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.

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.