Let’s solve the problem step by step:

Step 1: Determine the y-intercept of the linear function in the table

A linear function is written in the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Find the slope ($m$) of the function:

Using the points from the table:

• From $(2, 10)$ to $(4, 18)$, the slope is: $m = \frac{\Delta y}{\Delta x} = \frac{18 - 10}{4 - 2} = \frac{8}{2} = 4.$

So, $m = 4$.

Find the y-intercept ($b$):

Using the equation $f(x) = mx + b$ and substituting one point, e.g., $(2, 10)$: $10 = 4(2) + b \quad \Rightarrow \quad 10 = 8 + b \quad \Rightarrow \quad b = 2.$

Thus, the linear function is $f(x) = 4x + 2$, and the y-intercept is 2.

Step 2: Compare the y-intercepts of the given functions

Function A: $g(x) = 2x + 3$

The y-intercept of $g(x)$ is 3, which is greater than $2$. Answer: Yes.

Function B: $g(x) = 1 + 5x$

Rewriting as $g(x) = 5x + 1$, the y-intercept is 1, which is less than $2$. Answer: No.

Function C: $g(x) = -4(x - 1)$

Rewriting: $g(x) = -4x + 4.$ The y-intercept is 4, which is greater than $2$. Answer: Yes.

Final Answers:

• A: Yes
• B: No
• C: Yes

Do you want a detailed explanation of any part of the solution or clarification on any step?

Related Questions:

1. How do you calculate the slope of a line from a table?
2. How do you rewrite equations to identify the y-intercept?
3. What are the properties of linear functions in slope-intercept form?
4. Can the y-intercept be negative, and how does it affect the graph?
5. What is the difference between slope and y-intercept in real-world applications?

Tip:

Always write a function in $y = mx + b$ form to clearly identify the slope ($m$) and the y-intercept ($b$).