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Answer
Axis of symmetry:
Vertex:
Roots:
Max or Min: Minimum (at )
Vertex:
Roots:
Max or Min: Minimum (at
Solution
To analyze the quadratic function
, we can identify the following characteristics:
-
Axis of Symmetry: The axis of symmetry for a quadratic function in the form
is given by the formula . In this case, and . Therefore, the axis of symmetry is: -
Vertex: The vertex of the parabola can be found using the axis of symmetry. Since the axis of symmetry is
, we can find the y-coordinate of the vertex by substituting into the function: Thus, the vertex is at the point. -
Roots: To find the roots of the function, we set
: This simplifies to:Therefore, there is one root at. -
Max or Min: Since the coefficient
is positive, the parabola opens upwards. This means that the vertex represents a minimum point. Therefore, the function has a minimum value at the vertex, which is .
In summary:
- Axis of symmetry:
- Vertex:
- Roots:
- Max or Min: Minimum (at
)
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Mind Expander
The axis of symmetry for the equation
is the vertical line
. This is because the parabola opens upwards, and the vertex lies right on the y-axis. The vertex itself is at the point (0, 0), making it the lowest point on the graph.
As for the roots, this specific quadratic equation has only one root at
because it touches the x-axis at that point and then continues to rise. Since the parabola opens upward, it has a minimum value at the vertex, which is 0.