The graph of y = 3x^2 - 2x + 1 is a U-shaped parabola opening upward, with a minimum point.
The graph of the quadratic function y = 3x^2 - 2x + 1 is a parabola, a type of polynomial curve with a distinct U-shape. Specifically, since the leading coefficient (the coefficient of the x² term) is positive (3 in this case), the parabola opens upwards. This means that as you move along the x-axis in either direction, the values of y increase indefinitely, creating a concave shape.
The vertex of the parabola, which is the minimum point, occurs at the axis of symmetry. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. In this case, the axis of symmetry can be found using the formula (x =
), where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c. Once the axis of symmetry is determined, substituting this value back into the original equation yields the y-coordinate of the vertex.
The shape of the parabola also depends on the discriminant b^2 - 4ac of the quadratic equation. If the discriminant is positive, the parabola intersects the x-axis at two distinct points, creating a standard upward-opening parabola. If the discriminant is zero, the parabola touches the x-axis at exactly one point, and if it's negative, the parabola does not intersect the x-axis at all. These characteristics collectively contribute to the unique appearance of the graph y = 3x^2 - 2x + 1, reflecting the behavior of quadratic functions in general.
Question;
Is the graph of y=3x²+2x+1 a line, parabola, or some other shape?