Problem 93 - Precalculus Essentials

Question

Begin by graphing the absolute value function, \(f(x)=|x| .\) Then use transformations of this graph to graph the given function. $$g(x)=-2|x+4|+1$$

Step-by-Step Solution

Answer

After applying the transformations of translation 4 units to the left, vertical stretching by a factor of 2, reflection over the x-axis, and translation 1 unit upwards, the graph of \(g(x) = -2|x + 4| + 1 \) is an 'upside down V' with its vertex is at point \(-4,1\)

Step 1: Starting Function

Begin by drawing the graph of the absolute value function, \(f(x) = |x|\). The graph is a 'V' shape that intersects the origin (0,0) with the section in quadrants I and IV having a slope of 1, that is for every 1 unit move to the right, there's also exactly 1 unit move upwards (and vice versa for the parts in quadrants II and III.

Step 2: Horizontal Shift

The \(-4\) inside the absolute value function translates the graph 4 units to the left. Thus, the function becomes \(f(x) = |x + 4|\) and the vertex moves to \(-4,0\).

Step 3: Vertical Stretch and Reflection

The factor \(-2\) in front of the absolute value function stretches the graph vertically by a factor of 2 and reflects it over the x-axis (because the factor is negative). So we now have \(f(x) = -2|x + 4|\), and the graph opening downward with a wider angle.

Step 4: Vertical Shift

Afterwards, there's a vertical shift due to the \(+1\) added to the function, the entire graph moves 1 unit up. Therefore, the function is finalised as \(g(x) = -2|x + 4| + 1 \) with its vertex at \(-4,1\)

Step 5: Final Graph

Confirm the graph opens downward and the vertex of the curve is at point \(-4, 1\).