At what values of t is the vertex of the parabola y = 2tx ^ 2-4x + 3 located: below the Ox axis
At what values of t is the vertex of the parabola y = 2tx ^ 2-4x + 3 located: below the Ox axis, above the Ox axis, on the Ox axis.
Y = 2 * t * X ^ 2 – 4 * X + 3.t ≠ 0.
We use the formula for the coordinates of the vertex of the parabola.
Y0 = (4 * a * c – b ^ 2) / 4 * a;
Y0 = (4 * 2 * t * 3 – 16) / 4 * 2 * t = (3 * t – 2) / t;
If the vertex of the parabola lies above the OX axis, then Y0> 0.
(3 * t – 2) / t> 0;
{3 * t – 2> 0;
{t> 0;
t> 2/3;
t> 0.
General solution t> 2/3;
(3 * t – 2) / t <0;
{3 * t – 2 <0;
{t <0;
t <2/3;
t <0.
General solution t <0.
The vertex of the parabola is above the OX axis at 2/3 <t <0.
If the vertex of the parabola lies below the OX axis, then Y0 <0.
(3 * t – 2) / t <0;
{3 * t – 2> 0;
{t <0;
t> 2/3;
t <0.
There is no general meaning.
(3 * t – 2) / t <0;
{3 * t – 2 <0;
{t> 0;
t <2/3;
t> 0.
General solution 0 <t <2/3.
The vertex of the parabola is above the OX axis at 0 <t <2/3.
If the vertex of the parabola lies on the OX axis, then Y0 = 0.
(3 * t – 2) / t = 0; t ≠ 0.
3 * t – 2 = 0;
t = 2/3;
The vertex of the parabola lies on the OX axis at t = 2/3.
Answer: Above the axis at 2/3 <t <0, below the axis at 0 <t <2/3, on the axis at t = 2/3.
