I'm studying year 11 mathematics at the moment. I'm currently learning about parabolas and quadratics. I have just come across a chapter on the discriminant. In the beginning of the chapter, this is stated: https://gyazo.com/4b1f643a4228e8180433150175801ad5
I was wondering. If a parabola was created where you switch the x and y variables in the equation of, say, $y=x^2$ so it becomes $x=y^2$, would this not also create only one $x$-axis intercept?
On
The statement from your book is intended to only apply to parabola with a vertical axis of symmetry, which is by far the most common case studied in elementary texts.
If you orient the parabolas differently you get other situations. In addition to your idea of using a parabola with a horizontal axis of symmetry, you can also tilt it 45 degrees. Like this,
the parabola defined by the equation $$x+y-\frac14=(x-y)^2.$$
I invite you to prove yourself that it touches the $x$-axis only at the point $(1/2,0)$. Note that its vertex (= the turning point) is at the point $(1/8,1/8)$, so it does not fall under the umbrella of that statement either. Even though the curve is entirely above the $x$-axis.
We can use any tilt angle we want, and can still arrange the parabola to only have a single intercept with the $x$-axis. The case of a horizontal axis of symmetry is exceptional in the sense that only then can we have branches of the parabola on opposite sides of the $x$-axis as well as a single intercept.
It doesn't make much sense to do that. Of course, it depends strictly on the choice of the axis. It turns out that when you change the variable, the properties are still valid, however, it is necessary to change the axis.