The a value of a function in the form [tex]$f(x)=a x^2+b x+c$[/tex] is negative. Which statement must be true?
A. The vertex is a maximum.
B. The [tex]$y$[/tex]-intercept is negative.
C. The [tex]$x$[/tex]-intercepts are negative.
D. The axis of symmetry is to the left of zero.

The function is a quadratic f ( x ) = a x 2 + b x + c with a < 0 .
Since a < 0 , the parabola opens downwards, and the vertex is a maximum.
The y -intercept is c , which can be positive, negative, or zero.
The axis of symmetry is x = − 2 a b ​ , which can be to the left, right, or at zero.
Therefore, the vertex is a maximum. The vertex is a maximum. ​

Explanation

Understanding the Problem We are given a quadratic function in the form f ( x ) = a x 2 + b x + c , where a < 0 . We need to determine which of the given statements must be true.

Analyzing the Vertex Since a < 0 , the parabola opens downwards. This means that the vertex of the parabola is the highest point on the graph, so the vertex is a maximum.

Analyzing the y-intercept The y -intercept is the value of the function when x = 0 , which is f ( 0 ) = a ( 0 ) 2 + b ( 0 ) + c = c . The sign of c is independent of the sign of a , so the y -intercept is not necessarily negative.

Analyzing the x-intercepts The x -intercepts are the solutions to the equation a x 2 + b x + c = 0 . The discriminant of this quadratic equation is Δ = b 2 − 4 a c . If 0"> Δ > 0 , there are two x -intercepts. If Δ = 0 , there is one x -intercept. If Δ < 0 , there are no x -intercepts. The sign of the x -intercepts is not determined by the sign of a . For example, if f ( x ) = − x 2 + 5 x − 6 , then the x -intercepts are x = 2 and x = 3 , which are both positive. If f ( x ) = − x 2 − 5 x − 6 , then the x -intercepts are x = − 2 and x = − 3 , which are both negative.

Analyzing the Axis of Symmetry The axis of symmetry is the vertical line x = − 2 a b ​ . Since a < 0 , we have 0"> − 2 a > 0 . If 0"> b > 0 , then − 2 a b ​ < 0 , so the axis of symmetry is to the left of zero. If b < 0 , then 0"> − 2 a b ​ > 0 , so the axis of symmetry is to the right of zero. If b = 0 , then − 2 a b ​ = 0 , so the axis of symmetry is at zero. Therefore, the axis of symmetry is not necessarily to the left of zero.

Conclusion Therefore, the only statement that must be true is that the vertex is a maximum.


Examples
Understanding the properties of quadratic functions is crucial in various real-world applications. For instance, when designing a bridge, engineers use quadratic functions to model the parabolic shape of the bridge's arch. Knowing that a negative 'a' value results in a maximum point helps them determine the highest stress point on the arch, ensuring the bridge's stability and safety. Similarly, in projectile motion, the path of a ball thrown into the air can be modeled by a quadratic function, where the maximum height is determined by the vertex of the parabola.

Answered by GinnyAnswer | 2025-07-05