Math Problem Statement
Solution
Let's analyze the given task based on the visible image:
Part a:
- Range:
- Axis of symmetry:
- y-intercept:
This describes a downward-opening parabola (since ) with the vertex at . The parabola passes through , which is the y-intercept. The axis of symmetry is the vertical line .
Part b:
- Range:
- x-intercepts: The points and
- x-coordinate of the vertex:
This describes an upward-opening parabola (since ), and the vertex will have the form . It passes through the x-intercepts and .
Key Information for Graphing:
- For a: The parabola has its vertex at , opens downward, and passes through as one additional point. You could find another point by reflecting over the axis of symmetry or calculating based on the equation.
- For b: The parabola has its vertex at , opens upward, and passes through the x-intercepts and .
Do you want me to provide the equations for these parabolas? Let me know if you need more details!
Questions:
- How do you determine the vertex of a parabola given its axis of symmetry and range?
- How would you find additional points for the graph of a parabola based on its characteristics?
- Why is the range of the parabola important for determining its direction of opening?
- How do x-intercepts help in graphing the parabola?
- What is the significance of the axis of symmetry in a parabola?
Tip:
When graphing a parabola, the vertex and intercepts provide crucial reference points. Always ensure to check symmetry across the axis.
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Math Problem Analysis
Mathematical Concepts
Quadratic Functions
Parabolas
Graphing Parabolas
Formulas
Standard form of a parabola: y = ax^2 + bx + c
Vertex form of a parabola: y = a(x - h)^2 + k
Theorems
Symmetry of a parabola along its axis
Intercepts of a quadratic function
Suitable Grade Level
Grades 9-11
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