1. y= ax2 + bx + c, a>0 to determine what does b and c do.
2. Xv= -b/2a, where did the -b/2a come from.
1. In the equation y = ax^2 + bx + c, the variables b and c are coefficients that determine the shape and position of the quadratic curve.
- The coefficient b affects the linear term in the equation (bx), which represents the slope of the curve. If b is positive, the curve opens upward (concave up), and if b is negative, the curve opens downward (concave down). The larger the absolute value of b, the steeper the curve will be.
- The constant term c affects the y-intercept of the quadratic curve, which is the value of y when x = 0. If c is positive, the curve shifts upward, and if c is negative, the curve shifts downward. The larger the absolute value of c, the greater the shift from the origin.
2. The formula Xv = -b/2a is derived from the vertex form of a quadratic equation. The vertex form is given by y = a(x-h)^2 + k, where (h, k) represents the coordinates of the vertex (the maximum or minimum point) of the parabola.
To find the x-coordinate of the vertex, you can use the formula Xv = -b/2a. This formula is derived by taking the derivative of the quadratic equation and finding the value of x that makes the derivative zero. This corresponds to the x-coordinate of the vertex.
The "-b" part of the formula comes from negating the coefficient of the linear term in the equation (-b) so that it can be divided by 2a. The "2a" part is obtained by differentiating the equation and simplifying.
By substituting the values for a and b into the formula Xv = -b/2a, you can determine the x-coordinate of the vertex of the quadratic equation.