Problem 16 - Algebra A Combined Function

Question

Sketch the graph of each function. $$ f(x)=|x-3|+2 $$

Step-by-Step Solution

Answer

The graph is a V-shape with vertex at (3, 2), opening upwards.

Step 1: Identify the Basic Function

Our function is given as \( f(x) = |x-3| + 2 \). The basic parent function here is \( |x| \), which represents a V-shaped graph centered at the origin.

Step 2: Horizontal Shift

The expression \( |x-3| \) indicates a horizontal shift of the absolute value function. Specifically, the graph is shifted to the right by 3 units because of the \( -3 \) inside the absolute value.

Step 3: Vertical Shift

The \( +2 \) outside the absolute value function indicates a vertical shift. This will shift the entire graph upward by 2 units.

Step 4: Sketch the Vertex

With the horizontal shift of 3 units to the right and a vertical shift of 2 units up, the vertex of the absolute value function moves from the origin to the point \((3, 2)\).

Step 5: Draw the V-Shape

From the vertex \((3, 2)\), draw lines forming a V-shape. The slope on either side of the vertex is 1: the line to the left of the vertex rises with a slope of -1, and the line to the right of the vertex rises with a slope of 1.

Key Concepts

Vertex of a Function

In the world of functions, the vertex is a key point that can tell you a lot about the shape and position of a graph. For absolute value functions, the vertex is the point where the graph "turns". It acts as the tip of the V-shape that is characteristic of these types of functions.

For the function given, which is \( f(x) = |x-3| + 2 \), the vertex is not at the origin, like the basic \( |x| \) function. Instead, it has been moved to the point \((3, 2)\). This happens because the expression inside the absolute value, \( x - 3 \), causes a horizontal shift to the right by 3 units. The \(+2\) outside shifts the graph upward by 2 units.

So, whenever you have an absolute value function, you can find the vertex by looking at these shifts. The numbers inside the absolute value will affect the horizontal position, while numbers added or subtracted outside of it influence the vertical position.

Graph Transformations

Graph transformations allow you to effectively "move" your graph around and reshape it. In our function \( f(x) = |x-3| + 2 \), we see two main types of transformations: horizontal and vertical shifts.

• **Horizontal Shift:** The term \( x - 3 \) indicates a move rightward by 3 units. This is because the sign is negative. In general, if you have \( |x - h| \), the graph moves horizontally to \( h \). Whether \( h \) is positive or negative determines the direction of the shift.


• **Vertical Shift:** The \(+2\) causes a shift upward by 2 units. If it were minus, the graph would move downward. This is generally signified by a number outside the function, like \( + k \).

Remember, graph transformations can also include reflections, stretches or compressions, but in this function, we're mostly focusing on shifts. These transformations are important because they help us understand how the core shape of the graph "travels" in the coordinate system.

V-shaped Graph

The absolute value function often creates what is known as a V-shaped graph. This distinctive shape comes from the nature of the absolute value itself, which takes any input and outputs its non-negative counterpart.

For \( f(x) = |x-3| + 2 \), the V-shaped graph has its vertex at \((3, 2)\). It's where the two lines of the V meet, forming a point.

• On the **left side** of the vertex, the line heads downwards with a slope of \(-1\), capturing the decrease.


• On the **right side**, the line rises with a slope of \( 1 \), indicating an increase.

The symmetry of this V helps in sketching or predicting behavior around the vertex as you understand that both sides mirror one another. Just keep in mind that transformations change where this V appears on the graph but not the basic shape itself. Thus, even if you shift or scale it, the graph retains its V-shape.