To find the range of the function $f(x) = -2x^2 + cx$, where $c$ is a real number, we need to analyze the behavior of this quadratic function. The function is a parabola, and its range will depend on the value of the coefficient of the quadratic term and the linear term.

Step-by-Step Solution:

1. Identify the general form:
   The given function is $f(x) = -2x^2 + cx$. This is a quadratic function in the form $f(x) = ax^2 + bx + c$, where $a = -2$ and $b = c$.

2. Vertex form:
   Since the coefficient of $x^2$ is negative ($-2$), the parabola opens downwards. The range of a downward-opening parabola is determined by its vertex.

3. Find the vertex:
   For a quadratic function $f(x) = ax^2 + bx + c$, the x-coordinate of the vertex is given by: $x = \frac{-b}{2a}$ In our case: $x = \frac{-c}{2(-2)} = \frac{c}{4}$

4. Find the value of $f(x)$ at the vertex:
   Now substitute $x = \frac{c}{4}$ into the original function to find the corresponding $y$-value (which is the maximum value of $f(x)$ because the parabola opens downwards). $f\left( \frac{c}{4} \right) = -2 \left( \frac{c}{4} \right)^2 + c \left( \frac{c}{4} \right)$ Simplifying: $f\left( \frac{c}{4} \right) = -2 \cdot \frac{c^2}{16} + \frac{c^2}{4}$ $f\left( \frac{c}{4} \right) = -\frac{c^2}{8} + \frac{c^2}{4}$ $f\left( \frac{c}{4} \right) = \frac{c^2}{8}$

5. Determine the range:
   Since the parabola opens downwards, the maximum value of $f(x)$ occurs at the vertex, and the function decreases as $x$ moves away from $\frac{c}{4}$. Thus, the range of the function is: $(-\infty, \frac{c^2}{8}]$ Therefore, the range of $f(x)$ is all values less than or equal to $\frac{c^2}{8}$.

Final Answer:

The range of the function $f(x) = -2x^2 + cx$ is $(-\infty, \frac{c^2}{8}]$.

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Would you like more details on how the vertex was calculated, or do you have any other questions? Here are some related questions you might find interesting:

1. How does the value of $c$ affect the width of the parabola?
2. What happens if $c = 0$ in this function?
3. How can we use the discriminant to analyze the roots of this quadratic equation?
4. What is the vertex of the function when $c = 4$?
5. How can you find the axis of symmetry for this quadratic function?

Tip: The axis of symmetry for a quadratic function $ax^2 + bx + c$ is given by $x = \frac{-b}{2a}$, which is also the x-coordinate of the vertex.