Question

Graph the following groups of equations in each exercise, and determine how the numbers in the second and third equations affect the graph as compared to the first equation.
Y= x+5
y=2x+5
y=-2x+5

Answer 1

To graph the equations and analyze how the coefficients impact the graphs, let's start with each equation one at a time.

1. **Graph \( y = x + 5 \)**:
- This is a linear equation with a slope of 1 and a y-intercept of 5.
- It will cross the y-axis at (0, 5) and increases diagonally, moving upwards with a 45-degree angle in relation to the x-axis.

2. **Graph \( y = 2x + 5 \)**:
- This is also a linear equation but with a slope of 2 and a y-intercept of 5.
- The graph will cross the y-axis at (0, 5) as well, but it will be steeper than the first equation because the slope is greater than 1. For every increase of 1 in \(x\), \(y\) increases by 2.

3. **Graph \( y = -2x + 5 \)**:
- This equation has a slope of -2 and a y-intercept of 5.
- It also crosses the y-axis at (0, 5), but will decrease steeply as \(x\) increases. For every increase of 1 in \(x\), \(y\) decreases by 2. Thus, the line will slope downward at a steep angle.

### Graphing Summary:
- The first equation \( y = x + 5 \) has a moderate positive slope; the line rises gently and crosses the y-axis at 5.
- The second equation \( y = 2x + 5 \) is steeper, rising more sharply due to the larger slope (2). It's parallel to the first line but at a greater angle.
- The third equation \( y = -2x + 5 \) slopes downward due to the negative slope. It starts at the same y-intercept (5) but moves downwards as \(x\) increases.

### Effect of Numbers on the Graph:
- **Slope (Coefficient of \(x\))**:
- The slope determines the steepness of the graph. A larger positive slope results in a steeper line moving upwards, while a larger negative slope results in a steeper downward line.

- **Y-intercept (Constant term)**:
- The y-intercept determines where the line crosses the y-axis. In all three equations, the y-intercept is consistent at 5, resulting in all lines starting at the same point vertically.

### Visual Representation:
To visualize this:
- Use graphing paper or a graphing tool to plot:
- \( (0, 5) \) as the y-intercept for all three equations.
- For \( y = x + 5 \) where slope = 1, plot points like (1, 6) and (2, 7).
- For \( y = 2x + 5 \) where slope = 2, plot points (1, 7) and (2, 9).
- For \( y = -2x + 5 \) where slope = -2, plot points (1, 3) and (2, 1).

This should help visually distinguish how these slopes and the corresponding y-intercept affect the graph's behavior overall.