = 5|x - 2|\), we can rewrite it as \(y=5|x - 2|+0\). So, \(h = 2\) and \(k = 0\), and the vertex of the function is \((2,0)\). We plot the point \((2,0)\) on the coordinate plane.

Step2: Find another point for \(x>2\)

Let's choose \(x = 3\). Substitute \(x = 3\) into the equation \(y = 5|x - 2|\). Then \(y=5|3 - 2|=5\times|1| = 5\). So we get the point \((3,5)\).

Step3: Find another point for \(x<2\)

Let's choose \(x = 1\). Substitute \(x = 1\) into the equation \(y = 5|x - 2|\). Then \(y=5|1 - 2|=5\times|- 1|=5\). So we get the point \((1,5)\).

Step4: Draw the graph

The graph of an absolute - value function \(y = a|x - h|+k\) is a V - shaped graph. Since \(a = 5>0\), the graph opens upwards. We have the vertex \((2,0)\), and the two points \((1,5)\) and \((3,5)\). We can also find more points if needed, but with these three points, we can sketch the V - shaped graph. The left side of the vertex (for \(x<2\)) has a slope of \(- 5\) (because for \(x<2\), \(y = 5(2 - x)=10 - 5x\), slope \(m=-5\)) and the right side (for \(x > 2\)) has a slope of \(5\) (because for \(x>2\), \(y = 5(x - 2)=5x-10\), slope \(m = 5\)). We connect the points to form the graph of \(y = 5|x - 2|\).

(Note: Since the question is about graphing, the final answer is the graph with vertex at \((2,0)\), passing through \((1,5)\) and \((3,5)\) and having a V - shape opening upwards. But as per the format, we can describe the key points and the shape.)

Answer:

The graph of \(y = 5|x - 2|\) is a V - shaped graph with vertex at \((2,0)\), passing through \((1,5)\) and \((3,5)\) (and other points following the slope of \(-5\) for \(x<2\) and \(5\) for \(x > 2\)) and opening upwards.