To determine the slope and [tex]\( y \)[/tex]-intercept of the linear function passing through the points [tex]\((0, -3)\)[/tex] and [tex]\((6, 15)\)[/tex], we will follow a step-by-step approach.
### Step 1: Determine the slope (m)
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the given points [tex]\((x_1, y_1) = (0, -3)\)[/tex] and [tex]\((x_2, y_2) = (6, 15)\)[/tex] into the formula:
[tex]\[ m = \frac{15 - (-3)}{6 - 0} = \frac{15 + 3}{6} = \frac{18}{6} = 3 \][/tex]
So, the slope [tex]\( m \)[/tex] is 3.
### Step 2: Determine the [tex]\( y \)[/tex]-intercept
The [tex]\( y \)[/tex]-intercept of a linear function is the point where the line crosses the [tex]\( y \)[/tex]-axis. This happens when [tex]\( x = 0 \)[/tex].
From the given points, we already know that one of the points is [tex]\((0, -3)\)[/tex]. This means that the [tex]\( y \)[/tex]-intercept is [tex]\((0, -3)\)[/tex].
### Step 3: Verify the options
Given the slope [tex]\( m = 3 \)[/tex] and the [tex]\( y \)[/tex]-intercept as [tex]\((0, -3)\)[/tex], we can now verify the correct option:
- Option A states the slope is [tex]\(\frac{1}{3}\)[/tex] and the [tex]\( y \)[/tex]-intercept is [tex]\((-3,0)\)[/tex]. This is incorrect.
- Option B states the slope is [tex]\(\frac{1}{3}\)[/tex] and the [tex]\( y \)[/tex]-intercept is [tex]\((0,-3)\)[/tex]. This is incorrect.
- Option C states the slope is 3 and the [tex]\( y \)[/tex]-intercept is [tex]\((0,-3)\)[/tex]. This is correct.
- Option D states the slope is -3 and the [tex]\( y \)[/tex]-intercept is [tex]\((0,-3)\)[/tex]. This is incorrect.
Therefore, the correct answer is:
C. The slope is 3. The [tex]\( y \)[/tex]-intercept is [tex]\((0, -3)\)[/tex].
