question 1 (1 point)
compare the graph to the absolute value parent function ($y = |x|$) select all the transformations that apply.
graph of a v - shaped function with vertex at (0, 3), x - intercepts at (-3, 0) and (3, 0), and endpoints at (-6, -3) and (6, -3)
✓ 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

Question

Accepted Answer

# Brief Explanations:
1. **Reflected over the x - axis**: The parent function $y = |x|$ opens upwards (V - shape with vertex at $(0,0)$ and arms going up). The given graph opens downwards, which indicates a reflection over the $x$ - axis. So option a is correct.
2. **Horizontal Shift Left/Right**: The vertex of the parent function $y=|x|$ is at $(0,0)$. The vertex of the given graph is at $(0,3)$? Wait, no, looking at the graph, the vertex is at $(0,3)$? Wait, no, the graph intersects the $y$ - axis at $(0,3)$ and the $x$ - axis at $(- 3,0)$ and $(3,0)$. Wait, the parent function $y = |x|$ has $x$ - intercepts at $x = 0$ (vertex) and for $y=0$, $x = 0$. Wait, no, the given graph: let's find the equation. The slope of the left arm (from $(-6,-3)$ to $(0,3)$): slope $m=\frac{3 - (-3)}{0-(-6)}=\frac{6}{6} = 1$, but since it's opening down, the equation should be $y=-|x|$? Wait, no, the $x$ - intercepts are at $x=-3$ and $x = 3$, and the vertex is at $(0,3)$. Wait, the parent function $y = |x|$ has vertex at $(0,0)$, and for the given graph, when $x = 0$, $y = 3$, and when $y = 0$, $x=\pm3$. So the equation of the given graph is $y=-|x|+3$? Wait, no, if we consider the reflection first: reflection over $x$ - axis gives $y=-|x|$, then vertical shift up by 3 units? Wait, no, the vertex of the parent function $y = |x|$ is $(0,0)$, the vertex of the given graph is $(0,3)$? Wait, no, looking at the graph, the highest point (vertex) is at $(0,3)$, and it goes down to $(-6,-3)$ and $(6,-3)$. Wait, maybe my initial analysis was wrong. Let's re - examine the options:

- **Reflection over x - axis**: The parent function $y = |x|$ has a minimum at $(0,0)$, the given graph has a maximum at $(0,3)$, so it is reflected over the $x$ - axis (since the direction of opening is reversed), so option a is correct.
   - **Horizontal Shift**: The $x$ - intercepts of the parent function $y = |x|$ are at $x = 0$ (vertex). The $x$ - intercepts of the given graph are at $x=-3$ and $x = 3$. For the function $y=-|x - h|+k$, the horizontal shift is $h$. If we write the equation of the given graph, let's assume it is $y=-|x|+3$? Wait, no, when $x = 3$, $y=-|3|+3=0$, and when $x = 0$, $y = 3$. So the vertex is at $(0,3)$, and the $x$ - intercepts are at $x=\pm3$. The parent function $y = |x|$ has $x$ - intercept at $x = 0$ and for $y = 0$, $x = 0$. Wait, maybe the horizontal shift is not present? Wait, the original check - box for c (Horizontal Shift Right) was marked, but maybe that's incorrect. Wait, no, let's think again. The parent function $y = |x|$ has a vertex at $(0,0)$, and the given graph has a vertex at $(0,3)$ and $x$ - intercepts at $x=-3$ and $x = 3$. The equation of the given graph can be written as $y=-|x|+3$? Wait, no, if we consider the transformation from $y = |x|$:

- Reflection over $x$ - axis: $y=-|x|$
     - Vertical shift up by 3 units: $y=-|x| + 3$

But the options given: a is reflected over $x$ - axis (correct), d is vertical shift up (since we go from $y=-|x|$ (vertex at $(0,0)$) to $y=-|x|+3$ (vertex at $(0,3)$), so vertical shift up. Wait, but in the original check - boxes, a and c were marked, but maybe that's wrong. Wait, the $x$ - intercepts of the given graph are at $x=-3$ and $x = 3$, while the parent function $y = |x|$ has $x$ - intercept at $x = 0$. Wait, no, for $y = |x|$, when $y = 0$, $x = 0$. For the given graph, when $y = 0$, $x=\pm3$. So the graph of the given function is $y=-|x|+3$? Wait, no, if $y=-|x|+3$, when $x = 0$, $y = 3$; when $y = 0$, $|x|=3$, so $x=\pm3$, which matches. So the transformations are:

- Reflection over $x$ - axis (a: correct)
     - Vertical shift up (d: correct)

Wait, but the original check - boxes had a, c, f. But f is "Graph is Narrower". The parent function $y = |x|$ has a slope of $\pm1$ for its arms. For the given graph, the slope of the left arm (from $(-6,-3)$ to $(0,3)$): slope $m=\frac{3-(-3)}{0 - (-6)}=\frac{6}{6}=1$, and the right arm (from $(0,3)$ to $(6,-3)$): slope $m=\frac{-3 - 3}{6-0}=\frac{-6}{6}=-1$. The slope of $y = |x|$ is also $\pm1$, so the graph is not narrower or wider. So f is incorrect.

Wait, maybe I made a mistake in the equation. Let's take two points on the parent function $y = |x|$: $(0,0)$, $(1,1)$, $(-1,1)$. Two points on the given graph: $(0,3)$, $(3,0)$, $(-3,0)$. The distance from the vertex to the $x$ - intercept: for parent function, from $(0,0)$ to $(1,0)$ (wait, no, $y = |x|$ has $x$ - intercept only at $x = 0$). Wait, the given graph has $x$ - intercepts at $x=\pm3$ and vertex at $(0,3)$. So the "width" (the horizontal distance from the vertex to the $x$ - intercept) is 3 units, while for the parent function $y = |x|$, if we consider the same $y$ - value, say $y = 1$, $x=\pm1$, so the width is 1 unit. Wait, no, that's not the right way. The general form of the absolute - value function is $y=a|x - h|+k$, where $|a|$ determines the width: if $|a|>1$, the graph is narrower; if $|a|<1$, the graph is wider. For the parent function $y = |x|$, $a = 1$. For the given graph, let's find $a$. Using the vertex $(h,k)=(0,3)$ and the point $(3,0)$:

$0=a|3 - 0|+3$
     $0 = 3a+3$
     $3a=-3$
     $a=-1$

Since $|a| = 1$, same as the parent function, so the graph is not narrower or wider. So f is incorrect.

Now, horizontal shift: $h = 0$, so no horizontal shift. So b and c are incorrect.

Vertical shift: $k = 3$, so vertical shift up (d: correct).

Reflection over $x$ - axis: $a=-1$, so reflection over $x$ - axis (a: correct).

So the correct transformations are a (Reflected over the x - axis) and d (Vertical Shift Up). But wait, the original check - boxes had a, c, f marked, which is wrong. Let's re - evaluate:

- **Reflection over x - axis**: The parent function opens up, the given graph opens down, so reflection over $x$ - axis (a: correct).
   - **Vertical Shift Up**: The vertex of the parent function is at $(0,0)$, the vertex of the given graph is at $(0,3)$, so vertical shift up by 3 units (d: correct).
   - **Horizontal Shift**: The vertex is at $x = 0$, same as the parent function, so no horizontal shift (b and c: incorrect).
   - **Vertical Shift Down**: Vertex is higher, so no (e: incorrect).
   - **Graph is Narrower/Wider**: $|a| = 1$, same as parent, so neither (f and g: incorrect).

But maybe the initial analysis of the graph was wrong. Let's look at the graph again: the left end is at $(-6,-3)$ and the right end is at $(6,-3)$, vertex at $(0,3)$. Let's find the equation using two points: $(-6,-3)$ and $(0,3)$. The slope $m=\frac{3-(-3)}{0 - (-6)} = 1$. So the equation of the left arm is $y-3=1(x - 0)$, $y=x + 3$ for $x\leq0$. For the right arm, using $(0,3)$ and $(6,-3)$, slope $m=\frac{-3 - 3}{6-0}=-1$, so equation is $y-3=-1(x - 0)$, $y=-x + 3$ for $x\geq0$. So the overall equation is $y=-|x|+3$ (since for $x\geq0$, $y=-x + 3=-|x|+3$; for $x<0$, $y=x + 3=-|x|+3$ (because $|x|=-x$ when $x<0$, so $-|x|=-(-x)=x$, so $y=x + 3=-|x|+3$)).

So comparing to $y = |x|$:

- Reflection over $x$ - axis: because the sign of $|x|$ is negative ($y=-|x|+3$ vs $y = |x|$).
- Vertical shift up: because we add 3 to $-|x|$, so from $y=-|x|$ (vertex at $(0,0)$) to $y=-|x|+3$ (vertex at $(0,3)$).

So the correct options are a (Reflected over the x - axis) and d (Vertical Shift Up). But maybe the original problem's graph was misinterpreted. Wait, the user's graph: the vertex is at $(0,3)$, and the ends are at $(-6,-3)$ and $(6,-3)$. So the vertical distance from the vertex to the ends is $3-(-3)=6$ units over a horizontal distance of 6 units (from $x = 0$ to $x = 6$), so the slope is $\frac{-3 - 3}{6-0}=-1$, same as the parent function's slope ($\pm1$), so no change in width.

So the correct transformations are:

a. Reflected over the x - axis

d. Vertical Shift Up

But the original check - boxes had a, c, f marked, which is incorrect. Let's go back to the options:

a. Reflected over the x - axis: Correct, as the graph opens downwards.

d. Vertical Shift Up: Correct, as the vertex is at $(0,3)$ instead of $(0,0)$.

So the correct options are a and d.

# Answer:
a. Reflected over the x - axis, d. Vertical Shift Up

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