Problem 94 - Precalculus Essentials

Question

Begin by graphing the absolute value function, \(f(x)=|x| .\) Then use transformations of this graph to graph the given function. $$g(x)=-2|x+3|+2$$

Step-by-Step Solution

Answer

The process to graph the function \(g(x)=-2|x+3|+2\) includes three transformations. Start with the base graph of \(f(x) = |x|\), then apply a horizontal shift of 3 units to the left for the \(|x+3|\) portion, reflect over the x-axis due to the '-2' outside the absolute value sign, and finally move the graph upwards by 2 units resulting from '+2' outside the absolute value sign.

Step 1: Graph the Absolute Value Function

First begin by drawing the graph for \(f(x) = |x|\). This is a V-shaped graph which starts at the origin (0,0). The graph increases by a slope of 1 for all \(x \geq 0\) and decreases by a slope of -1 for all \(x < 0\).

Step 2: Understanding the Transformations

For the function \(g(x)=-2|x+3|+2\), there are three transformations to apply to the base absolute value function: \nThe graph is translated 3 units to the left, because of the '+3' inside the absolute value brackets. This means every \(x\) in \(f(x)\) will be replaced by \((x+3)\). This is a horizontal shift. \nThe graph is reflected over the x-axis, because of the '−2' coefficient of the absolute value. This means we multiply the whole function \(f(x)\) by -2, changing the direction of the V of the graph. \nLastly, the graph is translated 2 units up, due to '+2' at the end of the equation. This is a vertical shift, meaning the entire graph of \(f(x)\) is moved up by 2 units.

Step 3: Graphing the Function

Begin by moving every point in the base absolute value function 3 units to the left, altering if \(x\) is greater or smaller than 0. Then reflect the graph over the x-axis, changing the direction that the V opens. Lastly, move each point 2 units upwards. The graph that you draw should be the one ending up with the equation \(g(x) = -2|x+3| + 2\).