Question

the a value of a function in the form f(x) = ax² + bx + c is negative. which statement must be true? the vertex is a maximum. the y - intercept is negative. the x - intercepts are negative. the axis of symmetry is to the left of zero.

Explanation:

To solve this, we analyze the properties of a quadratic function \( f(x) = ax^2 + bx + c \):

1. Vertex and Direction of Parabola: The coefficient \( a \) determines if the parabola opens up (\( a > 0 \)) or down (\( a < 0 \)). If \( a \) is negative, the parabola opens downward, so the vertex is the highest point (a maximum).
2. Y - intercept: The y - intercept is found by setting \( x = 0 \), so \( f(0)=c \). The sign of \( a \) does not determine the sign of \( c \), so we can't say the y - intercept is negative.
3. X - intercepts: The x - intercepts depend on the discriminant \( b^2 - 4ac \) and the values of \( b \) and \( c \) as well. A negative \( a \) doesn't guarantee negative x - intercepts.
4. Axis of Symmetry: The axis of symmetry is \( x=-\frac{b}{2a} \). The sign of \( a \) (negative) and the sign of \( b \) determine the position of the axis of symmetry. We can't say it's to the left of zero without knowing \( b \).

Answer:

The vertex is a maximum.