Final Answer:
The equation of the quadratic is y = -2x + 3.
Step-by-step explanation:
Identify the axis of symmetry:
The given information states that the line of symmetry is x = 2. This means the vertex of the parabola lies on this line.
Utilize the given points:
We know that the parabola passes through the points (1,0) and (4, -6). We can use these points to form two equations and solve for the coefficients of the quadratic equation.
Form the equations:
The vertex is equidistant from both points on the parabola. Therefore:
Distance from vertex to (1,0): √((2 - 1)^2 + (0 - y)^2)
Distance from vertex to (4, -6): √((2 - 4)^2 + (-6 - y)^2)
Since these distances are equal, we can set the equations equal to each other:
√((2 - 1)^2 + (0 - y)^2) = √((2 - 4)^2 + (-6 - y)^2)
Solve for y:
Squaring both sides and simplifying, we get:
1 + y^2 = 36 + 12y + y^2
12y = -35
y = -2.92 (approximately)
Substitute y in the equation of the quadratic:
The standard equation of a parabola is y = ax^2 + bx + c. Since the vertex is at (2, -2.92), we can write the equation as:
y = a(x - 2)^2 - 2.92
Substitute the point (4, -6) to solve for a:
-6 = a(4 - 2)^2 - 2.92
-6 = 4a - 2.92
a = -0.99
Write the final equation:
Therefore, the equation of the quadratic is:
y = -0.99(x - 2)^2 - 2.92
Simplified, this becomes:
y = -2x + 3