Question

1. sketch the graph of the function $f(x) = |x|$.

Explanation:

Step1: Analyze the function definition

The absolute - value function \(f(x)=\vert x\vert\) can be written as a piece - wise function. By the definition of the absolute value, \(f(x)=

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

\)

Step2: Analyze the two pieces of the function

• For the part \(y = x\) when \(x\geq0\): This is a linear function with a slope of \(m = 1\) and a \(y\) - intercept of \(b = 0\). We can find some points on this line. When \(x = 0\), \(y=0\); when \(x = 1\), \(y = 1\); when \(x=2\), \(y = 2\), etc.
• For the part \(y=-x\) when \(x < 0\): This is a linear function with a slope of \(m=- 1\) and a \(y\) - intercept of \(b = 0\). We can find some points on this line. When \(x=-1\), \(y = 1\); when \(x=-2\), \(y = 2\); when \(x = - 3\), \(y=3\), etc.

Step3: Plot the points and draw the graph

• Plot the points for \(y = x(x\geq0)\): \((0,0)\), \((1,1)\), \((2,2)\) and draw a straight line with a slope of \(1\) starting from the origin \((0,0)\) and going to the right (in the first quadrant).
• Plot the points for \(y=-x(x < 0)\): \((- 1,1)\), \((-2,2)\), \((-3,3)\) and draw a straight line with a slope of \(- 1\) starting from the origin \((0,0)\) and going to the left (in the second quadrant).

The graph of \(y = \vert x\vert\) is a "V - shaped" graph with its vertex at the origin \((0,0)\). The left - hand side of the "V" (for \(x < 0\)) has a slope of \(-1\) and the right - hand side (for \(x\geq0\)) has a slope of \(1\).

Answer:

The graph of \(f(x)=\vert x\vert\) is a V - shaped graph with vertex at \((0,0)\), the right - hand branch (for \(x\geq0\)) is the line \(y = x\) and the left - hand branch (for \(x < 0\)) is the line \(y=-x\).