: In case of a quadratic function, why are there two x-intercepts and one y-intercept?
Let's say x^2 = y, which is a quadratic function.
For every value of y, there are two possibilities for x.
For example, if y is 4, x can either be 2 or -2.
To understand why there are two x-intercepts and one y-intercept in the case of a quadratic function, let's start by visualizing a quadratic function graphically.
A quadratic function is represented by a parabola, which is a U-shaped curve. This curve intersects the x-axis at two points, known as the x-intercepts or roots. These points are the values of x for which the function evaluates to zero. Graphically, these are the points where the parabola crosses or touches the x-axis.
On the other hand, the y-intercept is the point at which the parabola intersects the y-axis. It represents the value of the function when x is equal to zero.
Now, let's consider the equation x^2 = y, which is a simple quadratic function. When we solve this equation for y, we get y = x^2. In this case, the quadratic function is a perfect square, which means it has a vertex at the origin (0,0).
Since the vertex is at the origin, when x is equal to zero, y will also be equal to zero. This corresponds to the y-intercept, which is the point (0,0).
When we plug in different values of y, we can find the corresponding values of x. For example, if y is 4, we can take the square root of 4 to get x = ±2. This means that for y = 4, the quadratic function intersects the x-axis at x = 2 and x = -2. These are the x-intercepts.
Similarly, for any given y, there are two possibilities for x. This is because the quadratic function is symmetric about the y-axis, and the parabola touches the x-axis in two places.
In summary, in the case of a quadratic function, there are two x-intercepts because the parabola intersects the x-axis at two points. These points represent the solutions to the quadratic equation. There is also one y-intercept, which is the value of the function when x is equal to zero.