Find the parametrization for the left half of the parabola: y = x^2 - 4x + 3.
To find the parametrization for the left half of the parabola y = x^2 - 4x + 3, we need to determine the range of x-values that correspond to the left side of the parabola.
The left half of a parabola corresponds to the region where the x-values are less than the x-coordinate of the vertex.
Step 1: Determine the x-coordinate of the vertex
In general, the x-coordinate of the vertex of a parabola of the form y = ax^2 + bx + c can be found using the formula: x = -b/2a.
In this case, the equation of the parabola is y = x^2 - 4x + 3.
Comparing it to the general form, we have a = 1, b = -4, and c = 3.
The x-coordinate of the vertex is given by: x = -(-4) / (2*1) = 2.
Step 2: Determine the range of x-values for the left half
Since the vertex of the parabola is at x = 2, the left half of the parabola corresponds to the region where x < 2.
Step 3: Parametrize the left half of the parabola
To parametrize the left half of the parabola, we can choose a parameter t such that x = 2 - t, where t is a non-negative real number.
Substituting this value of x into the equation y = x^2 - 4x + 3, we get y = (2 - t)^2 - 4(2 - t) + 3.
Simplifying this equation, we have y = t^2 - 4t + 7.
Therefore, the parametrization of the left half of the parabola y = x^2 - 4x + 3 is given by:
x = 2 - t
y = t^2 - 4t + 7
Where t is a non-negative real number.