For all functions of the form $f(x)=a x^2+b x+c$, which is true when $b=0$?
A. The graph will always have zero $x$-intercepts.
B. The function will always have a minimum.
C. The $y$-intercept will always be the vertex.
D. The axis of symmetry will always be positive.

When b = 0 , the quadratic function simplifies to f ( x ) = a x 2 + c .
The x -intercepts are not always zero; they depend on the signs of a and c .
The function does not always have a minimum; it depends on the sign of a .
The y -intercept is always the vertex since the vertex occurs at x = 0 .
The axis of symmetry is always x = 0 , which is not positive. Therefore, the correct answer is: The y -intercept will always be the vertex. ​

Explanation

Analyzing the Problem We are given a quadratic function of the form f ( x ) = a x 2 + b x + c , and we want to determine which of the given statements is true when b = 0 . This simplifies the function to f ( x ) = a x 2 + c . Let's analyze each statement.

Analyzing Statement 1 Statement 1: The graph will always have zero x -intercepts.


To find the x -intercepts, we set f ( x ) = 0 , so a x 2 + c = 0 . This gives x 2 = − c / a . If − c / a < 0 , then there are no real x -intercepts. However, if 0"> − c / a > 0 , then there are two real x -intercepts, x = ± − c / a ​ .
For example, if a = 1 and c = 1 , then f ( x ) = x 2 + 1 , and x 2 = − 1 , which has no real solutions. In this case, there are zero x -intercepts. However, if a = 1 and c = − 1 , then f ( x ) = x 2 − 1 , and x 2 = 1 , which has solutions x = ± 1 . In this case, there are two x -intercepts. Therefore, the graph does not always have zero x -intercepts.

Analyzing Statement 2 Statement 2: The function will always have a minimum.

The function is f ( x ) = a x 2 + c . If 0"> a > 0 , the parabola opens upwards, and the function has a minimum value at its vertex. If a < 0 , the parabola opens downwards, and the function has a maximum value at its vertex. Therefore, the function does not always have a minimum. For example, if a = − 1 and c = 0 , then f ( x ) = − x 2 , which has a maximum at x = 0 .

Analyzing Statement 3 Statement 3: The y -intercept will always be the vertex.

The y -intercept is the value of f ( x ) when x = 0 . So, f ( 0 ) = a ( 0 ) 2 + c = c . The vertex of the parabola f ( x ) = a x 2 + c occurs at x = − b / ( 2 a ) . Since b = 0 , the vertex occurs at x = 0 . The y -coordinate of the vertex is f ( 0 ) = a ( 0 ) 2 + c = c . Therefore, the y -intercept is always the vertex.

Analyzing Statement 4 Statement 4: The axis of symmetry will always be positive.

The axis of symmetry is given by x = − b / ( 2 a ) . Since b = 0 , the axis of symmetry is x = − 0/ ( 2 a ) = 0 . Therefore, the axis of symmetry is always 0, which is not positive.

Conclusion Therefore, the only true statement is that the y -intercept will always be the vertex.

Examples
Understanding quadratic functions is crucial in various real-world applications. For instance, engineers use quadratic equations to model the trajectory of projectiles, such as the path of a ball thrown in the air. The vertex of the parabola represents the maximum height the ball reaches. Similarly, architects use quadratic functions to design arches and bridges, ensuring structural stability and optimal aesthetics. By analyzing the properties of quadratic functions, professionals can make informed decisions and create efficient designs.

Answered by GinnyAnswer | 2025-07-05