Problem 32

Question

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

Step-by-Step Solution

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Answer

The graph of function $g(x)=-|x+4|+2$ is a 'V' shape that opens downward, shifted 4 units to the left and 2 units up from the origin, such that its lowest point is at (-4,2).

1Step 1: Understanding the basic function

Start by graphing the function $f(x) = |x|$. This is a V-shaped graph that intersects the origin (0,0) and slopes upwards on either side.

2Step 2: Identifying transformations

In the transformed function $g(x)=-|x+4|+2$, three transformations can be identified: 1) A reflection on the x-axis due to the 'negative' introduced; 2) A horizontal shift to the left by 4 units represented by '+4' inside the absolute value; 3) A vertical shift upwards by 2 units represented by '+2' outside the absolute value.

3Step 3: Applying transformations to the graph

Now apply these transformations to the initial graph of $f(x)=|x|$. Because of the reflection, the 'V' that used to open upwards will now open downwards. The horizontal shift will move the lowest point of the 'V' from the origin to the point (-4,0). Lastly, with the upward shift of 2 units, this lowest point will move to (-4,2).

Key Concepts

Graph TransformationsReflection on the x-axisVertical and Horizontal ShiftsGraphing Techniques

Graph Transformations

Graph transformations are changes made to the basic graph of a function, altering its appearance without changing its overall shape. These changes can vary, including shifts, reflections, and stretching or compressing the graph. Understanding graph transformations is crucial for students learning algebra and calculus, as it allows us to visualize complex equations more easily.

Each transformation affects the position or orientation of the graph in a specific way.
Analyzing transformations involves breaking down the equation into parts to determine which transformation occurs.

In the context of absolute value functions, such as the base graph of $f(x) = |x|$, transformations will change its characteristic "V" shape by moving or flipping it rather than deleting or adding parts.

Reflection on the x-axis

A reflection on the x-axis means that the graph is flipped over the horizontal axis. This operation effectively changes the direction in which the graph opens. For the absolute value function, this reflection is induced by adding a negative sign in front of the absolute value.

If you initially have $f(x) = |x|$, reflecting it along the x-axis turns it into $-|x|$.
Visualize this as taking each point on the graph and mirroring it directly over the x-axis.

This means the "V" shape, which originally opens upward, will now open downward. This transformation is critical for turning functions that originally increase from a minimum point into functions that decrease from a maximum point.

Vertical and Horizontal Shifts

Vertical and horizontal shifts move the entire graph without altering its shape or orientation. These shifts help position the graph at its final location on the coordinate plane.

A horizontal shift happens when a constant is added or subtracted inside the absolute value symbols. For example, $f(x) = |x + 4|$ shifts the graph 4 units left.
A vertical shift occurs when a constant is added or subtracted outside the absolute value function. For instance, in $g(x) = |x| + 2$, the graph moves up 2 units.

In our specific example, the function $-|x+4|+2$ incorporates:

A horizontal shift left by 4 units (adjusting the vertex to $-4$)
A vertical shift up by 2 units (raising the vertex to an overall point of $-4, 2$)

These shifts are vital to correctly graphing and understanding where each point of the function resides.

Graphing Techniques

Graphing techniques are methods used to accurately plot functions using transformations and other algebraic tools. These techniques streamline the process of drawing function graphs and make it easier to understand the influence of each parameter on a function.