Problem 66 - Algebra and Trigonometry

Question

A verbal description of a function is given. Find (a) algebraic, (b) numerical, and (c) graphical representations for the function. To evaluate \(g(x),\) subtract 4 from the input and multiply the result by \(\frac{3}{4} .\)

Step-by-Step Solution

Answer

The function \( g(x) \) is represented as: (a) algebraic: \( g(x) = \frac{3}{4}(x-4) \), (b) numerical table: points (0, -3), (4, 0), (8, 3), (c) graph: line through these points.

Step 1: Understand the Problem

The function description asks us to find three representations—algebraic, numerical, and graphical—based on given operations to evaluate \( g(x) \). Specifically, subtract 4 from the input \( x \), then multiply the result by \( \frac{3}{4} \).

Step 2: Algebraic Representation

To express the function algebraically, follow the operations described: subtract 4 from \( x \) and multiply by \( \frac{3}{4} \). The algebraic expression becomes \( g(x) = \frac{3}{4}(x - 4) \).

Step 3: Evaluate the Function for Numerical Representation

Choose different values of \( x \) to evaluate the algebraic expression and construct a table of values.- If \( x = 0 \), then \( g(0) = \frac{3}{4}(0-4) = -3 \).- If \( x = 4 \), then \( g(4) = \frac{3}{4}(4-4) = 0 \).- If \( x = 8 \), then \( g(8) = \frac{3}{4}(8-4) = 3 \). This gives points (0, -3), (4, 0), (8, 3) for the numerical table.

Step 4: Graphical Representation

Use the points obtained in the numerical representation to graph the function. Plot the points (0, -3), (4, 0), and (8, 3) on a coordinate plane. Draw a straight line through these points because the function is linear, represented by the equation \( g(x) = \frac{3}{4}(x - 4) \).

Step 5: Verify the Solution

Check if all three representations are aligned and respect the operation described. The algebraic equation \( g(x) = \frac{3}{4}(x-4) \), the numerical values for selected \( x \), and the graph should consistently reflect subtracting 4 from \( x \) and then multiplying by \( \frac{3}{4} \).

Key Concepts

Algebraic Representation

Algebraic representation is a way to express functions using mathematical symbols and operations. In this exercise, we were given a verbal description: to evaluate the function \( g(x) \), subtract 4 from the input, and then multiply the result by \( \frac{3}{4} \). This can be translated into a mathematical equation. To find the algebraic representation, follow these steps:

• Identify the operations described: subtract and then multiply.
• Substitute these operations into a formula: first subtract 4 from \( x \), then multiply the entire result by \( \frac{3}{4} \).
• The resulting equation becomes \( g(x) = \frac{3}{4}(x - 4) \).

This translates the verbal steps into a clear, symbolic form that can be used for computation.

Numerical Representation

Numerical representation involves evaluating the algebraic expression at specific values of \( x \) to create a table of values. This approach helps visualize how the output of the function corresponds to different inputs.Here's how you can set up a numerical representation:

• Choose several values for \( x \), like 0, 4, and 8.
• For each \( x \), plug it into the function \( g(x) = \frac{3}{4}(x - 4) \) and calculate the result.
• Record these outputs to form pairs: e.g., \( x = 0 \) gives \( g(0) = -3 \), forming the pair (0, -3).

With these points, such as (0, -3), (4, 0), and (8, 3), we can establish a numerical understanding of how the function behaves across these values.

Graphical Representation

Graphical representation turns the insights from numerical data into a visual format, by plotting the function on a coordinate grid. This provides an intuitive way to see the behavior and shape of the function.To graph the function:

• Take the pairs from the numerical representation (e.g., (0, -3), (4, 0), (8, 3)).
• Plot each point on the coordinate plane by marking the x-value along the x-axis and corresponding y-value along the y-axis.
• Draw a line through these points. Since we have a linear equation, \( g(x) = \frac{3}{4}(x - 4) \), the graph will be a straight line.

The graph effectively visualizes the linear relationship described by the function, showing a consistent pattern as x increases.

Linear Functions

Linear functions represent the simplest class of functions, characterized by a constant rate of change, or slope. They can be recognized by their straight-line graphs and general form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.In our exercise, the function \( g(x) = \frac{3}{4}(x - 4) \) fits the profile of a linear function:

• The expression can be rewritten in the form \( y = mx + b \), revealing the slope \( m = \frac{3}{4} \) and shifting the function right by 4 units.
• The slope tells us for every unit increase in \( x \), \( g(x) \) increases by \( \frac{3}{4} \).
• By observing the graphical representation, this straight line confirms the linearity.

Linear functions like this one form a foundational concept in algebra, illustrating clear, predictable relationships between variables.