Math Problem Statement
Determine the equation of the quadratic function shown below in factored form.
Solution
To determine the equation of the quadratic function shown in the graph in factored form, let's analyze the given information:
-
Intercepts:
- The graph intersects the x-axis at and . These are the roots of the quadratic equation.
-
Factored Form:
- The general factored form of a quadratic equation is: where and are the roots (or x-intercepts) of the equation. Here, and , so the equation becomes:
-
Vertex Point:
- From the graph, the vertex is visible at . The vertex helps us determine the value of .
-
Substitute the Vertex into the Equation:
- Substituting and into the equation : Simplify:
-
Final Equation:
- Substituting back into the equation:
- Expand to verify:
Conclusion:
The equation of the quadratic function in factored form is:
Let me know if you have any questions!
Here are five related questions to deepen your understanding:
- How do we determine the vertex of a parabola from its graph?
- What role does the value of play in the shape of a parabola?
- How would the equation change if the parabola opened upwards instead of downwards?
- Can you find the standard form of a quadratic equation directly from the graph?
- How would you determine the axis of symmetry of a parabola based on its roots?
Tip: Always verify the final equation by substituting key points from the graph!
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Quadratic Equations
Factored Form
Vertex Form
Formulas
y = a(x - r1)(x - r2)
Substitution to find 'a' using vertex
Theorems
Vertex property of parabolas
Factored form of quadratic equations
Suitable Grade Level
Grades 8-10