Math Problem Statement
a parabola passes through the points (04), (1,0), and (-2,0). find the equation of the parabola
Solution
To find the equation of the parabola that passes through the points , , and , we will assume that the equation of the parabola is of the form:
Where , , and are constants we need to determine. Given that the parabola passes through the points, we can substitute the coordinates of the points into this equation and create a system of linear equations to solve for , , and .
Step 1: Use the point
Substitute and into the general form of the parabola:
So, the equation simplifies to:
Step 2: Use the point
Substitute and into the equation:
Step 3: Use the point
Substitute and into the equation:
Step 4: Solve the system of equations
Now, we have the following system of equations:
Solve Equation 1 for :
Substitute into Equation 2:
Substitute into :
Step 5: Write the final equation
Now that we have , , and , the equation of the parabola is:
Final Answer:
The equation of the parabola is:
Would you like more details on how to solve this, or do you have other questions?
Further Questions:
- What is the vertex of the parabola ?
- How can we find the axis of symmetry for this parabola?
- How do you determine the direction (upward or downward) of the parabola from the equation?
- How do you find the focus and directrix of this parabola?
- How can you use the quadratic formula to find the roots of the parabola equation?
Tip:
To find the vertex of a parabola given its equation , use the formula for the -coordinate of the vertex: .
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Systems of Equations
Formulas
y = ax^2 + bx + c
System of linear equations
Theorems
Substitution method for solving systems of equations
Suitable Grade Level
Grades 9-11
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