Question

consider the functions $y = |x|$ and $y = |x + 8| - 2$. (a) sketch the graph of the pair of functions using a standard window (b) describe the transformations used to obtain the graph of the second function from the first function (a) graph both functions on the same coordinate grid click to enlarge graph

Explanation:

Part (a) - Sketching the Graphs

for \( y = |x| \):
The parent function \( y = |x| \) is a V - shaped graph. Its vertex is at the origin \((0,0)\). For \( x\geq0\), the equation is \( y = x\), so when \( x = 0\), \( y = 0\); when \( x = 1\), \( y = 1\); when \( x=- 1\), \( y = 1\) (since for \( x<0\), \( y=-x\)).

for \( y=|x + 8|-2 \):

• For the function \( y=|x - h|+k\) (the vertex - form of an absolute - value function), the vertex is at \((h,k)\). For \( y = |x+8|-2=|x-(-8)|+(-2)\), the vertex is at \((-8,-2)\).
• To sketch \( y = |x + 8|-2\), we can use the transformation rules. But for the purpose of graphing:
• When \( x=-8\), \( y=| - 8 + 8|-2=-2\) (vertex point).
• When \( x=-7\), \( y=| - 7+8|-2=|1|-2=-1\).
• When \( x=-9\), \( y=| - 9 + 8|-2=|-1|-2=-1\).
• We plot the vertex \((-8,-2)\) and then use the slope of \(1\) for \(x\geq - 8\) (since for \(x\geq - 8\), \(y=(x + 8)-2=x + 6\), slope \(m = 1\)) and slope of \(-1\) for \(x<-8\) (since for \(x<-8\), \(y=-(x + 8)-2=-x-10\), slope \(m=-1\)) to draw the V - shaped graph.

Part (b) - Describing Transformations

Step 1: Horizontal Shift

For a function \(y = f(x)\), the function \(y=f(x + h)\) is a horizontal shift. If \(h>0\), the graph of \(y = f(x)\) is shifted \(h\) units to the left. For \(y = |x|\) and \(y=|x + 8|\), we have \(h = 8>0\), so the graph of \(y = |x|\) is shifted 8 units to the left to get the graph of \(y=|x + 8|\).

Step 2: Vertical Shift

For a function \(y = f(x)\), the function \(y=f(x)-k\) (where \(k>0\)) is a vertical shift down by \(k\) units. For \(y=|x + 8|\) and \(y=|x + 8|-2\), we have \(k = 2>0\), so the graph of \(y=|x + 8|\) is shifted 2 units down to get the graph of \(y=|x + 8|-2\).

Final Answers

(a) The graph of \(y = |x|\) has a vertex at \((0,0)\) and is a V - shaped graph opening upwards with slopes \(1\) (for \(x\geq0\)) and \(-1\) (for \(x<0\)). The graph of \(y=|x + 8|-2\) has a vertex at \((-8,-2)\) and is a V - shaped graph opening upwards with slopes \(1\) (for \(x\geq - 8\)) and \(-1\) (for \(x<-8\)). (The actual graphing is done by plotting the key points as described above).

(b) To obtain the graph of \(y = |x + 8|-2\) from the graph of \(y = |x|\), we first shift the graph of \(y = |x|\) 8 units to the left (because of the \(x+8\) inside the absolute - value) and then shift the resulting graph 2 units down (because of the \(-2\) outside the absolute - value).

For part (b) answer: The graph of \(y = |x|\) is shifted 8 units to the left and 2 units down to obtain the graph of \(y=|x + 8|-2\).

Answer:

for \( y=|x + 8|-2 \):

• For the function \( y=|x - h|+k\) (the vertex - form of an absolute - value function), the vertex is at \((h,k)\). For \( y = |x+8|-2=|x-(-8)|+(-2)\), the vertex is at \((-8,-2)\).
• To sketch \( y = |x + 8|-2\), we can use the transformation rules. But for the purpose of graphing:
• When \( x=-8\), \( y=| - 8 + 8|-2=-2\) (vertex point).
• When \( x=-7\), \( y=| - 7+8|-2=|1|-2=-1\).
• When \( x=-9\), \( y=| - 9 + 8|-2=|-1|-2=-1\).
• We plot the vertex \((-8,-2)\) and then use the slope of \(1\) for \(x\geq - 8\) (since for \(x\geq - 8\), \(y=(x + 8)-2=x + 6\), slope \(m = 1\)) and slope of \(-1\) for \(x<-8\) (since for \(x<-8\), \(y=-(x + 8)-2=-x-10\), slope \(m=-1\)) to draw the V - shaped graph.

Part (b) - Describing Transformations

Step 1: Horizontal Shift

For a function \(y = f(x)\), the function \(y=f(x + h)\) is a horizontal shift. If \(h>0\), the graph of \(y = f(x)\) is shifted \(h\) units to the left. For \(y = |x|\) and \(y=|x + 8|\), we have \(h = 8>0\), so the graph of \(y = |x|\) is shifted 8 units to the left to get the graph of \(y=|x + 8|\).

Step 2: Vertical Shift

For a function \(y = f(x)\), the function \(y=f(x)-k\) (where \(k>0\)) is a vertical shift down by \(k\) units. For \(y=|x + 8|\) and \(y=|x + 8|-2\), we have \(k = 2>0\), so the graph of \(y=|x + 8|\) is shifted 2 units down to get the graph of \(y=|x + 8|-2\).

Final Answers

(a) The graph of \(y = |x|\) has a vertex at \((0,0)\) and is a V - shaped graph opening upwards with slopes \(1\) (for \(x\geq0\)) and \(-1\) (for \(x<0\)). The graph of \(y=|x + 8|-2\) has a vertex at \((-8,-2)\) and is a V - shaped graph opening upwards with slopes \(1\) (for \(x\geq - 8\)) and \(-1\) (for \(x<-8\)). (The actual graphing is done by plotting the key points as described above).

(b) To obtain the graph of \(y = |x + 8|-2\) from the graph of \(y = |x|\), we first shift the graph of \(y = |x|\) 8 units to the left (because of the \(x+8\) inside the absolute - value) and then shift the resulting graph 2 units down (because of the \(-2\) outside the absolute - value).

For part (b) answer: The graph of \(y = |x|\) is shifted 8 units to the left and 2 units down to obtain the graph of \(y=|x + 8|-2\).