Question 1: Equation of the line in slope-intercept form

The slope-intercept form of a line is:

$y = mx + b$

Where:

• $m$ is the slope of the line
• $b$ is the y-intercept (the point where the line crosses the y-axis)

From the graph:

1. The line crosses the $y$-axis at $b = 2$.
2. To find the slope $m$, use two points on the line. Two clear points are:
   • $(0, 2)$ and $(2, 8)$
   • Slope formula:
     $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 2}{2 - 0} = \frac{6}{2} = 3$

Thus, the equation of the line is:

$y = 3x + 2$

Question 2: Classifying linear and nonlinear functions

To determine whether a function is linear or nonlinear:

• Linear functions can be written in the form $y = mx + b$, where $m$ and $b$ are constants.
• Nonlinear functions include powers, roots, or other non-linear terms.

From the given options:

1. Linear functions:

   • $y = x + \frac{25}{5}$ → Simplifies to $y = x + 5$ (linear form).
   • $y = \frac{7}{2}x - 19$ → Already in slope-intercept form (linear).
   • $13y = \frac{1}{3}x + 5$ → Solve for $y$:
     $y = \frac{1}{39}x + \frac{5}{13}$
     This is a linear function.

2. Nonlinear functions:

   • $3y = x^2$ → Contains $x^2$ (nonlinear).
   • $y^2 = x$ → Contains $y^2$ (nonlinear).

Final Classification:

Linear:

• $y = x + \frac{25}{5}$
• $y = \frac{7}{2}x - 19$
• $13y = \frac{1}{3}x + 5$

Nonlinear:

• $3y = x^2$
• $y^2 = x$

Do you want a detailed explanation of slope-intercept form or the classification process? Let me know!

Related Questions:

1. How do you calculate the slope of a line given two points?
2. What makes a function nonlinear?
3. How can you rewrite equations to check for slope-intercept form?
4. What is the difference between $x^2$ and linear terms?
5. How can you graph a line using slope and y-intercept?

Tip: Always check the powers of $x$ and $y$ in an equation to determine if a function is linear or nonlinear.