Which of the following statements is FALSE?
1.It is possible that for a parabola and a line to intersect exactly once
2.The y-intercept of the quadratic function is always higher than the y-coordinate of the vertex.
3.If a quadratic function has no x-intercept, then its discriminant must be less than 0.
4.Assuming a parabola has 2 x-intercepts, then the vertex would be exactly halfway between them.
help !
2. draw an upside down parabola (sheds water) with a vertex not on the y axis
To determine which statement is false, we need to evaluate each statement one by one:
1. It is possible that for a parabola and a line to intersect exactly once:
This statement is true. A parabola and a line can intersect at one point if they are not parallel and do not coincide.
2. The y-intercept of the quadratic function is always higher than the y-coordinate of the vertex:
This statement is false. The y-coordinate of the vertex can be higher or lower than the y-intercept, depending on the shape of the parabola.
3. If a quadratic function has no x-intercept, then its discriminant must be less than 0:
This statement is true. The discriminant (b^2 - 4ac) determines the nature of the roots. If there are no x-intercepts, it means the quadratic equation has no real roots, and the discriminant must be less than 0.
4. Assuming a parabola has 2 x-intercepts, then the vertex would be exactly halfway between them:
This statement is false. The vertex of a parabola does not necessarily lie exactly halfway between the x-intercepts. The position of the vertex depends on the coefficients of the quadratic equation.
Therefore, the false statement is statement number 2: "The y-intercept of the quadratic function is always higher than the y-coordinate of the vertex."
To determine which statement is false, let's analyze each one:
1. It is possible for a parabola and a line to intersect exactly once. This statement is true. The intersection point will occur when the values of both the parabola and the line are equal.
2. The y-intercept of a quadratic function is the value of y when x equals zero. The y-coordinate of the vertex is the highest or lowest point on the parabola, depending on whether it opens upwards or downwards. Generally, the y-intercept will not be higher than the y-coordinate of the vertex, but there may be cases where they overlap, which would make this statement possibly true. Thus, we cannot say this statement is always false.
3. The discriminant of a quadratic function is the value inside the square root of the quadratic formula, which is b^2 - 4ac. It determines the nature of the solutions of the quadratic equation or the number of x-intercepts. If a quadratic function has no x-intercept, it means it does not intersect the x-axis. In this case, the discriminant must be less than zero. Therefore, this statement is true.
4. Assuming a parabola has two x-intercepts, the vertex would lie on the axis of symmetry between these two points. In other words, the vertex would be exactly halfway between them. Thus, this statement is true.
Based on the above analysis, statement 2 is the one that cannot be proven false with certainty.