Question

which are characteristics of the graph of the parent absolute value function?
the graph is symmetric with respect to the y - axis. the graph is a \v\ with vertex at the point (0,0).
the two halves of the graph make 45° angles with the the two halves of the graph make 30° angles with the
x - axis. x - axis.
the graph is symmetric with respect to the x - axis.

Explanation:

Step1: Analyze symmetry about y - axis

For the parent absolute - value function \(y = |x|\), if we replace \(x\) with \(-x\), we get \(y=|-x|=|x|\). So, \(y = |x|\) is symmetric about the \(y\) - axis.

Step2: Analyze the shape and vertex

The parent absolute - value function \(y = |x|\) has the form \(y=

$$\begin{cases}x, & x\geq0\\-x, & x < 0\end{cases}$$

\). Its graph is a "V" - shape. When \(x = 0\), \(y=0\), so the vertex is at \((0,0)\).

Step3: Analyze the angle with the x - axis

For \(y = x\) (when \(x\geq0\)), the slope \(m = 1\). The angle \(\theta\) that the line \(y = x\) makes with the \(x\) - axis is given by \(\tan\theta=1\), so \(\theta = 45^{\circ}\). For \(y=-x\) (when \(x < 0\)), the slope \(m=- 1\), and the angle that the line makes with the positive \(x\) - axis (in the second quadrant) is also \(45^{\circ}\) (since the magnitude of the slope is \(1\)).

Step4: Analyze symmetry about x - axis

If a graph is symmetric about the \(x\) - axis, for every point \((x,y)\) on the graph, the point \((x, - y)\) is also on the graph. For \(y = |x|\), if we take a point \((1,1)\) (since \(y=|1| = 1\)), the point \((1,-1)\) is not on the graph \(y = |x|\) (because \(|-1|
eq - 1\)). So, the graph is not symmetric about the \(x\) - axis.

Answer:

The graph is symmetric with respect to the \(y\) - axis. The graph is a "V" with vertex at the point \((0,0)\). The two halves of the graph make \(45^{\circ}\) angles with the \(x\) - axis.