Question

graph the equation.
$y = 2|x| - 3$

Explanation:

Step1: Identify the vertex

The general form of an absolute - value function is \(y = a|x - h|+k\), where \((h,k)\) is the vertex. For the function \(y = 2|x|-3\), we can rewrite it as \(y=2|x - 0|-3\). So, the vertex \((h,k)=(0, - 3)\). We plot the point \((0,-3)\) on the coordinate plane.

Step2: Find two more points

• When \(x = 1\):

Substitute \(x = 1\) into the equation \(y=2|x|-3\). Since \(|1| = 1\), we have \(y=2\times1 - 3=2 - 3=-1\). So, the point \((1,-1)\) is on the graph.

• When \(x=- 1\):

Substitute \(x=-1\) into the equation \(y = 2|x|-3\). Since \(|-1| = 1\), we have \(y=2\times1-3=-1\). So, the point \((-1,-1)\) is on the graph.

• We can also find more points for a better graph. For example, when \(x = 2\), \(y=2|2|-3=4 - 3 = 1\), so the point \((2,1)\) is on the graph. When \(x=-2\), \(y=2|-2|-3 = 4-3=1\), so the point \((-2,1)\) is on the graph.

Step3: Draw the graph

The graph of an absolute - value function \(y = a|x - h|+k\) is a V - shaped graph. Since \(a = 2>0\), the graph opens upwards. We plot the vertex \((0,-3)\) and the other points we found (like \((1,-1)\), \((-1,-1)\), \((2,1)\), \((-2,1)\) etc.) and then draw two lines: one with a slope of \(2\) passing through the points with \(x\geq0\) (using the points \((0,-3)\), \((1,-1)\), \((2,1)\) etc.) and one with a slope of \(- 2\) passing through the points with \(x<0\) (using the points \((0,-3)\), \((-1,-1)\), \((-2,1)\) etc.).

(Note: Since the problem asks to graph the equation, the final answer is the graph constructed as above. But in text form, we can describe the key points and the shape. If we were to represent the graph, it is a V - shaped graph opening upwards with vertex at \((0,-3)\), passing through points like \((1,-1)\), \((-1,-1)\), \((2,1)\), \((-2,1)\) etc.)

Answer:

The graph is a V - shaped graph (absolute - value graph) opening upwards with vertex at \((0, - 3)\), passing through points such as \((-2,1)\), \((-1,-1)\), \((1,-1)\), \((2,1)\) (and other symmetric points) plotted on the given coordinate plane and connected with two straight lines (one for \(x\geq0\) with slope \(2\) and one for \(x < 0\) with slope \(-2\)).