Question

homework 1.6: transformations
sketch a graph of $f(x) = 2|x - 1| - 2$
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Explanation:

Step1: Identify the parent function

The parent function is \( y = |x| \), which has a V - shape with vertex at \((0,0)\), slope \(1\) for \(x\geq0\) and slope \(- 1\) for \(x < 0\).

Step2: Analyze the transformations

• Horizontal shift: The function \(f(x)=2|x - 1|-2\) has a horizontal shift. For a function \(y = |x - h|\), the graph is shifted \(h\) units to the right if \(h>0\). Here \(h = 1\), so the graph of \(y=|x|\) is shifted \(1\) unit to the right. The vertex of \(y = |x|\) at \((0,0)\) moves to \((1,0)\) after this shift.
• Vertical stretch: The coefficient \(a = 2\) (where the function is in the form \(y=a|x - h|+k\)) is greater than \(1\), so it vertically stretches the graph by a factor of \(2\). This means the slope of the lines in the V - shape will be \(2\) (for \(x\geq1\)) and \(- 2\) (for \(x<1\)) instead of \(1\) and \(- 1\) as in the parent function.
• Vertical shift: The constant \(k=-2\) means the graph is shifted \(2\) units down. So the vertex of the shifted (after horizontal shift) graph at \((1,0)\) will move to \((1,-2)\).

Step3: Find key points

• For \(x = 1\), \(f(1)=2|1 - 1|-2=2\times0 - 2=-2\) (vertex at \((1,-2)\)).
• For \(x=0\), \(f(0)=2|0 - 1|-2=2\times1 - 2 = 0\). So the point is \((0,0)\).
• For \(x = 2\), \(f(2)=2|2 - 1|-2=2\times1 - 2=0\). So the point is \((2,0)\).
• For \(x = 3\), \(f(3)=2|3 - 1|-2=2\times2 - 2 = 2\). So the point is \((3,2)\).
• For \(x=-1\), \(f(-1)=2|-1 - 1|-2=2\times2 - 2=2\). So the point is \((-1,2)\).

Step4: Sketch the graph

• Plot the vertex at \((1,-2)\).
• For \(x\geq1\), use the slope of \(2\). Starting from \((1,-2)\), when \(x\) increases by \(1\) (e.g., \(x = 2\)), \(y\) increases by \(2\) (from \(-2\) to \(0\) at \(x = 2\), and from \(0\) to \(2\) at \(x=3\)).
• For \(x<1\), use the slope of \(-2\). Starting from \((1,-2)\), when \(x\) decreases by \(1\) (e.g., \(x = 0\)), \(y\) increases by \(2\) (from \(-2\) to \(0\) at \(x = 0\), and from \(0\) to \(2\) at \(x=-1\)).
• Connect the points to form a V - shaped graph with vertex at \((1,-2)\), steeper (due to stretch) than the parent function \(y = |x|\) and shifted right by \(1\) unit and down by \(2\) units.

To sketch the graph:

1. Plot the vertex \((1,-2)\).
2. For \(x\geq1\), draw a line with slope \(2\) passing through \((1,-2)\), \((2,0)\), \((3,2)\), etc.
3. For \(x<1\), draw a line with slope \(-2\) passing through \((1,-2)\), \((0,0)\), \((-1,2)\), etc.

(Note: Since the question asks to sketch the graph, the above steps guide the process of plotting the points and drawing the lines. If we were to describe the final graph, it is a V - shaped graph with vertex at \((1,-2)\), opening upwards, with the left arm having a slope of \(-2\) and the right arm having a slope of \(2\).)

Answer:

The graph of \(f(x)=2|x - 1|-2\) is a V - shaped graph (absolute value graph) with vertex at \((1,-2)\), vertically stretched by a factor of \(2\) relative to \(y = |x|\), shifted \(1\) unit to the right and \(2\) units down. To sketch it, plot the vertex \((1,-2)\), and then draw two lines: one with slope \(2\) for \(x\geq1\) (passing through points like \((2,0)\), \((3,2)\)) and one with slope \(-2\) for \(x < 1\) (passing through points like \((0,0)\), \((-1,2)\)) and connect them.