https://file.aiquickdraw.com/mathbot/file/1726014130372ncc7el5e.jpg

Let's analyze the graph and the functions given to solve the required values. Here's the information provided:

### Functions:
- \( n(t) = m(t) + 2 \)
- \( k(t) = m(t + 1.5) \)
- \( w(t) = m(t - 0.5) - 2.5 \)
- \( p(t) = m(t - 1) \)

We are asked to find the following values:

1. \( n(-2) \)
2. \( n(1) = 3 \) (this is already given)
3. \( k(3.5) \)
4. \( w(1.5) = -1.5 \) (this is already given)
5. \( w(-1.5) \)
6. \( p(1.5) \)

To solve these, we need to use the graph of \( m(t) \) to get values at various points and then plug them into the respective function definitions.

I'll now read the graph and calculate each value based on the functions provided.

#### 1. \( n(-2) \)
- From the function \( n(t) = m(t) + 2 \), we need \( m(-2) \).
- Looking at the graph, \( m(-2) = -5 \).
- So, \( n(-2) = m(-2) + 2 = -5 + 2 = -3 \).

#### 2. \( n(1) = 3 \) (already given)

#### 3. \( k(3.5) \)
- \( k(t) = m(t + 1.5) \), so we need \( m(3.5 + 1.5) = m(5) \).
- From the graph, \( m(5) = 2 \).
- Therefore, \( k(3.5) = 2 \).

#### 4. \( w(1.5) = -1.5 \) (already given)

#### 5. \( w(-1.5) \)
- \( w(t) = m(t - 0.5) - 2.5 \), so we need \( m(-1.5 - 0.5) = m(-2) \).
- From the graph, \( m(-2) = -5 \).
- Therefore, \( w(-1.5) = m(-2) - 2.5 = -5 - 2.5 = -7.5 \).

#### 6. \( p(1.5) \)
- \( p(t) = m(t - 1) \), so we need \( m(1.5 - 1) = m(0.5) \).
- From the graph, \( m(0.5) = 0 \).
- Therefore, \( p(1.5) = 0 \).

### Final Answers:
1. \( n(-2) = -3 \)
2. \( n(1) = 3 \)
3. \( k(3.5) = 2 \)
4. \( w(1.5) = -1.5 \)
5. \( w(-1.5) = -7.5 \)
6. \( p(1.5) = 0 \)

Would you like further details on any step?

Here are 5 related questions to extend your understanding:
1. What would the value of \( k(2) \) be?
2. How does the transformation in the function \( w(t) \) affect the graph of \( m(t) \)?
3. What happens to \( p(t) \) as \( t \) increases beyond 5?
4. How would \( n(t) \) be affected if the function were \( n(t) = m(t) - 2 \) instead?
5. Can we predict the behavior of \( w(t) \) for values of \( t > 5 \)?

**Tip:** When solving for transformed functions, always be careful with horizontal shifts like \( t - 1 \), as these affect the graph's timeline and need precise graph reading!

Find the values of the functions n(t), k(t), w(t), and p(t) based on the graph of function m(t).

Explore how to interpret and calculate values from graph transformations for functions n(t), k(t), w(t), and p(t) using the graph of m(t). This tutorial is suitable for high school students studying advanced algebra concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation