Answer:
Explanation:
The vertex and x-intercepts of a quadratic function are important features of its graph. The vertex is the point at which the function reaches its maximum or minimum value, and the x-intercepts are the points at which the function crosses the x-axis. The quadratic formula can be used to algebraically find the x-intercepts and vertex of a quadratic function.
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic function ax^2 + bx + c. This formula gives us the roots, or x-intercepts, of the quadratic equation. We can use these roots to find the x-coordinates of the x-intercepts of the corresponding quadratic function.
To find the y-coordinate of the vertex, we need to find the axis of symmetry, which is given by:
x = -b / 2a
This formula gives us the x-coordinate of the vertex. We can then substitute this value of x into the quadratic function to find the corresponding y-coordinate.
So, how does the quadratic formula define the relative locations of the vertex and x-intercepts of the graph of a quadratic function?
First, let's consider the discriminant b^2 - 4ac that appears under the square root in the quadratic formula. If this quantity is positive, then the quadratic equation has two real roots, which means the corresponding quadratic function intersects the x-axis at two points, giving two x-intercepts. If the discriminant is zero, then the quadratic equation has one real root, which means the corresponding quadratic function is tangent to the x-axis at one point, giving one x-intercept. If the discriminant is negative, then the quadratic equation has two complex conjugate roots, which means the corresponding quadratic function does not intersect the x-axis at all, giving no x-intercepts.
The sign of the coefficient a also determines the shape of the graph of the quadratic function. If a is positive, then the graph opens upward and the vertex is at the minimum point of the function. If a is negative, then the graph opens downward and the vertex is at the maximum point of the function.
In summary, the quadratic formula defines the x-intercepts of a quadratic function algebraically by giving the roots of the corresponding quadratic equation. The discriminant of the quadratic formula determines the number of x-intercepts and whether they are real or complex. The sign of the coefficient a determines the shape of the graph of the quadratic function and the location of the vertex, which is given by the axis of symmetry formula.