Which of the following best describe the graph of a quadratic equation with one real root?
A: It has 2 x-intercepts
B: It does not cross the x-axis
C: Its vertex lies on the x-axis
D: Its vertex lies on the y-axis
Well, if a quadratic equation has one real root, that means it only touches the x-axis once. So, options A and B can be eliminated.
Now, let's think about the vertex. The vertex is the highest or lowest point on the graph, which lies on the axis of symmetry. For a quadratic equation with one real root, the vertex would be right there on the x-axis. So, option C is spot-on!
As for option D, the vertex is on the y-axis? That's a good one, but no. The vertex of a quadratic equation with one real root would definitely not be on the y-axis.
So, the answer is C: Its vertex lies on the x-axis.
The correct answer is C: Its vertex lies on the x-axis.
Explanation:
A quadratic equation represents a parabola, which can have one real root when the parabola touches the x-axis at a single point. This is also known as a tangent. In this case, the vertex of the parabola coincides with the x-axis.
To determine which of the given options best describe the graph of a quadratic equation with one real root, we need to understand the characteristics of such graphs.
A quadratic equation can be written in the form of y = ax^2 + bx + c, where a, b, and c are constants. The graph of this equation is a parabola.
In the case of a quadratic equation with one real root, the parabola intersects the x-axis at only one point. This implies that the parabola does not have two x-intercepts. Therefore, option A can be eliminated.
Since the graph of the equation intersects the x-axis at one point, it is not accurate to say that it does not cross the x-axis at all. Thus, option B can also be eliminated.
Now, let's consider the remaining options:
C: Its vertex lies on the x-axis
D: Its vertex lies on the y-axis
The vertex of a quadratic equation with one real root is a crucial point on the graph. It represents the lowest or highest point of the parabola, depending on whether the quadratic equation opens upward (a > 0) or downward (a < 0).
For a quadratic equation with one real root, the vertex lies on the x-axis. This is because the parabola intersects the x-axis at the point where the real root is located. Therefore, option C correctly describes the graph of a quadratic equation with one real root.
On the other hand, the vertex of a quadratic equation can never lie on the y-axis since the y-axis represents the vertical line where x = 0. The vertex only represents the lowest or highest point on the graph horizontally, not vertically. Thus, option D can be eliminated.
To summarize, the option that best describes the graph of a quadratic equation with one real root is C: Its vertex lies on the x-axis.
It could be either C or D.
x = ay^2 has a vertex on the y axis.
y = bx^2 has a vertex on the x axis.
a and b are constants