https://file.aiquickdraw.com/mathbot/file/1726074592460hm44zvd1.jpg

Let's break down and solve the questions one by one based on the image you uploaded:

### 1. Graph-related questions:
- **Problem 10: \( f(6) \)**
  To find \( f(6) \), locate \( x = 6 \) on the graph and check the corresponding \( y \)-value. The graph shows that when \( x = 6 \), the \( y \)-value is \( 8 \).
  \[
  f(6) = 8
  \]

- **Problem 11: \( f(3) \)**
  For \( f(3) \), check the point where \( x = 3 \). The graph indicates that when \( x = 3 \), \( y = -5 \).
  \[
  f(3) = -5
  \]

### 2. Equation-related questions:
- **Problem 12: If \( f(x) = -3 \), then \( x = \) ?**
  Check on the graph where \( y = -3 \). There are two points where this happens: \( x = 1 \) and \( x = 5 \).
  \[
  f(x) = -3 \quad \text{when} \quad x = 1 \quad \text{or} \quad x = 5
  \]

- **Problem 13: If \( f(x) = 7 \), then \( x = \) ?**
  Find where \( y = 7 \) on the graph. This occurs at \( x = 7 \).
  \[
  f(x) = 7 \quad \text{when} \quad x = 7
  \]

### 3. Story-based question (14):
- Hope is tracking her family's progress using the function \( d(t) = 78t \), where \( t \) is time.
  
  a. **What will Hope be finding if she writes \( d(4) = 312 \)?**
     This means that after 4 hours, the family has traveled 312 miles, as \( d(t) \) represents the distance traveled in miles after \( t \) hours.

  b. **What would the expression \( d(t) = 450 \) mean in the context of the story?**
     The expression \( d(t) = 450 \) means that the family has traveled 450 miles, and the question is asking for how many hours it took to reach this distance.

     To solve for \( t \):
     \[
     450 = 78t \implies t = \frac{450}{78} \approx 5.77 \, \text{hours}.
     \]
     So, it would take approximately 5.77 hours for the family to travel 450 miles.

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Would you like any further clarification or details on any part of this? Let me know!

Here are 5 follow-up questions for practice:
1. What is the value of \( f(0) \) based on the graph?
2. Can you solve for \( x \) if \( f(x) = -10 \) using the graph?
3. If \( f(2) = -10 \), how does this relate to the slope between points on the graph?
4. What does the slope of the graph between \( x = 3 \) and \( x = 6 \) tell you about the function’s behavior?
5. How would the equation for distance change if the family traveled at 90 miles per hour instead of 78?

**Tip**: When reading graphs, always ensure you align the \( x \)- and \( y \)-coordinates precisely to avoid mistakes in interpreting the function values.

The user has submitted math problems involving a graph and functions, including evaluating function values from a graph and interpreting a real-world scenario using a function for distance.

Learn how to evaluate function values using a graph and solve a real-world distance-time problem with the linear equation d(t) = 78t. Suitable for Grades 8-10.

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