Question

question 3 (1 point)
compare the graph to the absolute value parent function ($y = |x|$). select all the transformations that apply.
graph of a coordinate plane with a red absolute - value - like graph
\\(\square\\) a reflected over the x - axis
\\(\square\\) b horizontal shift left
\\(\square\\) c horizontal shift right
\\(\square\\) d vertical shift up
\\(\square\\) e vertical shift down
\\(\square\\) f graph is narrower
\\(\square\\) g graph is wider

Explanation:

Step1: Analyze Reflection over x - axis

The parent function \(y = |x|\) opens upwards (vertex at \((0,0)\) and the graph is above the x - axis for non - zero x). The given graph has a vertex at \((1,-1)\) and the graph is below the x - axis (since the y - values of the graph are negative). The transformation for reflecting over the x - axis is \(y=-|x|\), which changes the direction of the opening. So the graph is reflected over the x - axis.

Step2: Analyze Horizontal Shift

The vertex of the parent function \(y = |x|\) is at \((0,0)\). The vertex of the given graph is at \((1,-1)\). For a horizontal shift, if the vertex moves from \(x = 0\) to \(x=1\), the shift is to the right (since the x - coordinate of the vertex increases). The formula for a horizontal shift of \(y = |x|\) is \(y=|x - h|\) where \(h\) is the horizontal shift. Here \(h = 1\), so there is a horizontal shift right.

Step3: Analyze Vertical Shift

The y - coordinate of the vertex of the parent function is \(0\), and the y - coordinate of the vertex of the given graph is \(-1\). So the graph is shifted down by 1 unit. The formula for vertical shift is \(y=|x|+k\), where \(k=-1\) (so it's a shift down).

Step4: Analyze Width (Narrower/Wider)

The parent function \(y = |x|\) has a slope of \(1\) for \(x>0\) and \(- 1\) for \(x < 0\). Let's find the slope of the given graph. For the right - hand side of the vertex \((1,-1)\), when \(x = 2\), let's find \(y\). The graph goes from \((1,-1)\) to \((4,-6)\) (approximate, looking at the grid). The slope \(m=\frac{-6 - (-1)}{4 - 1}=\frac{-5}{3}\approx - 1.67\), and for the left - hand side, from \((1,-1)\) to \((-2,-6)\), slope \(m=\frac{-6-(-1)}{-2 - 1}=\frac{-5}{-3}=\frac{5}{3}\approx1.67\). The absolute value of the slope of the given graph is greater than \(1\). The general form of a transformed absolute - value function is \(y = a|x - h|+k\). If \(|a|>1\), the graph is narrower; if \(|a|<1\), the graph is wider. Since the slope of the given graph (in absolute value) is greater than \(1\) (compared to the slope of \(y = |x|\) which has a slope of \(1\) for \(x>0\) and \(-1\) for \(x < 0\)), the graph is narrower.

Step5: Analyze Vertical Shift Up

The y - coordinate of the vertex of the parent function is \(0\) and the y - coordinate of the vertex of the given graph is \(-1\), so it's a shift down, not up.

Answer:

a. Reflected over the x - axis, c. Horizontal Shift Right, e. Vertical Shift Down, f. Graph is Narrower