Let's work through the problems step-by-step.

1. What is the slope of the linear function?

• Problem 9: The equation is $y = \frac{1}{4}x - 3$. In slope-intercept form $y = mx + b$, the coefficient of $x$ (which is $\frac{1}{4}$) is the slope.
  Slope $m = \frac{1}{4}$.

• Problem 10: The equation is $y = -3x + 6$. Again, the coefficient of $x$ (which is $-3$) is the slope.
  Slope $m = -3$.

2. Are the following linear functions parallel, perpendicular, or neither?

• Problem 11: The equations are:

  1. $y = 3x - 4$
  2. $y = -\frac{1}{3}x + 6$

  To determine if the lines are parallel, perpendicular, or neither, we compare their slopes.

  • The slope of the first line is $m_1 = 3$.
  • The slope of the second line is $m_2 = -\frac{1}{3}$.

  If the product of the slopes is $-1$, the lines are perpendicular. Let's check: $3 \times \left( -\frac{1}{3} \right) = -1$ The product is $-1$, so these lines are perpendicular.

• Problem 12: The equations are:

  1. $y = \frac{2}{3}x + 6$
  2. $y = \frac{2}{3}x + 4$

  Both lines have the same slope $m_1 = m_2 = \frac{2}{3}$. Since their slopes are equal, the lines are parallel.

Let me know if you want any further details on the explanation!

Here are 5 related questions to practice further:

1. What is the slope of the line $y = -5x + 7$?
2. Are the lines $y = 4x - 3$ and $y = -\frac{1}{4}x + 5$ parallel, perpendicular, or neither?
3. Write the equation of a line parallel to $y = 2x + 1$ passing through the point (3, 4).
4. What is the slope of a line perpendicular to $y = \frac{1}{5}x - 2$?
5. How do you find the slope if the equation is in standard form $Ax + By = C$?

Tip: The slope of any line can be easily read from the coefficient of $x$ in the slope-intercept form $y = mx + b$.