Question

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

Explanation:

• For the parent absolute value function \(y = |x|\), if we replace \(x\) with \(-x\), we get \(y=|-x|=|x|\). So, the graph is symmetric about the \(y\) - axis.
• The graph of \(y = |x|\) is in the shape of a "V". When \(x = 0\), \(y=0\). So, the vertex is at \((0,0)\).
• The equation of the two - halves of \(y = |x|\) are \(y=x\) (for \(x\geq0\)) and \(y =-x\) (for \(x<0\)). The slope of \(y = x\) is \(m = 1=\tan45^{\circ}\) and the slope of \(y=-x\) is \(m=- 1\), and the angle that the line \(y = x\) (or \(y=-x\)) makes with the \(x\) - axis is \(45^{\circ}\).
• If a graph is symmetric about the \(x\) - axis, then for every point \((x,y)\) on the graph, the point \((x, - y)\) is also on the graph. For \(y = |x|\), if \((x,y)=(1,1)\) is on the graph, \((1,-1)\) is not on the graph since \(y = |x|\geq0\) for all real \(x\). So, it 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.