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 a detailed explanation of the solution:
Key Parts of a Parabola
A parabola is the graph of a quadratic function of the form \( ax^2 + bx + c \). The diagram shows various important features of the parabola. Let's identify and label each part step by step.
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1. Vertex
- The
vertex is the point where the parabola changes direction. It is the lowest point on the graph if the parabola opens upwards (as in this case) or the highest point if it opens downwards.
- In the given diagram, the vertex is located at the point \((2, -1)\).
- The vertex is labeled as the
minimum point since the parabola opens upwards.
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2. Axis of Symmetry
- The
axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. It passes through the vertex.
- In the diagram, the axis of symmetry is the dashed vertical line \( x = 2 \).
- This line is crucial because any point on one side of the parabola has a corresponding point on the other side that is equidistant from the axis of symmetry.
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3. Roots/Zeros
- The
roots or
zeros of the parabola are the points where the graph intersects the \( x \)-axis. These are the solutions to the equation \( ax^2 + bx + c = 0 \).
- In the diagram, the roots are at \( x = 1 \) and \( x = 3 \). These points are where the parabola crosses the \( x \)-axis.
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4. Y-Intercept
- The
y-intercept is the point where the parabola intersects the \( y \)-axis. This occurs when \( x = 0 \).
- In the diagram, the y-intercept is at the point \((0, 3)\).
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5. Direction of Opening
- The parabola opens upwards because the coefficient of \( x^2 \) (denoted as \( a \)) is positive. If \( a \) were negative, the parabola would open downwards.
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6. Focus and Directrix
- Although not explicitly shown in the diagram, every parabola has a
focus (a point inside the parabola) and a
directrix (a line outside the parabola). These are key geometric properties of a parabola, but they are not typically labeled in basic diagrams like this one.
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Labeling the Diagram
Based on the above analysis, we can label the parts of the parabola as follows:
1.
The parabola: The entire curve itself.
2.
Vertex: The point \((2, -1)\).
3.
Axis of symmetry: The vertical line \( x = 2 \).
4.
Roots/Zeros: The points \((1, 0)\) and \((3, 0)\).
5.
Y-intercept: The point \((0, 3)\).
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Final Answer
The labels for the parts of the parabola are:
\[
\boxed{\text{Vertex, Axis of Symmetry, Roots/Zeros, Y-intercept}}
\]
Parent Tip: Review the logic above to help your child master the concept of identifying parts of a parabola worksheet.
