Let's analyze the properties of a quadratic function given in the form [tex]\( f(x) = ax^2 + bx + c \)[/tex] where the coefficient [tex]\( a \)[/tex] is negative.
1. The Vertex is a Maximum:
When the coefficient [tex]\( a \)[/tex] is negative, the parabola opens downwards. This means that the highest point on the graph, the vertex, represents the maximum value of the function. Hence, we can conclude that the vertex is indeed a maximum point. Therefore, this statement is true.
2. The [tex]\( y \)[/tex]-intercept is Negative:
The [tex]\( y \)[/tex]-intercept of the quadratic function is given by the value of [tex]\( c \)[/tex] (i.e., [tex]\( f(0) = c \)[/tex]). The sign of [tex]\( c \)[/tex] is independent of the sign of [tex]\( a \)[/tex], so we cannot conclude that the [tex]\( y \)[/tex]-intercept is negative based solely on [tex]\( a \)[/tex] being negative. Therefore, this statement is not necessarily true.
3. The [tex]\( x \)[/tex]-intercepts are Negative:
The [tex]\( x \)[/tex]-intercepts of the quadratic function are the solutions to the equation [tex]\( ax^2 + bx + c = 0 \)[/tex]. The sign of these intercepts depends on the values of [tex]\( a, b, \)[/tex] and [tex]\( c \)[/tex]. Since the sign and existence of [tex]\( x \)[/tex]-intercepts depend on the discriminant [tex]\( b^2 - 4ac \)[/tex], we cannot assert that the [tex]\( x \)[/tex]-intercepts are negative just because [tex]\( a \)[/tex] is negative. Therefore, this statement is not necessarily true.
4. The Axis of Symmetry is to the Left of Zero:
The equation for the axis of symmetry of the quadratic function is given by [tex]\( x = -\frac{b}{2a} \)[/tex]. Whether this value is to the left of zero (negative) depends on the sign of [tex]\( b \)[/tex] and the magnitude of [tex]\( a \)[/tex]. Since we cannot determine the position of the axis of symmetry solely based on [tex]\( a \)[/tex] being negative, this statement is not necessarily true.
Based on these analyses, the statement that must be true when the coefficient [tex]\( a \)[/tex] is negative is: The vertex is a maximum.
Therefore, the correct answer is:
The vertex is a maximum.
