A quadratic function f is given by f(x) = ax^(2) + bx + c where a is not 0 . Select all the statements that must be true about the graph of f. (A) The y-intercept of the graph is at (0,c). (B) The graph has an x-intercept at (c,0). (C) When a

Question

A quadratic function f is given by f(x) = ax^(2) + bx + c where a is not 0 . Select all the statements that must be true about the graph of f. (A) The y-intercept of the graph is at (0,c). (B) The graph has an x-intercept at (c,0). (C) When a < 0 the graph opens downward. (D) The graph has two x-intercepts. (E) If b=0, then the vertex is on the y-axis.

Bytelearn · Accepted Answer

Y-Intercept Definition: The y-intercept of a graph is the point where the graph crosses the y-axis. This occurs when x = 0. Substituting x = 0 into the equation f(x) = ax^2 + bx + c gives f(0) = a(0)^2 + b(0) + c = c. Therefore, the y-intercept is at (0, c).X-Intercepts and Statement B: The x-intercepts of a graph are the points where the graph crosses the x-axis. These occur when f(x) = 0. Setting f(x) = 0 gives ax^2 + bx + c = 0. The statement B suggests that there is an x-intercept at (c, 0), which would imply that f(c) = 0. However, f(c) = ac^2 + bc + c, which is not necessarily zero unless a, b, and c satisfy a specific relationship. Therefore, statement B is not necessarily true.Parabola Direction: When a < 0, the parabola opens downward because the coefficient a determines the direction of the opening. If a is positive, the parabola opens upward, and if a is negative, the parabola opens downward. This is a fundamental property of quadratic functions.Quadratic Function X-Intercepts: The graph of a quadratic function can have zero, one, or two x-intercepts depending on the discriminant (b^2 - 4ac). If the discriminant is positive, there are two real and distinct x-intercepts. If it is zero, there is exactly one x-intercept (the vertex). If it is negative, there are no real x-intercepts. Therefore, statement D is not necessarily true.Quadratic Function Vertex: If b = 0, the quadratic function simplifies to f(x) = ax^2 + c. The vertex of this parabola is at the point where the derivative of f with respect to x is zero. Taking the derivative gives f'(x) = 2ax. Setting f'(x) = 0 gives x = 0. Therefore, the vertex is at x = 0, which is on the y-axis.

Full solution

More problems from Write equations of cosine functions using properties

Related topics