Question

compare the graph to the absolute value parent function ($y = |x|$). select all the transformations that apply
options:
a. reflected over the x - axis
b. horizontal shift left
c. horizontal shift right
d. vertical shift up
e. vertical shift down
f. graph is narrower
g. graph is wider

Explanation:

Step1: Analyze Reflection over x - axis

The parent function \(y = |x|\) has a V - shape opening upwards. The given graph also has a V - shape opening upwards (the vertex is at the bottom and the arms go up), so it is not reflected over the x - axis (a reflection over the x - axis would make the V - shape open downwards). So option a is not correct.

Step2: Analyze Horizontal Shift

The vertex of the parent function \(y=|x|\) is at \((0,0)\). The vertex of the given graph is at \((- 2,-1)\)? Wait, no, looking at the graph, the vertex is at \((-2,-1)\)? Wait, no, let's re - examine. Wait, the parent function \(y = |x|\) has vertex at \((0,0)\). The given graph's vertex is at \((-2,-1)\)? Wait, no, maybe I made a mistake. Wait, the graph: let's see the vertex. The left arm goes from \((-6,3)\) to \((-2,-1)\), and the right arm goes from \((-2,-1)\) to \((5,6)\). Wait, the vertex is at \((-2,-1)\). The parent function \(y = |x|\) has vertex at \((0,0)\). To get from \((0,0)\) to \((-2,-1)\), we shift left 2 units (horizontal shift left) and down 1 unit (vertical shift down). Wait, but let's check the options. Wait, the options are: a) Reflected over x - axis (no, as the graph opens upwards like \(y = |x|\)), b) Horizontal Shift Left (yes, because the vertex is at \(x=-2\) instead of \(x = 0\), so shifting left 2 units), c) Horizontal Shift Right (no, since it's shifted left), d) Vertical Shift Up (no, the vertex is lower than \((0,0)\) in y - coordinate), e) Vertical Shift Down (yes, the vertex's y - coordinate is \(-1\) instead of \(0\), so shifted down 1 unit), f) Graph is Narrower (the slope of the parent function \(y = |x|\) has a slope of 1 for \(x\geq0\) and - 1 for \(x<0\). Let's check the slope of the right arm of the given graph. From \((-2,-1)\) to \((5,6)\), the slope \(m=\frac{6 - (-1)}{5-(-2)}=\frac{7}{7}=1\), same as the parent function \(y = |x|\) (slope 1 for \(x\geq0\)). The left arm: from \((-6,3)\) to \((-2,-1)\), slope \(m=\frac{-1 - 3}{-2-(-6)}=\frac{-4}{4}=-1\), same as the parent function \(y = |x|\) (slope - 1 for \(x<0\)). So the graph is not narrower or wider (slope is same as parent function, so the width is same as parent function, so f and g are not correct). Wait, but let's re - check the vertex. Wait, maybe I misread the graph. Wait, the graph: the vertex is at \((-2,-1)\). So compared to \(y = |x|\) (vertex at \((0,0)\)), to get to \((-2,-1)\), we shift left 2 units (horizontal shift left, option b) and down 1 unit (vertical shift down, option e). Wait, but let's check the options again. Wait, the options are: a) Reflected over x - axis (no), b) Horizontal Shift Left (yes, because the vertex is at \(x=-2\) instead of \(x = 0\), so horizontal shift left), e) Vertical Shift Down (yes, because the vertex is at \(y=-1\) instead of \(y = 0\), so vertical shift down). Wait, but let's confirm the horizontal shift. The parent function \(y = |x|\) has its vertex at \(x = 0\). The given graph's vertex is at \(x=-2\), so we shift the graph of \(y = |x|\) left by 2 units (since \(x\) value of vertex is decreased by 2, so horizontal shift left). And the \(y\) value of the vertex is \(-1\) instead of \(0\), so vertical shift down by 1 unit. Also, the slope of the lines: for \(y = |x|\), the slope of the right arm is 1, and for the given graph, the slope of the right arm (from \((-2,-1)\) to \((5,6)\)) is \(\frac{6-(-1)}{5 - (-2)}=\frac{7}{7}=1\), same as parent function. The slope of the left arm (from \((-6,3)\) to \((-2,-1)\)) is \(\frac{-1 - 3}{-2-(-6)}=\frac{-4}{4}=-1\), same as parent function. So the graph is not narrower or wider (so f and g are out). Reflected over x - axis? No, because the graph opens upwards, same as \(y = |x|\). So the correct transformations are Horizontal Shift Left (b) and Vertical Shift Down (e). Wait, but let's check the options again. Wait, the options are:

a) Reflected over the x - axis: No, as the graph opens upwards like \(y = |x|\).

b) Horizontal Shift Left: Yes, because the vertex is at \(x=-2\) (parent at \(x = 0\), so shift left 2 units).

c) Horizontal Shift Right: No.

d) Vertical Shift Up: No, vertex is lower than \((0,0)\) in y - direction.

e) Vertical Shift Down: Yes, vertex is at \(y=-1\) (parent at \(y = 0\), so shift down 1 unit).

f) Graph is Narrower: No, slope is same as parent function, so width is same.

g) Graph is Wider: No, slope is same as parent function, so width is same.

Answer:

b. Horizontal Shift Left, e. Vertical Shift Down