Question

Find the direction in which the parabola \( y = ax^2 + bx - 2 \) is facing.
1. a = b
2. a < 0

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question ask
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Hint:

For quadratic functions of the form \(y = ax^2 + bx + c\), remember: the 'a' value controls the vertical stretching/compression and the direction of opening. The 'b' value influences the position of the vertex, and 'c' is the y-intercept. To know the direction, you only need to know the sign of 'a'.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The direction a parabola opens is determined by the sign of the leading coefficient, which is the coefficient of the \( x^2 \) term. For a parabola given by the equation \( y = ax^2 + bx + c \):
- If \( a>0 \), the parabola opens upwards.
- If \( a<0 \), the parabola opens downwards.
The question asks for the direction, so we need to determine the sign of \( a \).
Step 2: Detailed Explanation:
Analyzing Statement (1): a = b
This statement tells us that the coefficients \( a \) and \( b \) are equal. However, it does not give any information about their sign.
- If \( a = b = 3 \), then \( a>0 \), and the parabola opens upwards.
- If \( a = b = -3 \), then \( a<0 \), and the parabola opens downwards.
Since the direction can be either upwards or downwards, this statement is not sufficient.
Analyzing Statement (2): a < 0
This statement directly tells us that the coefficient \( a \) is negative.
When \( a<0 \), the parabola always opens downwards.
This provides a definitive answer to the question. Therefore, statement (2) alone is sufficient.
Step 3: Final Answer:
Statement (2) alone is sufficient to answer the question, but statement (1) alone is not.