To find the equation of the linear function represented by the table in slope-intercept form $y = mx + b$, follow these steps:

Step 1: Find the slope $m$

The slope $m$ is calculated using the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

From the table, pick two points:
$(x_1, y_1) = (0, -8)$ and $(x_2, y_2) = (1, -3)$.

Substitute into the slope formula:

$m = \frac{-3 - (-8)}{1 - 0} = \frac{-3 + 8}{1} = \frac{5}{1} = 5$

Thus, the slope $m = 5$.

Step 2: Write the equation using the slope-intercept form

The slope-intercept form is:

$y = mx + b$

We know $m = 5$, so:

$y = 5x + b$

Step 3: Solve for $b$ (the y-intercept)

To find $b$, substitute one point into the equation $y = 5x + b$.
Using the point $(0, -8)$, where $x = 0$ and $y = -8$:

$-8 = 5(0) + b$

$b = -8$

Final Equation:

The equation of the linear function is:

$y = 5x - 8$

Verification:

To confirm, substitute the other points into the equation $y = 5x - 8$:

• At $x = 1$: $y = 5(1) - 8 = -3$ ✔
• At $x = 2$: $y = 5(2) - 8 = 2$ ✔
• At $x = 3$: $y = 5(3) - 8 = 7$ ✔
• At $x = 4$: $y = 5(4) - 8 = 12$ ✔

The equation is correct.

Do you want me to explain any step further or verify another method? Let me know!

Here are 5 related questions for practice:

1. Find the slope and y-intercept for $y = 3x + 2$.
2. Write the equation of a line with slope $-2$ and y-intercept $5$.
3. Given two points $(2, 1)$ and $(4, 5)$, find the equation of the line.
4. How do you determine if a table represents a linear function?
5. What does the slope tell you about the graph of a line?

Tip: The slope $m$ tells you how much $y$ changes for every unit increase in $x$.