Answer :
Given a quadratic function in the form [tex]\( f(x) = ax^2 + bx + c \)[/tex] where the coefficient [tex]\( a \)[/tex] is negative, we need to determine which statement must be true.
### Step-by-Step Solution:
1. Understanding the Vertex of a Parabola:
- The general form of a quadratic function is [tex]\( f(x) = ax^2 + bx + c \)[/tex].
- For a quadratic function, [tex]\( a \)[/tex] determines the direction of the parabola.
2. Direction of Parabola:
- If [tex]\( a > 0 \)[/tex], the parabola opens upwards.
- If [tex]\( a < 0 \)[/tex], the parabola opens downwards. In our case, [tex]\( a \)[/tex] is negative, so the parabola opens downwards.
3. Vertex Point:
- The vertex of a parabola represented by the function [tex]\( f(x) = ax^2 + bx + c \)[/tex] occurs at [tex]\( x = -\frac{b}{2a} \)[/tex].
- Since the parabola opens downwards (because [tex]\( a \)[/tex] is negative), the vertex represents the highest or maximum point on the graph.
4. Analyzing the Statements:
- Statement 1: "The vertex is a maximum."
- This statement must be true because the parabola opens downwards, meaning the highest point (vertex) is a maximum.
- Statement 2: "The [tex]\( y \)[/tex]-intercept is negative."
- The [tex]\( y \)[/tex]-intercept is the point where the graph crosses the [tex]\( y \)[/tex]-axis, which occurs at [tex]\( x = 0 \)[/tex]. The [tex]\( y \)[/tex]-intercept is simply [tex]\( c \)[/tex]. There is no information given about [tex]\( c \)[/tex], so we cannot definitively determine this.
- Statement 3: "The [tex]\( x \)[/tex]-intercepts are negative."
- [tex]\( x \)[/tex]-intercepts are the points where the function crosses the [tex]\( x \)[/tex]-axis (i.e., the solutions to [tex]\( ax^2 + bx + c = 0 \)[/tex]). The signs of [tex]\( x \)[/tex]-intercepts depend on [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] and can’t be determined solely by knowing [tex]\( a \)[/tex] is negative.
- Statement 4: "The axis of symmetry is to the left of zero."
- The axis of symmetry is the vertical line passing through the vertex, given by [tex]\( x = -\frac{b}{2a} \)[/tex]. Whether this is to the left of zero depends on the sign of [tex]\( b \)[/tex]. This cannot be determined just from knowing [tex]\( a \)[/tex] is negative.
### Conclusion:
Considering the nature of the parabola when [tex]\( a \)[/tex] is negative, only Statement 1 ("The vertex is a maximum") must be true.
Therefore, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
### Step-by-Step Solution:
1. Understanding the Vertex of a Parabola:
- The general form of a quadratic function is [tex]\( f(x) = ax^2 + bx + c \)[/tex].
- For a quadratic function, [tex]\( a \)[/tex] determines the direction of the parabola.
2. Direction of Parabola:
- If [tex]\( a > 0 \)[/tex], the parabola opens upwards.
- If [tex]\( a < 0 \)[/tex], the parabola opens downwards. In our case, [tex]\( a \)[/tex] is negative, so the parabola opens downwards.
3. Vertex Point:
- The vertex of a parabola represented by the function [tex]\( f(x) = ax^2 + bx + c \)[/tex] occurs at [tex]\( x = -\frac{b}{2a} \)[/tex].
- Since the parabola opens downwards (because [tex]\( a \)[/tex] is negative), the vertex represents the highest or maximum point on the graph.
4. Analyzing the Statements:
- Statement 1: "The vertex is a maximum."
- This statement must be true because the parabola opens downwards, meaning the highest point (vertex) is a maximum.
- Statement 2: "The [tex]\( y \)[/tex]-intercept is negative."
- The [tex]\( y \)[/tex]-intercept is the point where the graph crosses the [tex]\( y \)[/tex]-axis, which occurs at [tex]\( x = 0 \)[/tex]. The [tex]\( y \)[/tex]-intercept is simply [tex]\( c \)[/tex]. There is no information given about [tex]\( c \)[/tex], so we cannot definitively determine this.
- Statement 3: "The [tex]\( x \)[/tex]-intercepts are negative."
- [tex]\( x \)[/tex]-intercepts are the points where the function crosses the [tex]\( x \)[/tex]-axis (i.e., the solutions to [tex]\( ax^2 + bx + c = 0 \)[/tex]). The signs of [tex]\( x \)[/tex]-intercepts depend on [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] and can’t be determined solely by knowing [tex]\( a \)[/tex] is negative.
- Statement 4: "The axis of symmetry is to the left of zero."
- The axis of symmetry is the vertical line passing through the vertex, given by [tex]\( x = -\frac{b}{2a} \)[/tex]. Whether this is to the left of zero depends on the sign of [tex]\( b \)[/tex]. This cannot be determined just from knowing [tex]\( a \)[/tex] is negative.
### Conclusion:
Considering the nature of the parabola when [tex]\( a \)[/tex] is negative, only Statement 1 ("The vertex is a maximum") must be true.
Therefore, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
