Question

graph the equation.
\\( y = -4|x| \\)

Explanation:

Step1: Recall the parent function

The parent function of absolute value is \( y = |x| \), which is a V - shaped graph with vertex at the origin \((0,0)\), opening upwards, and passing through \((1,1)\) and \(( - 1,1)\).

Step2: Analyze the transformation

For the function \( y=-4|x| \), we have two transformations:

• The negative sign reflects the graph of \( y = |x| \) over the \( x \) - axis, so the graph will open downwards.
• The coefficient \( 4 \) is a vertical stretch by a factor of \( 4 \). This means that for a given \( x \) - value, the \( y \) - value of \( y=-4|x| \) is \( 4 \) times the \( y \) - value of \( y=-|x| \) (after reflection) or \( 4 \) times the negative of the \( y \) - value of \( y = |x| \).

Step3: Find key points

• When \( x = 0 \): Substitute \( x = 0 \) into \( y=-4|x| \), we get \( y=-4|0|=0 \). So the vertex is at \((0,0)\).
• When \( x = 1 \): Substitute \( x = 1 \) into \( y=-4|x| \), we get \( y=-4|1|=-4 \). So the point is \((1, - 4)\).
• When \( x=-1 \): Substitute \( x = - 1 \) into \( y=-4|x| \), we get \( y=-4|-1|=-4 \). So the point is \(( - 1, - 4)\).
• We can also find more points, for example, when \( x = 2 \), \( y=-4|2|=-8 \) (point \((2, - 8)\)) and when \( x=-2 \), \( y=-4|-2|=-8 \) (point \(( - 2, - 8)\)).

Step4: Graph the function

Plot the vertex \((0,0)\), then plot the points \((1, - 4)\), \(( - 1, - 4)\), \((2, - 8)\), \(( - 2, - 8)\) etc. Then draw a V - shaped graph (opening downwards) connecting these points. The graph will have its vertex at the origin, open downwards, and be steeper than the graph of \( y=-|x| \) (since it is stretched by a factor of 4).

Answer:

To graph \( y = - 4|x| \):

1. Plot the vertex at \((0,0)\).
2. Plot the points \((1, - 4)\), \(( - 1, - 4)\), \((2, - 8)\), \(( - 2, - 8)\) (and other points as needed).
3. Draw a V - shaped graph opening downwards, passing through these points, with the vertex at the origin.