For all functions of the form [tex]f(x)=a x^2+b x+c[/tex], which is true when [tex]b=0[/tex]?
A. The graph will always have zero [tex]x[/tex]-intercepts.
B. The function will always have a minimum.
C. The [tex]y[/tex]-intercept will always be the vertex.
D. The axis of symmetry will always be positive.
Answer (2)
When b = 0 in the quadratic function f ( x ) = a x 2 + c , the y -intercept is always the vertex of the parabola, occurring at the point ( 0 , c ) . This means that the correct statement among the given options is that the y -intercept will always be the vertex, which identifies option C as the true answer.
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The function is simplified to f ( x ) = a x 2 + c when b = 0 .
The graph does not always have zero x -intercepts; it depends 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 because the vertex occurs at x = 0 , and f ( 0 ) = c .
The axis of symmetry is x = 0 , which is not positive.
Therefore, the correct answer is that the y -intercept will always be the vertex, so the answer is T h e y − I n t erce pt w i ll a lw a ys b e t h e v er t e x .
Explanation
Understanding the Problem We are given a quadratic function of the form f ( x ) = a x 2 + b x + c , and we are told that b = 0 . This simplifies the function to f ( x ) = a x 2 + c . We need to determine which of the given statements is always true for this type of function.
Analyzing Each Statement Let's analyze each statement:
The graph will always have zero x -intercepts. This means the equation a x 2 + c = 0 has no real solutions. This is equivalent to a x 2 = − c , or x 2 = − c / a . If a and c have the same sign (both positive or both negative), then − c / a is negative, and there are no real solutions. However, if a and c have opposite signs, then − c / a is positive, and there are two real solutions (x-intercepts). Therefore, this statement is not always true.
The function will always have a minimum. The function f ( x ) = a x 2 + c is a parabola. If 0"> a > 0 , the parabola opens upwards, and the function has a minimum. If a < 0 , the parabola opens downwards, and the function has a maximum. Therefore, this statement is not always true.
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 ) = c . Thus, the vertex is at ( 0 , c ) , which is the same as the y -intercept. Therefore, this statement is true.
The axis of symmetry will always be positive. The axis of symmetry is the vertical line x = − b / ( 2 a ) . Since b = 0 , the axis of symmetry is x = 0 , which is not positive. Therefore, this statement is false.
Conclusion Based on the analysis above, only the statement "The y -intercept will always be the vertex" is true for all functions of the form f ( x ) = a x 2 + c .
Examples
Consider a simple parabolic reflector used in car headlights. The shape of the reflector is described by a quadratic function. When the coefficient 'b' is zero, it ensures the light source is positioned directly at the vertex of the parabola, maximizing the focus and intensity of the beam. This principle is crucial in designing efficient lighting systems, solar concentrators, and satellite dishes, where precise focus is essential for optimal performance.
