Problem 5 - Algebra 2 and Trigonometry

Question

In \(3-6,\) find the coordinates of the ordered pair with the smallest value of \(y\) for each function. $$ f(x)=\left|\frac{x}{2}-7\right| $$

Step-by-Step Solution

Answer

The ordered pair with the smallest \( y \) value is \((14, 0)\).

Step 1: Understanding the Function

The function given is a piecewise function because it involves an absolute value: \( f(x) = \left| \frac{x}{2} - 7 \right| \). Absolute value functions create a 'V' shape graph, and the point of the 'V' (vertex) is where the smallest value of \( y \) occurs.

Step 2: Finding the Vertex of the Absolute Value

The expression inside the absolute value, \( \frac{x}{2} - 7 \), equals zero at the vertex. To find \( x \), set \( \frac{x}{2} - 7 = 0 \). Solving for \( x \), we get \( x = 14 \).

Step 3: Calculating the Corresponding y-coordinate

Substitute the \( x \)-value found into the function to find \( y \): \( f(14) = \left| \frac{14}{2} - 7 \right| = 0\).

Step 4: Identifying the Coordinates of the Vertex

The vertex, which is the point with the smallest \( y \)-value on the graph, is \( (14, 0) \).

Key Concepts

Piecewise Function

Absolute value functions, like the one given, are examples of piecewise functions. Piecewise functions are defined by different expressions based on different intervals of the domain. For an absolute value function, the domain is typically split into two intervals depending on the input value, resulting in a 'V' shape on the graph.
These functions can be expressed as two linear expressions:

• If the value inside the absolute value is non-negative, the function behaves like a linear function without the absolute value.
• If the value inside is negative, the absolute value changes the sign, converting it back to a positive, again creating a linear expression with a different slope.

Understanding piecewise functions is crucial because it helps in sketching the graph, finding vertices, and solving for specific values like in the exercise.

Vertex of a Function

The vertex of an absolute value function is the critical point where the graph changes direction, forming the point of the 'V'.
For the function \( f(x) = \left| \frac{x}{2} - 7 \right| \), the vertex occurs when the expression inside the absolute value is zero. This is because zero is the turning point where the output switches from decreasing to increasing or vice-versa.

To find the vertex:

• Set the inside of the absolute value to zero: \( \frac{x}{2} - 7 = 0 \).
• Solve for \( x \), resulting in \( x = 14 \).
• Plug \( x = 14 \) back into the function to find \( y \): \( f(14) = \left| 0 \right| = 0 \).

The vertex is thus \( (14, 0) \), representing the smallest \( y \)-value in this context.

Coordinate Geometry

Coordinate geometry helps visualize mathematical functions. It involves plotting points, lines, and curves on a coordinate plane to understand better how a function behaves.
For absolute value functions, specifically, coordinate geometry allows us to see the unique 'V' shape formed by the piecewise linear expressions.

To graph such functions:

• Identify critical points such as the vertex, which serves as a turning point.
• Plot points on either side of the vertex to see how the function behaves in different intervals.

By understanding coordinate geometry and practicing plotting these points and pieces, we can efficiently graph and analyze functions like \( f(x) = \left| \frac{x}{2} - 7 \right| \), making it easier to determine features like minimum values and intercepts.