https://file.aiquickdraw.com/mathbot/file/17249885758075jmiakuk.jpg

Jade: f(t)=.....(mi)
Jari: g(t)=.....(mi)
find f(t) and g(t) based on the graph

Let's find the equations \( f(t) \) for Jade and \( g(t) \) for Jari based on the graph provided.

### 1. **Jade's Function \( f(t) \):**

Jade's graph is a straight line, so the equation will be in the form:
\[
f(t) = mt + b
\]
where:
- \( m \) is the slope of the line (rate of change of distance with respect to time).
- \( b \) is the y-intercept (the distance at \( t = 0 \)).

**Slope Calculation for Jade:**
Using the points \((0, 10)\) and \((6, 16)\),
\[
m_\text{Jade} = \frac{16 - 10}{6 - 0} = \frac{6}{6} = 1 \text{ mile per minute}
\]
So, \( m = 1 \).

**Y-intercept \( b \):**
From the graph, the line intersects the y-axis at \( y = 10 \) when \( t = 0 \).

Thus, the equation for Jade is:
\[
f(t) = 1t + 10 = t + 10 \text{ (in miles)}
\]

### 2. **Jari's Function \( g(t) \):**

Similarly, for Jari, the equation is:
\[
g(t) = mt + b
\]

**Slope Calculation for Jari:**
Using the points \((0, 5)\) and \((6, 9)\),
\[
m_\text{Jari} = \frac{9 - 5}{6 - 0} = \frac{4}{6} = \frac{2}{3} \text{ miles per minute}
\]
So, \( m = \frac{2}{3} \).

**Y-intercept \( b \):**
From the graph, the line intersects the y-axis at \( y = 5 \) when \( t = 0 \).

Thus, the equation for Jari is:
\[
g(t) = \frac{2}{3}t + 5 \text{ (in miles)}
\]

### Final Equations:

- **Jade**: \( f(t) = t + 10 \)
- **Jari**: \( g(t) = \frac{2}{3}t + 5 \)

Would you like any further details or have more questions?

Here are 5 related questions to explore further:

1. How far would Jade have traveled after 12 minutes?
2. At what time will Jade and Jari have traveled the same distance?
3. If Jade continues at the same speed, how far will she be at 20 minutes?
4. How much faster is Jade compared to Jari in terms of miles per minute?
5. What would happen to the graph if Jari increased his speed by 0.5 miles per minute?

**Tip**: Knowing how to find the slope and intercepts of a line is crucial for writing linear equations.

Learn how to find the linear equations f(t) and g(t) from given graphs. This problem involves interpreting graphs, calculating slopes, and determining y-intercepts to write equations in the form y = mx + b. Suitable for students in Grades 7-9.

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