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:
- Function a: \( y = \frac{5}{6}x + 2 \)
- Function b: Given discrete points
- \( (0, 2.50) \)
- \( (2, 4.03) \)
- \( (4, 5.56) \)
- \( (6, 7.09) \)
- 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):
-
Function a:
- \( x = 0: \quad y = \frac{5}{6}(0) + 2 = 2 \)
- \( x = 2: \quad y = \frac{5}{6}(2) + 2 = \frac{10}{6} + 2 = 3.67 \)
- \( x = 4: \quad y = \frac{5}{6}(4) + 2 = \frac{20}{6} + 2 = 5.33 \)
- \( x = 6: \quad y = \frac{5}{6}(6) + 2 = \frac{30}{6} + 2 = 7 \)
-
Function b:
- \( x = 0: \quad y = 2.50 \)
- \( x = 2: \quad y = 4.03 \)
- \( x = 4: \quad y = 5.56 \)
- \( x = 6: \quad y = 7.09 \)
-
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.