Certainly! Let's break down the linear function [tex]\( y = 2x - 5 \)[/tex]:
### Step-by-Step Description:
1. Identifying the Slope and Y-Intercept:
- The given equation is 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.
- For this particular function, [tex]\( m = 2 \)[/tex] and [tex]\( b = -5 \)[/tex].
2. Y-Intercept ([tex]\( b \)[/tex]):
- The y-intercept is the point where the line crosses the y-axis. This happens when [tex]\( x = 0 \)[/tex].
- So, if you replace [tex]\( x \)[/tex] with 0 in the equation, [tex]\( y = 2(0) - 5 = -5 \)[/tex]. Hence, the line crosses the y-axis at the point (0, -5).
3. Slope ([tex]\( m \)[/tex]):
- The slope of the line is [tex]\( 2 \)[/tex]. The slope represents the rate of change of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex].
- This means that for every 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 2 units. Conversely, for every 1 unit decrease in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 2 units.
4. Verbal Description:
- This linear function describes a line that crosses the y-axis at -5.
- As you move along the x-axis, for every 1 unit that you move to the right, the value of [tex]\( y \)[/tex] increases by 2 units.
- Conversely, for every 1 unit you move to the left on the x-axis, the value of [tex]\( y \)[/tex] decreases by 2 units.
### Conclusion:
- The function [tex]\( y = 2x - 5 \)[/tex] represents a straight line.
- It has a y-intercept at -5, meaning it crosses the y-axis at the point [tex]\( (0, -5) \)[/tex].
- The slope of the line is 2, meaning the line rises 2 units vertically for every 1 unit it moves horizontally.
This comprehensive description should give you a clear understanding of how the function behaves graphically and algebraically.
