Problem 29 - College Algebra

Question

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

Step-by-Step Solution

Answer

The graph of the function \(h(x)=-|x+4|\) is a 'V' shape opening downwards due to the negative coefficient, and it is shifted 4 units to the left because of the \('+4'\) inside absolute value. The vertex is at the point \((-4,0)\).

Step 1: Graph the absolute value function \(f(x) = |x|\)

Plot the graph of the function \(f(x)=|x|\), which is in the shape of a 'V'. The vertex is at the origin (0,0), it increases to the right with a slope of 1 and to the left with a slope of -1. This will serve as the foundational graph.

Step 2: Plot vertical transformation of function \(h(x)=-|x+4|\)

Now, turning towards the given function \(h(x)=-|x+4|\), we can see that function \(f(x)\) is reflected vertically because of the negative sign. Now, the 'V' opens downward instead of upward. The function value is negative whenever the input is nonzero.

Step 3: Conduct the horizontal transformation

For the Horizontal shift, consider the \(x+4\) inside the absolute value part. It moves (or shifts) all the points on the function \(4\) units to the left. The vertex of the 'V' now becomes \(-4, 0\). Plot the respective graph, keeping in mind the effects of both transformations. The final graph should be a 'V' shape opening downwards with the vertex at the point \((-4, 0)\).

Key Concepts

Absolute Value Function

The absolute value function is a crucial aspect of many mathematical concepts, particularly when dealing with transformations. It's often written as \(f(x) = |x|\). The graph of this function is characterized by a distinctive 'V' shape. The vertex of the graph sits at the origin, which is the point \((0,0)\). On a graph:

• The right arm of the 'V' climbs upwards with a slope of 1, meaning it creates a 45-degree angle with the horizontal axis.
• The left arm descends with a slope of -1, mirroring the right side in the opposite direction.

The absolute value function inherently ensures that all output values are non-negative, creating a mirror effect around the y-axis. It's this basic graph that acts as a starting point before applying any transformations. Understanding the unaltered absolute value function provides the foundation needed to comprehend more complex transformations.

Graphing Functions

Graphing functions is a fundamental skill in mathematics that visualizes how a function behaves across different values of \(x\). It involves plotting a series of points that satisfy the function and then connecting these points to see how the function progresses. When dealing with absolute value functions like \(f(x) = |x|\), start by identifying key characteristics:

• The vertex, which for the basic absolute value function is \((0,0)\).
• The symmetry around the y-axis which helps ensure that both sides of the graph are identical.
• Different slopes on each side of the vertex, where one side will have a positive slope and the other a negative slope.

Graphing is not just about connecting dots, but rather unveiling the underlying structure of the function. This graphical representation aids in understanding how various transformations affect the function's shape and position on the graph.

Horizontal Shifts

Transformations often involve shifting the entire graph horizontally. In the function \(h(x) = -|x+4|\), the expression \(x+4\) is the key to understanding the horizontal shift. Whenever you see a transformation within the function, as in \(x + 4\), it indicates a displacement along the x-axis:

• The sign of the number inside affects the direction of the shift. In this case, a \(+4\) translates the graph 4 units to the left.
• Had it been \(x-4\), the shift would have moved the graph 4 units to the right.

To visualize this on the graph, note that the vertex moves from \((0,0)\) to \((-4,0)\). Each point on the graph similarly moves left, maintaining the graph’s shape but changing its position. This horizontal movement doesn't affect the vertical positions of points, only their horizontal alignment.

Vertical Reflections

A vertical reflection is a transformation strategy that involves flipping the graph over a horizontal axis, typically the x-axis. For the function \(h(x) = -|x+4|\), the negative sign in front of the absolute value signals such a reflection. Here's what happens:

• The original 'V' shape of the \(f(x)=|x|\) graph is flipped upside down.
• This transformation changes the direction in which the 'V' opens from upward to downward.

When you reflect a function vertically, it’s like looking at the function in a mirror placed along the x-axis. Each point on the graph is inverted over this axis. The y-values of the function turn negative whenever it would naturally be positive, ensuring that the 'V' is oriented downwards. This transformation is easy to identify as it primarily affects the direction of the graph rather than its size or position.