Question

score on last try: 0 of 1 pts. see details for more.
hint question get a similar question you can retry this que
sketch a graph of ( f(x) = -2|x| )
coordinate grid with axes from -5 to 5, drawing tools below
clear all draw: line, parabola, v-shape, circle, dot icons
question help: video message instructor
submit question

Explanation:

Step1: Analyze the parent function

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

Step2: Analyze the transformations

For the function \( f(x)=-2|x| \), 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 \( 2 \) vertically stretches the graph of \( y = |x| \) by a factor of \( 2 \). So the points \((1,1)\) and \((-1,1)\) on \( y = |x| \) will be transformed to \((1,-2)\) and \((-1,-2)\) on \( f(x)=-2|x| \), and the vertex remains at \((0,0)\).

Step3: Plot the points and draw the graph

• Plot the vertex at \((0,0)\).
• Plot the points \((1, - 2)\) and \((-1,-2)\).
• Since the absolute value function is composed of two linear pieces, for \( x\geq0 \), \( f(x)=-2x \) (a line with slope - 2 passing through \((0,0)\) and \((1,-2)\)), and for \( x < 0 \), \( f(x)=2x \) (a line with slope 2 passing through \((0,0)\) and \((-1,-2)\)). Draw the two line segments connecting these points to form a V - shaped graph opening downwards with vertex at the origin and passing through \((1,-2)\) and \((-1,-2)\).

(Note: Since this is a graph - sketching problem, the final answer is the graph of \( f(x)=-2|x| \) which is a downward - opening V - shape with vertex at \((0,0)\), passing through \((1, - 2)\) and \((-1,-2)\).)

Answer:

Step1: Analyze the parent function

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

Step2: Analyze the transformations

For the function \( f(x)=-2|x| \), 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 \( 2 \) vertically stretches the graph of \( y = |x| \) by a factor of \( 2 \). So the points \((1,1)\) and \((-1,1)\) on \( y = |x| \) will be transformed to \((1,-2)\) and \((-1,-2)\) on \( f(x)=-2|x| \), and the vertex remains at \((0,0)\).

Step3: Plot the points and draw the graph

• Plot the vertex at \((0,0)\).
• Plot the points \((1, - 2)\) and \((-1,-2)\).
• Since the absolute value function is composed of two linear pieces, for \( x\geq0 \), \( f(x)=-2x \) (a line with slope - 2 passing through \((0,0)\) and \((1,-2)\)), and for \( x < 0 \), \( f(x)=2x \) (a line with slope 2 passing through \((0,0)\) and \((-1,-2)\)). Draw the two line segments connecting these points to form a V - shaped graph opening downwards with vertex at the origin and passing through \((1,-2)\) and \((-1,-2)\).

(Note: Since this is a graph - sketching problem, the final answer is the graph of \( f(x)=-2|x| \) which is a downward - opening V - shape with vertex at \((0,0)\), passing through \((1, - 2)\) and \((-1,-2)\).)