The image provided is a diagram of a parabola, and the task appears to be identifying and labeling the parts of the parabola. Below is an explanation of the key components of a parabola and how they relate to the diagram:
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Key Parts of a Parabola
1.
Vertex: The vertex is the highest or lowest point of the parabola, depending on whether it opens upwards or downwards. In this case, the parabola opens upwards, so the vertex is the lowest point.
- In the diagram, the vertex is located at the point \((2, -1)\).
2.
Axis of Symmetry: This is a vertical line that divides the parabola into two symmetric halves. It passes through the vertex.
- In the diagram, the axis of symmetry is the dashed vertical line \(x = 2\).
3.
Focus: The focus is a fixed point inside the parabola that helps define its shape. For a standard parabola \(y = ax^2 + bx + c\), the focus lies along the axis of symmetry.
- Although not explicitly marked in the diagram, the focus can be calculated using the formula for the focus of a parabola.
4.
Directrix: The directrix is a horizontal line that is equidistant from the vertex as the focus but lies on the opposite side of the vertex.
- The directrix is not explicitly marked in the diagram but can be inferred based on the position of the focus.
5.
Roots/Zeros: These are the points where the parabola intersects the x-axis. They are the solutions to the quadratic equation \(ax^2 + bx + c = 0\).
- In the diagram, the roots are at \(x = 1\) and \(x = 3\).
6.
Y-intercept: This is the point where the parabola intersects the y-axis. It corresponds to the value of \(c\) in the quadratic equation \(y = ax^2 + bx + c\).
- In the diagram, the y-intercept is at \((0, 3)\).
7.
Latus Rectum: This is a line segment perpendicular to the axis of symmetry, passing through the focus, and whose endpoints lie on the parabola.
- The latus rectum is not explicitly marked in the diagram but can be determined based on the focus and the shape of the parabola.
8.
Parabola: The entire curve itself is referred to as the parabola.
- In the diagram, the curved line is labeled as "The parabola."
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Labeling the Diagram
Based on the above explanations, the labels for the blank boxes in the diagram can be filled as follows:
1.
Top Left Box: This box is near the y-intercept. The label should be:
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Y-intercept
2.
Middle Right Box: This box is near the horizontal line segment across the parabola. The label should be:
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Latus Rectum
3.
Bottom Right Box: This box is near the x-axis, indicating the roots. The label should be:
-
Roots (or Zeros)
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Final Answer
The completed labels for the diagram are:
- Top Left:
Y-intercept
- Middle Right:
Latus Rectum
- Bottom Right:
Roots (or Zeros)
Thus, the final answer is:
\[
\boxed{\text{Y-intercept, Latus Rectum, Roots}}
\]
Parent Tip: Review the logic above to help your child master the concept of parts of a parabola worksheet.
