7. The graph of a quadratic function is shown in the xy -plane above. The x-intercepts are 3 and -1, and the y-intercept is 2. What is the y-coordinate of the vertex?
Answer: E
The y-coordinate of the vertex is 9/4.
To find the y-coordinate of the vertex of a quadratic function given its x-intercepts, we can use the formula for the vertex y-coordinate, which is calculated as \( f\left(\frac{x_1 + x_2}{2}\right) \), where \( x_1 \) and \( x_2 \) are the x-intercepts. Here, the intercepts are 3 and -1, which gives a midpoint of 1, and evaluating the function at this point yields a vertex y-coordinate of 9/4.
A) 5/2
This option is incorrect because 5/2 does not correspond to the calculated vertex y-coordinate. To confirm, substituting the vertex calculation shows that the value is lower than what we derived.
B) 8/3
8/3 is also incorrect as it does not reflect the vertex y-coordinate derived from the function's properties. The calculations for the vertex yield a higher value than this option.
C) 7/3
While 7/3 is a plausible value, it is not the correct y-coordinate of the vertex. Evaluating the function at the midpoint of the x-intercepts indicates a value that exceeds 7/3.
D) 11/4
This option is incorrect because 11/4 exceeds the calculated y-coordinate of the vertex. The vertex calculation reveals that the correct value is less than this option.
E) 9/4
This is the correct answer, derived from calculating the vertex's y-coordinate using the average of the x-intercepts. The function's evaluation at the midpoint shows that the vertex indeed has a y-coordinate of 9/4.
Conclusion
The correct choice is 9/4 because it accurately represents the y-coordinate of the vertex calculated from the midpoint of the x-intercepts. All other options fail to reflect the vertex's position based on the properties of the quadratic function given the specified x-intercepts.