Problem 53 - Precalculus with Limits

Question

In Exercises 45-56, identify any intercepts and test for symmetry. Then sketch the graph of the equation. \( y = |x-6| \)

Step-by-Step Solution

Answer

The x-intercept of the function is at (6,0) and y-intercept is at (0,6). The graph is symmetric about the vertical line \( x = 6 \). The graph of the function is a V shape that opens upward.

Step 1: Find Intercepts

For the x-intercept, we set \( y = 0 \) and solve for \( x \). So, \( 0 = |x - 6| \) which gives \( x = 6 \). This implies that the x-intercept is at (6,0). For the y-intercept, we set \( x = 0 \) and solve for \( y \). So, \( y = |0 - 6| = 6 \), which gives y-intercept as (0,6).

Step 2: Test for Symmetry

The equation \( y = |x - 6| \) shows a symmetric property around the vertical line \( x = 6 \) as the absolute value function is always symmetric about the vertical line \( x = h \) in the general form \( y = |x - h| \).

Step 3: Sketch the Graph

The absolute value function \( y = |x - 6| \) has a V shape that opens upward, with the point (6,0) at the bottom of the 'V'. Draw the graph starting at the x-intercept and extending up and out in the positive and negative x-directions. At the y-intercept (0,6), the graph will show a point and continue parallel to the x-axis in negative y-direction.

Key Concepts

Symmetry in Absolute Value Functions

Absolute value functions, like the equation \( y = |x - 6| \), inherently possess a special kind of symmetry. This symmetry is centered around a vertical line. In the general form of an absolute value function \( y = |x - h| \), symmetry is around the line \( x = h \). For this particular equation, the line of symmetry is \( x = 6 \).

This means that if you were to fold the graph along this line, the two halves would match exactly. The absolute nature of the function ensures that values on either side of the line are equal in distance, creating a mirror effect on the graph.

Understanding this symmetry helps to quickly sketch graphs and predict their behavior. This reflects a fundamental property of absolute value functions and makes analyzing these graphs intuitive.

Finding Intercepts of the Graph

Identifying intercepts is a crucial step in graphing equations. It involves determining where the graph touches the x-axis and the y-axis.

• **X-intercept**: To find the x-intercept, set \( y = 0 \). For our function \( y = |x - 6| \), setting it equal to 0 gives \( 0 = |x - 6| \), which simplifies to \( x = 6 \). Thus, the graph intersects the x-axis at (6,0).
  
  
• **Y-intercept**: To find the y-intercept, set \( x = 0 \). Substituting gives \( y = |0 - 6| = 6 \), leading to the point (0,6) on the y-axis.

These intercepts are critical for constructing the graph as they provide the initial points to plot the basic shape of the equation.

Sketching the Graph of Absolute Value Equations

Graphing absolute value equations like \( y = |x - 6| \) involves a few straightforward steps:Firstly, locate the intercepts. These provide a grounding framework for the graph. Here, we determined that the x-intercept is at (6,0) and the y-intercept is at (0,6).

The shape of these graphs is characteristically a 'V'. The vertex of this 'V' is at the point (6,0), which is also the minimum point of the graph since the graph opens upwards. From this vertex, consider each side of the 'V'. The lines extend out evenly along the positive and negative x-directions.

With this, the V-shape forms around the symmetry line \( x = 6 \), which acts as a central mirror. Sketching allows for a visual representation of the function's behavior, and identifying key points like intercepts guides the drawing of accurate graphs.