Problem 20 - Algebra 2

Question

For Exercises 20 and \(21,\) use the following information. Carlos is walking on a moving walkway. His speed is given by the function \(C(x)=3 x^{2}+3 x-4,\) and the speed of the walkway is \(W(x)=x^{2}-4 x+7\) What is his total speed as he walks along the moving walkway?

Step-by-Step Solution

Answer

The total speed function is \( T(x) = 4x^2 - x + 3 \).

Step 1: Identify Functions

Carlos's speed is represented by the function \( C(x) = 3x^2 + 3x - 4 \). The speed of the walkway is given by \( W(x) = x^2 - 4x + 7 \). We need to determine the total speed as Carlos walks on the moving walkway.

Step 2: Add the Speeds

To find the total speed, we add the two functions together: \[ C(x) + W(x) = (3x^2 + 3x - 4) + (x^2 - 4x + 7) \]

Step 3: Combine Like Terms

Combine the like terms in the expression: - Combine the \( x^2 \) terms: \( 3x^2 + x^2 = 4x^2 \).- Combine the \( x \) terms: \( 3x - 4x = -x \).- Combine the constant terms: \( -4 + 7 = 3 \).

Step 4: Write the Total Speed Function

The total speed as Carlos walks on the moving walkway is given by the function: \[ T(x) = 4x^2 - x + 3 \]

Key Concepts

Function Addition

When dealing with algebraic functions, one powerful operation you can perform is **function addition**. This means you're literally adding two functions together to create a new one. In the exercise, Carlos's walking speed and the moving walkway's speed each have their own separate functions. But when you want to find out his overall speed on the walkway, you need to add these functions together.
The functions in this case are:

• Carlos's speed: \( C(x) = 3x^2 + 3x - 4 \)
• Walkway's speed: \( W(x) = x^2 - 4x + 7 \)

To find out his total speed, you perform addition:

• Combine these functions: \( T(x) = C(x) + W(x) \)
• This becomes \( T(x) = (3x^2 + 3x - 4) + (x^2 - 4x + 7) \)

By combining like terms from both functions, you effectively create a new function that represents the total speed Carlos has while walking on the moving walkway.

Polynomial Functions

**Polynomial Functions** are mathematical expressions that have variables raised to different powers, and they can include different terms like constants, variables, and coefficients. They are both versatile and common in algebra.
The functions given in the problem are examples of polynomials. Carlos's speed function is a quadratic polynomial:

• \( C(x) = 3x^2 + 3x - 4 \)

The walkway's speed function is also quadratic:

• \( W(x) = x^2 - 4x + 7 \)

So, both of these functions have terms with the variable \( x \) raised to the power of \( 2 \), which is typical for quadratic functions. Polynomial functions are used to model a wide variety of real-world situations because they are flexible and can fit many different kinds of data. In this context, they help describe speeds as functions of time or other variables.

Speed Calculation

Calculating speed when dealing with different contributions, such as a person's speed and another moving object's speed, is important and practical. In this exercise, you calculate the total speed by combining the individual speeds.
When Carlos is on the moving walkway, his effective speed is the sum of his speed \( C(x) \) and the walkway's speed \( W(x) \). Calculating it involves:

• Adding the function expressions: \( T(x) = C(x) + W(x) \)

Once you have that expression, the next task is to **combine like terms**. This process simplifies the equation, providing a clearer view of how various components contribute to the overall speed.

• For example: In \( (3x^2 + 3x - 4) + (x^2 - 4x + 7) \), you combine to get \( 4x^2 - x + 3 \)

The final expression \( T(x) = 4x^2 - x + 3 \) gives a comprehensive understanding of how Carlos’s speed changes with different values, revealing the interaction between personal movement and external conditions like the moving walkway.