key term - Corner Point

Definition

A corner point is a specific point on the graph of an absolute value function where the function changes direction, forming an angle or 'corner' shape. This point is crucial in understanding the behavior and characteristics of absolute value functions.

5 Must Know Facts For Your Next Test

1. The corner point of an absolute value function occurs at the point where the function changes from increasing to decreasing or vice versa.
2. The x-coordinate of the corner point is the value of x where the function is equal to zero, i.e., the solution to the equation |x| = 0.
3. The y-coordinate of the corner point is the absolute value of the x-coordinate, i.e., the distance of the x-coordinate from zero.
4. The corner point divides the graph of an absolute value function into two linear segments, one increasing and one decreasing.
5. Identifying the corner point is crucial for sketching the graph of an absolute value function and understanding its key features, such as the domain, range, and transformations.

Review Questions

• Explain how the corner point relates to the behavior of an absolute value function.
  • The corner point of an absolute value function is the point where the function changes direction, transitioning from increasing to decreasing or vice versa. This point is significant because it divides the graph into two linear segments, one increasing and one decreasing. Understanding the location and coordinates of the corner point is essential for analyzing the domain, range, and transformations of the absolute value function.
• Describe the process of finding the coordinates of the corner point for an absolute value function.
  • To find the coordinates of the corner point for an absolute value function, $f(x) = |x - h| + k$, follow these steps: 1) Set the function equal to zero to find the x-coordinate of the corner point: $|x - h| + k = 0$, which simplifies to $x - h = 0$. 2) Solve for $x$ to find the x-coordinate of the corner point, which is $x = h$. 3) Substitute the x-coordinate back into the original function to find the y-coordinate of the corner point, which is $f(h) = |h - h| + k = k$.
• Analyze how the location of the corner point affects the graph and key features of an absolute value function.
  • The location of the corner point on the graph of an absolute value function $f(x) = |x - h| + k$ has a significant impact on the function's behavior. The x-coordinate $h$ determines the horizontal position of the corner point, while the y-coordinate $k$ determines the vertical position. Changes in the values of $h$ and $k$ result in transformations of the graph, such as horizontal and vertical shifts. Furthermore, the coordinates of the corner point directly influence the domain and range of the function, as well as the points where the function intersects the x-axis and y-axis. Understanding the relationship between the corner point and these key features is crucial for sketching and analyzing absolute value functions.

Related terms

Absolute Value Function:

A function that describes the distance of a number from zero on the number line, denoted as |x|.

Domain:

The set of all possible input values for a function.

Range:

The set of all possible output values for a function.