Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Get quick and reliable solutions to your questions from a community of seasoned experts on our user-friendly platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
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]
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.