Let be a quadratic function.

Suppose we have given the following information:

and .

Which of the following answers are correct (Please note that exactly two are correct). If you cannot find the formula of the function, you still can find the shape of its graph by using a graphical argument.

Question

Let  be a quadratic function.

Suppose we have given the following information:

and .

Which of the following answers are correct (Please note that exactly two are correct).  If you cannot find the formula of the function, you still can find the shape of its graph by using a graphical argument.

BlackTom AI · Accepted Answer

The problem statement as provided is incomplete, which makes it impossible to definitively determine which answer options are correct. Option 1: "" (an empty option). This is not a meaningful statement about a quadratic function, so it cannot be evaluated as true or false. It should be disregarded in a proper multiple-choice analysis. Option 2: "" (another empty option). Like option 1, this does not convey a valid mathematical claim and cannot be judged as correct. Option 3: "" (yet another empty option). Same issue as the previous empty options; no mathematical content to assess. Option 4: "" (still empty). Again, no substantive statement to analyze. Option 5: "is u-shaped." In general, a quadratic function of the form f(x) = ax^2 + bx + c is U-shaped when the leading coefficient a > 0. This is true under that condition, and false if a < 0 (in which case the graph is an inverted U). Without knowing the sign of a, we cannot assert that the graph is U-shaped in all cases. Option 6: "in inverse u-shaped" (inverted U). This describes the case where the leading coefficient a < 0, giving a downward-opening parabola. Again, without knowing a's sign, we cannot guarantee this as the correct characterization. Key issue: For a quadratic, the graph is U-shaped if a > 0 and inverted U-shaped if a < 0. These are mutually exclusive possibilities depending on the sign of the leading coefficient. Therefore, both statements cannot be unconditionally correct at the same time unless additional information about a is provided. Given that the instruction states exactly two options are correct, the provided data appears inconsistent: you cannot simultaneously guarantee both an always-U-shaped and an always-inverted-U-shaped graph without specifying the sign of a. If the data intended to say something like "the graph is either U-shaped or inverted U-shaped depending on a" it would be a different framing, but as written, options 5 and 6 refer to opposite cases and cannot both be universally correct. In summary: - The empty options (1–4) are not meaningful statements to evaluate. - Option 5 is conditionally correct when a > 0, but not universally true for all quadratics. - Option 6 is conditionally correct when a < 0, but not universally true for all quadratics. - The claim that exactly two options are correct cannot be verified from the given information due to the missing value or sign of the leading coefficient a, leading to an inconsistency in the stated "two correct" condition.

23508 Quantitative Methods in Economics and Business - Spring 2025 Quiz #2

Let be a quadratic function. Suppose we have given the following information: and . Which of the following answers are correct (Please note that exactly two are correct). If you cannot find the formula of the function, you still can find the shape of its graph by using a graphical argument.

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