To determine which must be true for a quadratic function whose vertex is the same as its [tex]\( y \)[/tex]-intercept, let's delve into the properties and structure of quadratic functions.
A quadratic function generally has the form [tex]\( y = ax^2 + bx + c \)[/tex]. The vertex form of a quadratic function can also be written as [tex]\( y = a(x-h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
Now, let's examine the properties given in the question:
1. The vertex is the same as its [tex]\( y \)[/tex]-intercept:
- The [tex]\( y \)[/tex]-intercept is the point where the graph intersects the [tex]\( y \)[/tex]-axis. At this point, [tex]\( x = 0 \)[/tex].
- If the vertex is also on the [tex]\( y \)[/tex]-axis, then the coordinates of the vertex are [tex]\((0, k)\)[/tex] for some value [tex]\( k \)[/tex]. This implies that the vertex form of the quadratic function simplifies to [tex]\( y = ax^2 + k \)[/tex], since [tex]\( h = 0 \)[/tex].
2. Analyzing the given options:
- Option 1: The axis of symmetry for the function is [tex]\( x = 0 \)[/tex]:
- The axis of symmetry in a quadratic function is a vertical line that passes through the vertex.
- Given that the vertex [tex]\( (0, k) \)[/tex] is on the [tex]\( y \)[/tex]-intercept (where [tex]\( x = 0 \)[/tex]), it implies that the axis of symmetry must indeed be [tex]\( x = 0 \)[/tex].
- Option 2: The axis of symmetry for the function is [tex]\( y = 0 \)[/tex]:
- This would mean a horizontal line passing through [tex]\( y = 0 \)[/tex], which does not make sense for a standard quadratic function that opens either upward or downward.
- Option 3: The function has no [tex]\( x \)[/tex]-intercepts:
- Whether the function intersects the [tex]\( x \)[/tex]-axis depends on the value of [tex]\( k \)[/tex] and the coefficient [tex]\( a \)[/tex]. So, this cannot generally be determined without further information.
- Option 4: The function has 1 [tex]\( x \)[/tex]-intercept:
- Again, this is not a definitive property without knowing the specific values of [tex]\( a \)[/tex] and [tex]\( k \)[/tex].
Given these explanations, we can confirm that the correct and insightful insight is that the quadratic function, whose vertex is at the same place as its y-intercept, must have the axis of symmetry at [tex]\( x = 0 \)[/tex].
Thus, the correct statement is:
- The axis of symmetry for the function is [tex]\( x = 0 \)[/tex].
