To find a quadratic function whose range is described as \( y \leq 10.5 \), we note that a quadratic function can be written in the standard form:
\[ f(x) = ax^2 + bx + c \]
For a quadratic function to have a range limited to values less than or equal to a certain maximum, the leading coefficient \( a \) must be negative (indicating that the parabola opens downwards).
The vertex of the quadratic function will provide the maximum value since the vertex is the highest point for a downward-opening parabola. The y-coordinate of the vertex can be found using the formula:
\[ y = -\frac{D}{4a} \]
where \( D \) is the discriminant of the quadratic (given by \( D = b^2 - 4ac \)), or simply using vertex form.
To ensure that the maximum \( y \)-value is \( 10.5 \), we can set the vertex at that value. The vertex form of a quadratic function is:
\[ f(x) = a(x - h)^2 + k \]
where \( (h, k) \) is the vertex. We can choose \( k = 10.5 \) and any \( h \) value, while ensuring \( a < 0 \).
For example, we can choose \( h = 0 \) and \( a = -1 \), yielding:
\[ f(x) = -1(x - 0)^2 + 10.5 \] \[ f(x) = -x^2 + 10.5 \]
This function has its vertex at \( (0, 10.5) \) and opens downward. Thus, the range of the function is \( y \leq 10.5 \).
In conclusion, one example of a quadratic function that has a range described as \( y \leq 10.5 \) is:
\[ f(x) = -x^2 + 10.5 \]
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