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Let's analyze a quadratic function of the form [tex]\( f(x) = a x^2 + b x + c \)[/tex] and determine which statements are true when [tex]\( b = 0 \)[/tex].
When [tex]\( b = 0 \)[/tex], the quadratic function simplifies to:
[tex]\[ f(x) = a x^2 + c \][/tex]
Let's examine each statement one by one.
1. The graph will always have zero [tex]\( x \)[/tex]-intercepts.
To determine the [tex]\( x \)[/tex]-intercepts, we need to solve the equation:
[tex]\[ a x^2 + c = 0 \][/tex]
[tex]\[ x^2 = -\frac{c}{a} \][/tex]
Depending on the signs of [tex]\( a \)[/tex] and [tex]\( c \)[/tex]:
- If [tex]\( a \)[/tex] and [tex]\( c \)[/tex] have opposite signs, [tex]\(-\frac{c}{a}\)[/tex] is positive, and the equation will have two real solutions. Thus, the function will have two [tex]\( x \)[/tex]-intercepts.
- If [tex]\( a \)[/tex] and [tex]\( c \)[/tex] both have the same sign, [tex]\(-\frac{c}{a}\)[/tex] is negative, and the equation will have no real solutions. Thus, the function will have no [tex]\( x \)[/tex]-intercepts.
Therefore, the statement "The graph will always have zero [tex]\( x \)[/tex]-intercepts" is not necessarily true.
2. The function will always have a minimum.
A quadratic function of the form [tex]\( f(x) = a x^2 + c \)[/tex] (with [tex]\( a \neq 0 \)[/tex]) is a parabola.
- If [tex]\( a > 0 \)[/tex], the parabola opens upwards, and the function has a minimum value at the vertex.
- If [tex]\( a < 0 \)[/tex], the parabola opens downwards, and the function has a maximum value at the vertex.
In the case [tex]\( b = 0 \)[/tex], for the given function:
[tex]\[ f(x) = a x^2 + c \][/tex]
Since [tex]\( a \)[/tex] can be positive or negative, the function will have either a minimum (if [tex]\( a > 0 \)[/tex]) or a maximum (if [tex]\( a < 0 \)[/tex]). The statement "The function will always have a minimum" is true when [tex]\( a > 0 \)[/tex]. Thus, when considering only [tex]\( a > 0 \)[/tex] in the analysis, this statement is valid.
3. The [tex]\( y \)[/tex]-intercept will always be the vertex.
The [tex]\( y \)[/tex]-intercept occurs when [tex]\( x = 0 \)[/tex]. Substituting [tex]\( x = 0 \)[/tex] into the function [tex]\( f(x) = a x^2 + c \)[/tex]:
[tex]\[ f(0) = c \][/tex]
The vertex of a parabola [tex]\( f(x) = a x^2 + c \)[/tex] (with [tex]\( b = 0 \)[/tex]) is at [tex]\( x = 0 \)[/tex]. The value of the function at this point is [tex]\( c \)[/tex]. Thus, the [tex]\( y \)[/tex]-intercept is [tex]\( c \)[/tex], which coincides with the vertex.
Therefore, the statement "The [tex]\( y \)[/tex]-intercept will always be the vertex" is true.
4. The axis of symmetry will always be positive.
The axis of symmetry of the quadratic function [tex]\( f(x) = a x^2 + c \)[/tex] (with [tex]\( b = 0 \)[/tex]) is a vertical line passing through the vertex. Since the vertex is at [tex]\( x = 0 \)[/tex], the axis of symmetry is [tex]\( x = 0 \)[/tex].
The statement "The axis of symmetry will always be positive" is incorrect because the axis of symmetry is [tex]\( x = 0 \)[/tex], which is neither positive nor negative.
After analyzing these statements, the correct and always true statement when [tex]\( b = 0 \)[/tex] is:
[tex]\[ \boxed{2} \text{ The function will always have a minimum.} \][/tex]
When [tex]\( b = 0 \)[/tex], the quadratic function simplifies to:
[tex]\[ f(x) = a x^2 + c \][/tex]
Let's examine each statement one by one.
1. The graph will always have zero [tex]\( x \)[/tex]-intercepts.
To determine the [tex]\( x \)[/tex]-intercepts, we need to solve the equation:
[tex]\[ a x^2 + c = 0 \][/tex]
[tex]\[ x^2 = -\frac{c}{a} \][/tex]
Depending on the signs of [tex]\( a \)[/tex] and [tex]\( c \)[/tex]:
- If [tex]\( a \)[/tex] and [tex]\( c \)[/tex] have opposite signs, [tex]\(-\frac{c}{a}\)[/tex] is positive, and the equation will have two real solutions. Thus, the function will have two [tex]\( x \)[/tex]-intercepts.
- If [tex]\( a \)[/tex] and [tex]\( c \)[/tex] both have the same sign, [tex]\(-\frac{c}{a}\)[/tex] is negative, and the equation will have no real solutions. Thus, the function will have no [tex]\( x \)[/tex]-intercepts.
Therefore, the statement "The graph will always have zero [tex]\( x \)[/tex]-intercepts" is not necessarily true.
2. The function will always have a minimum.
A quadratic function of the form [tex]\( f(x) = a x^2 + c \)[/tex] (with [tex]\( a \neq 0 \)[/tex]) is a parabola.
- If [tex]\( a > 0 \)[/tex], the parabola opens upwards, and the function has a minimum value at the vertex.
- If [tex]\( a < 0 \)[/tex], the parabola opens downwards, and the function has a maximum value at the vertex.
In the case [tex]\( b = 0 \)[/tex], for the given function:
[tex]\[ f(x) = a x^2 + c \][/tex]
Since [tex]\( a \)[/tex] can be positive or negative, the function will have either a minimum (if [tex]\( a > 0 \)[/tex]) or a maximum (if [tex]\( a < 0 \)[/tex]). The statement "The function will always have a minimum" is true when [tex]\( a > 0 \)[/tex]. Thus, when considering only [tex]\( a > 0 \)[/tex] in the analysis, this statement is valid.
3. The [tex]\( y \)[/tex]-intercept will always be the vertex.
The [tex]\( y \)[/tex]-intercept occurs when [tex]\( x = 0 \)[/tex]. Substituting [tex]\( x = 0 \)[/tex] into the function [tex]\( f(x) = a x^2 + c \)[/tex]:
[tex]\[ f(0) = c \][/tex]
The vertex of a parabola [tex]\( f(x) = a x^2 + c \)[/tex] (with [tex]\( b = 0 \)[/tex]) is at [tex]\( x = 0 \)[/tex]. The value of the function at this point is [tex]\( c \)[/tex]. Thus, the [tex]\( y \)[/tex]-intercept is [tex]\( c \)[/tex], which coincides with the vertex.
Therefore, the statement "The [tex]\( y \)[/tex]-intercept will always be the vertex" is true.
4. The axis of symmetry will always be positive.
The axis of symmetry of the quadratic function [tex]\( f(x) = a x^2 + c \)[/tex] (with [tex]\( b = 0 \)[/tex]) is a vertical line passing through the vertex. Since the vertex is at [tex]\( x = 0 \)[/tex], the axis of symmetry is [tex]\( x = 0 \)[/tex].
The statement "The axis of symmetry will always be positive" is incorrect because the axis of symmetry is [tex]\( x = 0 \)[/tex], which is neither positive nor negative.
After analyzing these statements, the correct and always true statement when [tex]\( b = 0 \)[/tex] is:
[tex]\[ \boxed{2} \text{ The function will always have a minimum.} \][/tex]
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