Question

graph the equation. y = -2|x + 4|

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 \((h,k)\) is the vertex. For the function \(y=-2|x + 4|\), we can rewrite it as \(y=-2|x-(-4)|+0\). So, the vertex \((h,k)\) is \((-4,0)\). We plot the point \((-4,0)\) on the coordinate plane.

Step2: Determine the direction and the slope of the branches

The coefficient \(a=-2\) in the function \(y = a|x - h|+k\). Since \(a=-2<0\), the graph of the absolute - value function opens downwards.
For the absolute - value function \(y = a|x - h|+k\), when \(x\geq h\), the function can be written as \(y=a(x - h)+k\), and when \(x < h\), the function can be written as \(y=-a(x - h)+k\).
For our function \(y=-2|x + 4|\), when \(x\geq - 4\), \(y=-2(x + 4)=-2x-8\). The slope of this line (for \(x\geq - 4\)) is \(-2\). When \(x=-4\), \(y = 0\). When \(x=-3\), \(y=-2(-3 + 4)=-2(1)=-2\). So, we can find another point on the right - hand branch by moving 1 unit to the right from \(x=-4\) (to \(x=-3\)) and 2 units down (since the slope is - 2) from \(y = 0\) to \(y=-2\). So, the point \((-3,-2)\) is on the right - hand branch.
When \(x < - 4\), \(y=-2(-(x + 4))=2x + 8\). The slope of this line (for \(x < - 4\)) is \(2\). When \(x=-5\), \(y=2(-5)+8=-10 + 8=-2\). So, we can find a point on the left - hand branch by moving 1 unit to the left from \(x=-4\) (to \(x=-5\)) and 2 units down (since the slope is 2, but since \(x\) is decreasing, a change in \(x\) of - 1 (from \(x=-4\) to \(x=-5\)) gives a change in \(y\) of \(2\times(-1)=-2\)) from \(y = 0\) to \(y=-2\). So, the point \((-5,-2)\) is on the left - hand branch.

Step3: Draw the graph

We use the vertex \((-4,0)\) and the two points \((-3,-2)\) (for the right - hand branch) and \((-5,-2)\) (for the left - hand branch) to draw the two branches of the absolute - value graph. The left - hand branch (for \(x < - 4\)) has a slope of 2 and passes through \((-5,-2)\) and \((-4,0)\), and the right - hand branch (for \(x\geq - 4\)) has a slope of - 2 and passes through \((-4,0)\) and \((-3,-2)\). We connect the points to form the V - shaped (but opening downwards) graph of the function \(y=-2|x + 4|\).

(Note: Since the problem is about graphing, the final answer is the graph of the function \(y = - 2|x + 4|\) with vertex at \((-4,0)\), opening downwards, and passing through points like \((-3,-2)\) and \((-5,-2)\) as described above. If we were to represent the key features, the graph is an absolute - value graph with vertex \((-4,0)\), opening downward, with a vertical stretch factor of 2.)

Answer:

Step1: Identify the vertex of the absolute - value function

The general form of an absolute - value function is \(y = a|x - h|+k\), where \((h,k)\) is the vertex. For the function \(y=-2|x + 4|\), we can rewrite it as \(y=-2|x-(-4)|+0\). So, the vertex \((h,k)\) is \((-4,0)\). We plot the point \((-4,0)\) on the coordinate plane.

Step2: Determine the direction and the slope of the branches

The coefficient \(a=-2\) in the function \(y = a|x - h|+k\). Since \(a=-2<0\), the graph of the absolute - value function opens downwards.
For the absolute - value function \(y = a|x - h|+k\), when \(x\geq h\), the function can be written as \(y=a(x - h)+k\), and when \(x < h\), the function can be written as \(y=-a(x - h)+k\).
For our function \(y=-2|x + 4|\), when \(x\geq - 4\), \(y=-2(x + 4)=-2x-8\). The slope of this line (for \(x\geq - 4\)) is \(-2\). When \(x=-4\), \(y = 0\). When \(x=-3\), \(y=-2(-3 + 4)=-2(1)=-2\). So, we can find another point on the right - hand branch by moving 1 unit to the right from \(x=-4\) (to \(x=-3\)) and 2 units down (since the slope is - 2) from \(y = 0\) to \(y=-2\). So, the point \((-3,-2)\) is on the right - hand branch.
When \(x < - 4\), \(y=-2(-(x + 4))=2x + 8\). The slope of this line (for \(x < - 4\)) is \(2\). When \(x=-5\), \(y=2(-5)+8=-10 + 8=-2\). So, we can find a point on the left - hand branch by moving 1 unit to the left from \(x=-4\) (to \(x=-5\)) and 2 units down (since the slope is 2, but since \(x\) is decreasing, a change in \(x\) of - 1 (from \(x=-4\) to \(x=-5\)) gives a change in \(y\) of \(2\times(-1)=-2\)) from \(y = 0\) to \(y=-2\). So, the point \((-5,-2)\) is on the left - hand branch.

Step3: Draw the graph

We use the vertex \((-4,0)\) and the two points \((-3,-2)\) (for the right - hand branch) and \((-5,-2)\) (for the left - hand branch) to draw the two branches of the absolute - value graph. The left - hand branch (for \(x < - 4\)) has a slope of 2 and passes through \((-5,-2)\) and \((-4,0)\), and the right - hand branch (for \(x\geq - 4\)) has a slope of - 2 and passes through \((-4,0)\) and \((-3,-2)\). We connect the points to form the V - shaped (but opening downwards) graph of the function \(y=-2|x + 4|\).

(Note: Since the problem is about graphing, the final answer is the graph of the function \(y = - 2|x + 4|\) with vertex at \((-4,0)\), opening downwards, and passing through points like \((-3,-2)\) and \((-5,-2)\) as described above. If we were to represent the key features, the graph is an absolute - value graph with vertex \((-4,0)\), opening downward, with a vertical stretch factor of 2.)