Question

graph the equation shown below by transforming the given graph of the parent function.
$y = |x + 3| + 2$

Explanation:

Step1: Identify Parent Function

The parent function of absolute - value functions is \(y = |x|\). Its vertex is at \((0,0)\), and it has a V - shape with slopes of \(1\) (for \(x\geq0\)) and \(- 1\) (for \(x < 0\)).

Step2: Analyze Horizontal Shift

For a function of the form \(y=|x - h|+k\), the horizontal shift is determined by the value of \(h\). In the given function \(y = |x + 3|+2\), we can rewrite \(x + 3\) as \(x-(-3)\). So, \(h=-3\). This means the graph of the parent function \(y = |x|\) will be shifted 3 units to the left.

Step3: Analyze Vertical Shift

The value of \(k\) in the function \(y = |x - h|+k\) determines the vertical shift. Here, \(k = 2\), which means the graph will be shifted 2 units up.

Step4: Determine the New Vertex

The vertex of the parent function \(y = |x|\) is \((0,0)\). After a horizontal shift of 3 units to the left (\(x\) - coordinate: \(0-3=-3\)) and a vertical shift of 2 units up (\(y\) - coordinate: \(0 + 2=2\)), the new vertex of the function \(y=|x + 3|+2\) is \((-3,2)\).

Step5: Analyze the Shape and Slopes

The shape of the absolute - value function remains the same (a V - shape) because there are no reflections or stretches/compressions. The slope of the right - hand side (for \(x\geq - 3\)) is still \(1\) and the slope of the left - hand side (for \(x < - 3\)) is still \(-1\). To graph the function, we can take points from the parent function, apply the transformations. For example, a point on \(y = |x|\) like \((0,0)\) becomes \((-3,2)\) after transformation. A point \((1,1)\) on \(y = |x|\) becomes \((1-3,1 + 2)=(-2,3)\) and a point \((-1,1)\) becomes \((-1-3,1 + 2)=(-4,3)\) and so on. Then we can plot these transformed points and draw the V - shaped graph with the vertex at \((-3,2)\), slope \(1\) for \(x\geq - 3\) and slope \(-1\) for \(x < - 3\).

Answer:

To graph \(y = |x + 3|+2\), shift the graph of \(y = |x|\) 3 units left and 2 units up. The vertex is at \((-3,2)\), with a V - shape, slope \(1\) for \(x\geq - 3\) and slope \(-1\) for \(x < - 3\).