To determine which graph represents the given linear function, we first need to find the slope and y-intercept of the function using the provided ordered pairs. Let's analyze the points:
The ordered pairs are:
- (-1, 3.5)
- (0, 4)
- (1, 4.5)
- (2, 5)
From the points above, we can calculate the slope \( m \) using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using two points, for example (0, 4) and (1, 4.5):
\[ m = \frac{4.5 - 4}{1 - 0} = \frac{0.5}{1} = 0.5 \]
Now we need to find the y-intercept \( b \). Since the point (0, 4) indicates that when \( x = 0 \), \( y = 4 \), we can directly state that:
\[ b = 4 \]
Thus, the equation of the line representing this linear function is:
\[ y = 0.5x + 4 \]
Next, we can verify a few points to ensure the accuracy of this equation:
-
For \( x = -1 \): \[ y = 0.5(-1) + 4 = -0.5 + 4 = 3.5 \] (matches the point)
-
For \( x = 1 \): \[ y = 0.5(1) + 4 = 0.5 + 4 = 4.5 \] (matches the point)
-
For \( x = 2 \): \[ y = 0.5(2) + 4 = 1 + 4 = 5 \] (matches the point)
The function is linear, with a positive slope of 0.5 and a y-intercept of 4, meaning the line will slope upwards.
Now let's analyze the options given for the graph:
-
Graph A: It has an upward slope and roughly agrees with the calculated values. It passes through (3, 5) which is not in our points but could be correct.
-
Graph B: This does not match as the slope appears steeper than 0.5.
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Graph C: This has an upward slope but the points it claims to pass (1, 0) and (2, 5) do not align with our function.
-
Graph D: This has a downward slope, which is incorrect based on our function.
From this analysis, the best match seems to be Graph A, which features an upward slope which is consistent with the linear function we've derived.