To determine which graph represents the equation [tex]\( y - 4 = -3(x + 5) \)[/tex], let's rewrite the equation in the slope-intercept form, which is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Here are the steps to rewrite the equation:
1. Distribute the [tex]\(-3\)[/tex] on the right-hand side:
[tex]\[
y - 4 = -3(x + 5)
\][/tex]
[tex]\[
y - 4 = -3x - 15
\][/tex]
2. Add 4 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y - 4 + 4 = -3x - 15 + 4
\][/tex]
[tex]\[
y = -3x - 11
\][/tex]
Now, the equation is in slope-intercept form ([tex]\( y = mx + b \)[/tex]), where the slope [tex]\( m \)[/tex] is [tex]\(-3\)[/tex], and the y-intercept [tex]\( b \)[/tex] is [tex]\(-11\)[/tex].
### Key Features of the Graph:
- Slope ([tex]\( m \)[/tex]): [tex]\(-3\)[/tex]
- This means the line decreases steeply, falling 3 units in [tex]\( y \)[/tex] for every 1 unit increase in [tex]\( x \)[/tex].
- Y-Intercept ([tex]\( b \)[/tex]): [tex]\(-11\)[/tex]
- This is the point where the line crosses the y-axis.
### Drawing the Graph:
1. Y-Intercept: Start by plotting the y-intercept. The point [tex]\((0, -11)\)[/tex] is on the y-axis.
2. Slope: From the y-intercept, use the slope to find another point.
- Since the slope is [tex]\(-3\)[/tex], move 1 unit to the right and 3 units down. This gives you the point [tex]\((1, -14)\)[/tex].
3. Line: Draw the line passing through these points. The line should be straight and extend through both points plotted (and beyond).
Thus, the graph of the equation [tex]\( y - 4 = -3(x + 5) \)[/tex] is a straight line with a steep negative slope passing through the points [tex]\((0, -11)\)[/tex] (y-intercept) and [tex]\((1, -14)\)[/tex]. It continues infinitely in both directions.
### Example Points on the Line:
Besides the two points used:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[
y = -3(0) - 11 = -11
\][/tex]
- When [tex]\( x = 1 \)[/tex]:
[tex]\[
y = -3(1) - 11 = -14
\][/tex]
These points support our graph's slope and y-intercept.
### Recap:
- The equation [tex]\( y - 4 = -3(x + 5) \)[/tex] describes a line with:
- Slope: [tex]\(-3\)[/tex]
- Y-intercept: [tex]\(-11\)[/tex]
- The line decreases steeply as you move from left to right.
- Plotting points, such as [tex]\((0, -11)\)[/tex] and [tex]\((1, -14)\)[/tex], helps illustrate the line visually.
