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Let's analyze the properties of the given quadratic function [tex]\( f(x) = a x^2 + b x + c \)[/tex] when [tex]\( b = 0 \)[/tex]. The function simplifies to [tex]\( f(x) = a x^2 + c \)[/tex].
1. Statement: The graph will always have zero [tex]\(x\)[/tex]-intercepts.
This statement is not true. The number of [tex]\(x\)[/tex]-intercepts (roots) depends on the values of [tex]\(a\)[/tex] and [tex]\(c\)[/tex]:
- If [tex]\(a > 0\)[/tex] and [tex]\(c > 0\)[/tex], the parabola opens upwards and the vertex is above the [tex]\(x\)[/tex]-axis, so there are no [tex]\(x\)[/tex]-intercepts.
- If [tex]\(a < 0\)[/tex] and [tex]\(c < 0\)[/tex], the parabola opens downwards and the vertex is below the [tex]\(x\)[/tex]-axis, so there are no [tex]\(x\)[/tex]-intercepts.
- If [tex]\(c = 0\)[/tex], the vertex is on the [tex]\(x\)[/tex]-axis, resulting in exactly one [tex]\(x\)[/tex]-intercept irrespective of the value of [tex]\(a\)[/tex].
- If [tex]\(a > 0\)[/tex] and [tex]\(c < 0\)[/tex] or [tex]\(a < 0\)[/tex] and [tex]\(c > 0\)[/tex], the parabola will have two [tex]\(x\)[/tex]-intercepts because it will intersect the [tex]\(x\)[/tex]-axis twice.
Therefore, the graph does not always have zero [tex]\(x\)[/tex]-intercepts.
2. Statement: The function will always have a minimum.
This statement is only true if [tex]\(a > 0\)[/tex]. When [tex]\(a > 0\)[/tex], the parabola opens upwards, and the vertex represents the minimum point of the function. If [tex]\(a < 0\)[/tex], the parabola opens downwards, and the vertex represents a maximum point. Thus, the function does not always have a minimum unless it is specified that [tex]\(a > 0\)[/tex].
3. Statement: The [tex]\(y\)[/tex]-intercept will always be the vertex.
This statement is false. For the function [tex]\( f(x) = a x^2 + c \)[/tex], the [tex]\(y\)[/tex]-intercept occurs where [tex]\(x = 0\)[/tex], i.e., [tex]\(f(0) = c\)[/tex]. The vertex of the function [tex]\(f(x) = a x^2 + c\)[/tex] is [tex]\((0, c)\)[/tex], which means the [tex]\(y\)[/tex]-intercept and the vertex coincide. However, this is not always the case for all quadratic equations; it only holds true when [tex]\(b = 0\)[/tex].
4. Statement: The axis of symmetry will always be positive.
This statement is irrelevant to the value of [tex]\(a\)[/tex] or [tex]\(c\)[/tex] and only involves the coefficient [tex]\(b\)[/tex]. Specifically, when [tex]\( b = 0 \)[/tex], the axis of symmetry of the parabola is [tex]\(x = 0\)[/tex], which is the [tex]\(y\)[/tex]-axis. This is not dependent on any specific values but rather on the structure being [tex]\(b = 0\)[/tex].
Given the conclusions above, the correct assertion when [tex]\( b = 0 \)[/tex] for the quadratic function [tex]\( f(x) = a x^2 + c \)[/tex] is:
- The y-intercept will always be the vertex.
Among the given statements, the one that holds true consistently is:
- The y-intercept will always be the vertex when [tex]\( b = 0 \)[/tex].
1. Statement: The graph will always have zero [tex]\(x\)[/tex]-intercepts.
This statement is not true. The number of [tex]\(x\)[/tex]-intercepts (roots) depends on the values of [tex]\(a\)[/tex] and [tex]\(c\)[/tex]:
- If [tex]\(a > 0\)[/tex] and [tex]\(c > 0\)[/tex], the parabola opens upwards and the vertex is above the [tex]\(x\)[/tex]-axis, so there are no [tex]\(x\)[/tex]-intercepts.
- If [tex]\(a < 0\)[/tex] and [tex]\(c < 0\)[/tex], the parabola opens downwards and the vertex is below the [tex]\(x\)[/tex]-axis, so there are no [tex]\(x\)[/tex]-intercepts.
- If [tex]\(c = 0\)[/tex], the vertex is on the [tex]\(x\)[/tex]-axis, resulting in exactly one [tex]\(x\)[/tex]-intercept irrespective of the value of [tex]\(a\)[/tex].
- If [tex]\(a > 0\)[/tex] and [tex]\(c < 0\)[/tex] or [tex]\(a < 0\)[/tex] and [tex]\(c > 0\)[/tex], the parabola will have two [tex]\(x\)[/tex]-intercepts because it will intersect the [tex]\(x\)[/tex]-axis twice.
Therefore, the graph does not always have zero [tex]\(x\)[/tex]-intercepts.
2. Statement: The function will always have a minimum.
This statement is only true if [tex]\(a > 0\)[/tex]. When [tex]\(a > 0\)[/tex], the parabola opens upwards, and the vertex represents the minimum point of the function. If [tex]\(a < 0\)[/tex], the parabola opens downwards, and the vertex represents a maximum point. Thus, the function does not always have a minimum unless it is specified that [tex]\(a > 0\)[/tex].
3. Statement: The [tex]\(y\)[/tex]-intercept will always be the vertex.
This statement is false. For the function [tex]\( f(x) = a x^2 + c \)[/tex], the [tex]\(y\)[/tex]-intercept occurs where [tex]\(x = 0\)[/tex], i.e., [tex]\(f(0) = c\)[/tex]. The vertex of the function [tex]\(f(x) = a x^2 + c\)[/tex] is [tex]\((0, c)\)[/tex], which means the [tex]\(y\)[/tex]-intercept and the vertex coincide. However, this is not always the case for all quadratic equations; it only holds true when [tex]\(b = 0\)[/tex].
4. Statement: The axis of symmetry will always be positive.
This statement is irrelevant to the value of [tex]\(a\)[/tex] or [tex]\(c\)[/tex] and only involves the coefficient [tex]\(b\)[/tex]. Specifically, when [tex]\( b = 0 \)[/tex], the axis of symmetry of the parabola is [tex]\(x = 0\)[/tex], which is the [tex]\(y\)[/tex]-axis. This is not dependent on any specific values but rather on the structure being [tex]\(b = 0\)[/tex].
Given the conclusions above, the correct assertion when [tex]\( b = 0 \)[/tex] for the quadratic function [tex]\( f(x) = a x^2 + c \)[/tex] is:
- The y-intercept will always be the vertex.
Among the given statements, the one that holds true consistently is:
- The y-intercept will always be the vertex when [tex]\( b = 0 \)[/tex].
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