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?
The y-coordinate of the vertex is 9/4.
To find the y-coordinate of the vertex of a quadratic function, we can use the formula ( y = f(x) ), where ( x ) is the average of the x-intercepts. Given the x-intercepts at 3 and -1, the vertex's x-coordinate is ( x = frac{3 + (-1)}{2} = 1 ). Evaluating the function at this x-coordinate yields a y-coordinate of ( f(1) = -2(1-3)(1+1) + 2 = 9/4 ).
This value is incorrect because it does not accurately reflect the calculated y-coordinate of the vertex based on the given x-intercepts and y-intercept. The vertex's y-coordinate must be derived from the function's evaluation at the vertex's x-coordinate, which is not 5/2.
This option is also incorrect, as it fails to match the determined y-coordinate of the vertex. The calculation shows that when substituting the correct x-coordinate into the function, the outcome is not 8/3.
Similarly, this choice does not correspond with the actual vertex's y-coordinate. The method of calculating the vertex's position based on the average of the x-intercepts leads to a different value than 7/3.
While this value is in the right range, it is still incorrect. The vertex's y-coordinate was specifically calculated as 9/4, and thus does not equal 11/4.
The y-coordinate of the vertex of the quadratic function is calculated through evaluating the function at the average of the x-intercepts. In this case, the x-intercepts at 3 and -1 yield a vertex y-coordinate of 9/4, confirming that this choice aligns with the properties of the quadratic function represented in the graph. Other options do not satisfy the vertex's calculation, emphasizing the significance of evaluating at the correct x-coordinate.
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