Final answer:
The correct quadratic function that fits the points (-1,-4), (0,0), and (2,-10) is f(x) = -3x² + x, which corresponds to option (c).
Step-by-step explanation:
The student has asked to identify the quadratic function that contains the points (-1,-4), (0,0), and (2,-10). These points will fit into a quadratic equation of the form f(x) = ax² + bx + c. To determine the correct function, we can substitute the given points into the possible function options and check which one satisfies all three points. However, since one of the points is the y-intercept (0,0), we can immediately rule out options with a non-zero constant term (since c represents the y-intercept of the quadratic function). Thus, we only need to plug the other two points into the remaining equations.
Let's substitute the point (-1, -4) into the possible functions:
f(x) = 3x² - x
For the first option, f(-1) = 3(-1)² - (-1) = 3 + 1 = 4, which does not equal -4. For the second option, f(-1) = -3(-1)² - (-1) = -3 + 1 = -2, which also does not equal -4. Therefore, neither of these functions is the right answer.
However, the task can be done in a simpler way. Since (0, 0) is one of the points, we know c = 0 in the quadratic function f(x) = ax² + bx + c. Therefore, our equation will have the form f(x) = ax² + bx. Now, let's consider the point (2, -10). It will help us identify which one from the given options (all having c = 0) is the correct function. Substituting 2 for x, we should get -10 for f(x).
Again, we substitute the point (2, -10) into the possible functions:
f(x) = 3x² - x: f(2) = 3(2)² - 2 = 12 - 2 = 10, which is not equal to -10.
- f(x) = 3x² + x: f(2) = 3(2)² + 2 = 12 + 2 = 14, also incorrect.
- f(x) = -3x² + x: f(2) = -3(2)² + 2 = -12 + 2 = -10, which matches the point (2, -10).
- f(x) = -3x² - x: This option is not correct as we already found the matching function.
Therefore, the correct quadratic function is f(x) = -3x² + x, matching option (c).