Question

9-26 Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Table 1.2.3, and then applying the appropriate transformations.

21.\({\mathop{\rm y}\nolimits} = 2 - \left| x \right|\)

Short answer

Reflect the graph of \(y = \left| x \right|\) about the \(x\)-axis, then shift 2 units upward to obtain the graph of \(y = 2 - \left| x \right|\).

Step-by-step solution

Condition of vertical and horizontal shifts

If \(c > 0\) and \(y = f\left( x \right)\) is the original function, then for the graph of \(y = f\left( x \right) + c\) shift the graph \(c\) units upward. For \(y = f\left( x \right) - c\), shift the graph of \(y = f\left( x \right)\), \(c\) units downward. For \(y = f\left( {x - c} \right)\), shift the graph of \(y = f\left( x \right)\), c units to the right. And, for \(y = f\left( {x + c} \right)\), shift the graph of \(y = f\left( x \right)\), c units to the left.

Condition for vertical and horizontal stretching and reflecting

Let \(c > 0\). To obtain the graph of \(y = cf\left( x \right)\), stretch the graph of \(y = f\left( x \right)\) verticallyby factor c. For \(y = \left( {\frac{1}{x}} \right)f\left( x \right)\), shrink the graph of \(y = f\left( x \right)\) vertically by factor c. For \(y = f\left( {cx} \right)\), shrink the graph of \(y = f\left( x \right)\) horizontally by factor c. For \(y = f\left( {\frac{x}{c}} \right)\), stretch the graph of \(y = f\left( x \right)\) horizontally by factor c. For\(y = - f\left( x \right)\), reflect the graph of \(y = f\left( x \right)\) about the \(x\)-axis. And, for \(y = f\left( { - x} \right)\), reflect the graph of \(y = f\left( x \right)\) about the \(y\)-axis.

Draw the graph of the function 

Begin with the graph of \(y = \left| x \right|\) as shown below:

Reflect the graph of \(y = \left| x \right|\) about the \(x\)-axis, then shift 2 units upward to obtain the graph of \(y = 2 - \left| x \right|\) as shown below:

Thus, the graph of the function is obtained.