Problem 26 - College Algebra

Question

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

Step-by-Step Solution

Answer

The graph of \(g(x)=|x+3|\) is a V-shaped graph that starts from the point (-3,0) and extends upwards to the left and right, analogous to the absolute function graph \(f(x)=|x|\), but shifted three units to the left.

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

On a coordinate plane, draw a V-shaped line that begins at the origin (0,0) and extends upwards to the left and right. The line will pass through the points (-1,1), (0,0), and (1,1). This is because the value of \(f(x)\) is always positive (or zero) for every \(x\). The function intersects the x-axis at x=0.

Step 2: Understanding how to transform the graph

The graph of \(g(x)\) is the same as the graph of \(f(x)\), shifted three units to the left. This is because including '+3' inside the absolute value sign has the effect of shifting the graph horizontally. In general, if we have \(|x-h|\), the graph will shift h units to the right if h > 0 and left if h < 0. Since in our case h = -3, the graph will shift 3 units to the left.

Step 3: Graphing the function \(g(x)=|x+3|\)

Take each point on the original \(f(x)=|x|\) graph and move it three units to the left. For example, the origin (0,0) moves to (-3,0), the point (-1,1) moves to (-4,1), and the point (1,1) moves to (-2,1). Joining those points and extending the lines yields the V-shaped graph of \(g(x)=|x+3|\).

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves using a set of horizontal and vertical axes. These axes are typically labeled as the x-axis (horizontal) and the y-axis (vertical), intersecting at a point known as the origin, which has the coordinates (0, 0).

Each point on this plane can be specified by an ordered pair of numbers \( (x, y) \) where \( x \) represents the horizontal distance from the origin, and \( y \) the vertical distance. In the context of graphing absolute value functions, the coordinate plane allows us to visualize how the function behaves for different values of \( x \) and to understand the relationship between the algebraic expression and its graphical representation.

Transformations of Graphs

One of the powerful aspects of algebra is that we can apply transformations to graphs to easily create new functions from existing ones. Transformations include shifting, reflecting, stretching, and compressing. A shift moves the graph horizontally or vertically, while reflections flip the graph over an axis. Stretching or compressing alters the shape of the graph without changing its overall direction.

For absolute value functions like \( f(x) = |x| \), transformations can drastically change the appearance of the graph. By understanding these transformations, students can more easily graph complex functions, as they can start with the basic \( |x| \) shape and then apply the necessary changes.

Absolute Value Transformation

Absolute value transformations specifically refer to changes in the graph of an absolute value function. When the absolute value function \( f(x) = |x| \) is transformed, we can see a shift, stretch, or reflection.

In our exercise, the transformation involves a horizontal shift. The presence of \( +3 \) within the absolute value, \( g(x) = |x+3| \) indicates a horizontal shift to the left by 3 units. If it were \( -3 \) instead, this would mean a shift to the right. Recognizing how to apply these transformations allows students to quickly and accurately graph absolute value functions on a coordinate plane.

Function Intersection

Function intersection refers to the points where two or more functions share the same coordinates on the coordinate plane. When graphing absolute value functions, identifying intersection points, especially with the axes, can be crucial for understanding the function's behavior.

With absolute value functions like \( f(x) = |x| \), the function's graph intersects the x-axis at the origin because \( |0| = 0 \). After applying transformations, such as the horizontal shift seen in \( g(x) = |x+3| \) for our exercise, the graph's intersection with the x-axis will change accordingly. In this case, the intersection occurs at \( x = -3 \) where \( g(x) = 0 \) because the shift moves the 'V' shape of the absolute value function three units to the left.