Let's break this problem into parts:

Step 1: Understand the problem

We need to determine:

1. The initial values (y-intercepts) of functions $G$ and $H$.
2. Whether Kelly's statement that the initial value of $G$ is greater than that of $H$ is correct.

Step 2: Find the y-intercept of $G$

For $G$, the slope is given as $\frac{1}{3}$, and the point $(0, -1)$ is given. Since the point $(0, -1)$ already lies on the line, the initial value (y-intercept) of $G$ is: $b_G = -1$

Step 3: Find the equation and y-intercept of $H$

For $H$, two points are given: $(8, 3)$ and $(10, 4)$. First, calculate the slope: $m_H = \frac{4 - 3}{10 - 8} = \frac{1}{2}$

Now, use the point-slope formula to find the equation of the line. Starting with the point $(8, 3)$: $y - 3 = \frac{1}{2}(x - 8)$ Simplify to get: $y = \frac{1}{2}x - 4 + 3$ $y = \frac{1}{2}x - 1$

From this equation, the y-intercept (initial value) of $H$ is: $b_H = -1$

Step 4: Compare the initial values

The initial value of $G$ is $-1$, and the initial value of $H$ is also $-1$. Therefore, Kelly's statement that the initial value of $G$ is greater than the initial value of $H$ is incorrect.

Final Answer

• Function H has an initial value of: $-1$.
• The initial value of function G is: $-1$.
• Kelly is: Incorrect.

Would you like me to explain any part of this further? 😊

Related Questions

1. How can you find the equation of a line given two points?
2. What does the initial value of a function represent in real-world terms?
3. Why does the point-slope form help when writing equations of a line?
4. How does comparing slopes help us analyze lines geometrically?
5. What are the differences between slope-intercept and point-slope forms?

Tip

Always start by calculating the slope when working with linear functions based on two points!