Question

graph the equation.
$y = -4|x|$

Explanation:

Step1: Analyze the parent function

The parent function of an absolute - value function is \(y = |x|\), which has a V - shape with the vertex at the origin \((0,0)\), and the graph opens upwards. For \(x\geq0\), \(y = x\); for \(x<0\), \(y=-x\).

Step2: Analyze the transformation

The given function is \(y=-4|x|\). The negative sign in front of the absolute - value function reflects the graph of \(y = |x|\) over the \(x\) - axis, so the graph will open downwards. The coefficient \(4\) is a vertical stretch factor. It means that for each \(x\) - value, the corresponding \(y\) - value of \(y = |x|\) is multiplied by \(4\) and then by \(- 1\).

Step3: Find key points

• When \(x = 0\): Substitute \(x = 0\) into \(y=-4|x|\), we get \(y=-4|0|=0\). So the vertex of the graph is at \((0,0)\).
• When \(x = 1\): Substitute \(x = 1\) into \(y=-4|x|\), we have \(y=-4|1|=-4\). So the point is \((1, - 4)\).
• When \(x=-1\): Substitute \(x=-1\) into \(y=-4|x|\), we have \(y=-4|-1|=-4\). So the point is \((-1, - 4)\).
• When \(x = 2\): Substitute \(x = 2\) into \(y=-4|x|\), we get \(y=-4|2|=-8\). So the point is \((2, - 8)\).
• When \(x=-2\): Substitute \(x=-2\) into \(y=-4|x|\), we get \(y=-4|-2|=-8\). So the point is \((-2, - 8)\).

Step4: Plot the points and draw the graph

Plot the vertex \((0,0)\), and the points \((1, - 4)\), \((-1, - 4)\), \((2, - 8)\), \((-2, - 8)\) (and other points if needed). Then draw a V - shaped graph (opening downwards) that passes through these points. The left side of the V (for \(x<0\)) has a slope of \(4\) (since for \(x<0\), \(y=-4|x|=-4(-x) = 4x\), so the slope \(m = 4\)) and the right side (for \(x\geq0\)) has a slope of \(- 4\) (since for \(x\geq0\), \(y=-4|x|=-4x\), so the slope \(m=-4\)).

To graph it:

1. Start at the vertex \((0,0)\).
2. For the right - hand side (where \(x\geq0\)): From the vertex \((0,0)\), move 1 unit to the right (increase \(x\) by 1) and 4 units down (decrease \(y\) by 4) to get to the point \((1, - 4)\). Then from \((1, - 4)\), move 1 unit to the right and 4 units down to get to \((2, - 8)\), and so on.
3. For the left - hand side (where \(x<0\)): From the vertex \((0,0)\), move 1 unit to the left (decrease \(x\) by 1) and 4 units down (decrease \(y\) by 4) to get to the point \((-1, - 4)\). Then from \((-1, - 4)\), move 1 unit to the left and 4 units down to get to \((-2, - 8)\), and so on.

Answer:

The graph of \(y = - 4|x|\) is a V - shaped graph opening downwards with the vertex at \((0,0)\), passing through points like \((1, - 4)\), \((-1, - 4)\), \((2, - 8)\), \((-2, - 8)\) etc. (The actual drawing should be done on the given coordinate plane by plotting these points and drawing the two lines of the V - shape.)