Problem 93 - College Algebra

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

The graph of \(g(x) = -2|x+4| + 1\) is a transformation of the graph of \(f(x) = |x|\) that has been shifted up by 1 unit, shifted left by 4 units, stretched vertically by a factor of 2, and flipped over the x-axis.

Step 1: Graph the basic function

Start with graphing the basic absolute value function \(f(x) = |x|\). This function makes a V shape with the point of the V at the origin.

Step 2: Understand the transformations

Now look at the function \(g(x) = -2|x+4| + 1\). The +1 added after the absolute value function indicates it is shifted up by 1 unit. The +4 inside the absolute value function indicates it is shifted left by 4 units. The -2 as a factor indicates a vertical stretch by a factor of 2 and a vertical reflection, meaning the graph is flipped over the x-axis.

Step 3: Apply the transformations

Apply the transformations to the graph of \(f(x) = |x|\) to create the graph of \(g(x) = -2|x+4| + 1\). Shift the entire graph of \(f(x) = |x|\) up by 1 unit, left by 4 units, stretch it vertically by a factor of 2, and flip it over the x-axis.

Key Concepts

Transformations of Absolute Value Graphs

When graphing absolute value functions, understanding the concept of transformations is key. Transformations involve shifting, stretching, compressing, and reflecting the graph of the parent function, which, in this case, is the basic absolute value function given by f(x) = |x|.

The graph of this parent function creates a 'V' shape with the vertex at the origin (0,0). Every transformation applied to this graph will move or change that 'V' in some manner without losing its basic shape. Think of the graph as a piece of pliable material that you can shift to a new position, stretch or compress to make it wider or narrower, and finally flip it over a line to reflect it.

Using these transformations, more complex absolute value functions can be graphed. For example, a function like g(x) = a|bx+c|+d has four transformation parameters: a, b, c, and d. The value d dictates vertical shifts, c influences horizontal shifts, a handles vertical stretches or reflections, and b could indicate a horizontal stretch or compression (though it's not present in our example problem). By analyzing these parameters, one can systematically alter the parent graph to obtain the desired function's graph.

Vertical Stretches and Reflections

Vertical stretches and reflections change the way an absolute value graph appears in relation to the x-axis. A vertical stretch makes the graph taller, while a vertical reflection flips it over the x-axis.

For the function g(x) from our exercise, the coefficient -2 directly in front of the absolute value causes these two transformations. The number 2 indicates that the graph will be stretched so that it is twice as tall; this will effectively make the 'V' shape of the graph narrower.

The negative sign signifies a reflection. This causes the graph's 'V' to flip upside down, making the parts that were once above the x-axis now lie below it, and vice versa. It's important to keep in mind that reflections are not the same as rotations – the 'V' shape still maintains its left-right orientation but is inverted vertically.

When applying these transformations, one needs to stretch the graph before reflecting it. By accurately performing these steps, the vertical aspects of the function's graph will be correctly depicted, leading to a sound visual representation of the transformed function.

Horizontal Translations

Horizontal translations shift the absolute value graph left or right on the coordinate plane. This is often one of the easiest transformations to spot in an equation, as it involves the addition or subtraction of a number inside the absolute value function.

In our example, the variable x inside the absolute value is modified by +4, which tells us that we need to shift the graph to the left by 4 units. A positive number inside the absolute value (with x) always indicates a shift in the opposite direction – it’s a common initial misconception that a positive number would move the graph to the right.

It's crucial to apply this transformation accurately since it changes the function's vertex – the lowest or highest point of the 'V' – to a new location on the grid. It's one of the first things a student should look out for when beginning to graph an absolute value function that's been altered from its parent function. By recognizing the direction and magnitude of this horizontal shift, students can accurately establish the new starting point from which other transformations, such as vertical shifts and stretches or reflections, will take effect.