Math Problem Statement
Sketch the graph of the quadratic function with the given vertex and that passes through the given point. Vertex (3,4) ; Point (5,8)
Solution
To find the equation of the quadratic function and sketch the graph, we can use the vertex form of a quadratic equation, which is:
where is the vertex. For this problem, the vertex is given as , so the equation becomes:
We are also given that the function passes through the point . To find the value of , substitute and into the equation:
Simplifying:
Now that we have , the equation of the quadratic function is:
Sketch of the Graph:
- The vertex is at .
- The parabola opens upwards because , which is positive.
- It passes through the point .
- The axis of symmetry is the vertical line .
Would you like details on how to sketch it step by step or explore further?
Here are 5 related questions for practice:
- What is the axis of symmetry of a quadratic function?
- How does changing the value of affect the graph of a quadratic function?
- How do you find the x-intercepts of a quadratic function?
- Can you determine the direction of a parabola based on the sign of ?
- How can you convert a quadratic function from vertex form to standard form?
Tip: The vertex form of a quadratic function is especially useful when you know the vertex, as it simplifies the process of graphing and solving equations.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Functions
Graphing Parabolas
Formulas
Vertex form of a quadratic function: y = a(x - h)^2 + k
Theorems
Axis of symmetry of a parabola
Properties of quadratic functions
Suitable Grade Level
Grades 8-10