Question

put these equations in order from least to greatest
function a: y = 5/6x + 2
function b:
x 0 2 4 6
y 2.50 4.03 5.56 7.09

function c:
y = 1.3x + 1

Answer 1

To compare the functions and order them from least to greatest, we can evaluate each function at specific x-values and calculate corresponding y-values.

### Functions:
1. **Function a:** \( y = \frac{5}{6}x + 2 \)
2. **Function b:** Given discrete points
- \( (0, 2.50) \)
- \( (2, 4.03) \)
- \( (4, 5.56) \)
- \( (6, 7.09) \)
3. **Function c:** \( y = 1.3x + 1 \)

### Choosing Specific x-Values
Let's evaluate these functions at \( x = 0 \), \( x = 2 \), \( x = 4 \), and \( x = 6 \) (the same values used in function b):

1. **Function a:**
- \( x = 0: \quad y = \frac{5}{6}(0) + 2 = 2 \)
- \( x = 2: \quad y = \frac{5}{6}(2) + 2 = \frac{10}{6} + 2 = 3.67 \)
- \( x = 4: \quad y = \frac{5}{6}(4) + 2 = \frac{20}{6} + 2 = 5.33 \)
- \( x = 6: \quad y = \frac{5}{6}(6) + 2 = \frac{30}{6} + 2 = 7 \)

2. **Function b:**
- \( x = 0: \quad y = 2.50 \)
- \( x = 2: \quad y = 4.03 \)
- \( x = 4: \quad y = 5.56 \)
- \( x = 6: \quad y = 7.09 \)

3. **Function c:**
- \( x = 0: \quad y = 1.3(0) + 1 = 1 \)
- \( x = 2: \quad y = 1.3(2) + 1 = 2.6 + 1 = 3.6 \)
- \( x = 4: \quad y = 1.3(4) + 1 = 5.2 + 1 = 6.2 \)
- \( x = 6: \quad y = 1.3(6) + 1 = 7.8 + 1 = 8.8 \)

### Summary of Calculated Values
Now, we can summarize the results for the critical points:

- At \( x = 0: \)
- **a:** 2, **b:** 2.50, **c:** 1 → Order: \( c < a < b \)

- At \( x = 2: \)
- **a:** 3.67, **b:** 4.03, **c:** 3.6 → Order: \( c < a < b \)

- At \( x = 4: \)
- **a:** 5.33, **b:** 5.56, **c:** 6.2 → Order: \( a < b < c \)

- At \( x = 6: \)
- **a:** 7, **b:** 7.09, **c:** 8.8 → Order: \( a < b < c \)

### Overall Order
When considering the results at all x-values, we can see a consistent pattern:

- For \( x = 0: \) \( c < a < b \)
- For \( x = 2: \) \( c < a < b \)
- For \( x = 4: \) \( a < b < c \)
- For \( x = 6: \) \( a < b < c \)

Thus, we can conclude that the overall order from least to greatest is approximately:

**c < a < b**

This consistent relationship suggests that function c has the lowest values, followed by function a, and then function b has the highest values.