To determine which parabolas Lauren could be describing, we need to understand the relationship between the standard form of a parabola and its focus.
Parabolas can be written in different forms, but we are particularly interested in parabolas of the form [tex]\( y^2 = 4ax \)[/tex]. For such parabolas, the focus is located at [tex]\( (a, 0) \)[/tex]. Here are the conditions:
1. If [tex]\( a > 0 \)[/tex], the focus lies on the positive x-axis, which means that the parabola opens to the right.
2. If [tex]\( a < 0 \)[/tex], the focus lies on the negative x-axis, which means that the parabola opens to the left.
Given the problem, we need to identify which of the provided parabolas have a focus with a positive, nonzero x-coordinate.
Let's analyze each equation:
1. [tex]\( x^2 = 4y \)[/tex]:
- This is not in the form [tex]\( y^2 = 4ax \)[/tex]. Instead, it is of the form [tex]\( x^2 = 4ay \)[/tex], which is a parabola that opens upwards and the focus is along the y-axis rather than the x-axis. Thus, it does not meet the requirement.
2. [tex]\( x^2 = -6y \)[/tex]:
- Similarly, this is also of the form [tex]\( x^2 = 4ay \)[/tex]. Here, [tex]\( a \)[/tex] would be negative, and the parabola opens downwards. The focus is also on the y-axis, not on the x-axis.
3. [tex]\( y^2 = x \)[/tex]:
- Rewriting this as [tex]\( y^2 = 1 \cdot x \)[/tex], we see it matches the form [tex]\( y^2 = 4ax \)[/tex] with [tex]\( a = 1/4 \)[/tex], which is positive. Therefore, the focus is [tex]\( (1/4, 0) \)[/tex], satisfying the positive x-coordinate condition.
4. [tex]\( y^2 = 10x \)[/tex]:
- This matches the form [tex]\( y^2 = 4ax \)[/tex] with [tex]\( a = 10/4 = 2.5 \)[/tex], which is positive. Thus, the focus is [tex]\( (2.5, 0) \)[/tex], satisfying the positive x-coordinate condition.
5. [tex]\( y^2 = -3x \)[/tex]:
- This matches the form [tex]\( y^2 = 4ax \)[/tex] with [tex]\( a = -3/4 \)[/tex], which is negative. Therefore, the focus is [tex]\( (-3/4, 0) \)[/tex], not satisfying the condition of having a positive x-coordinate.
6. [tex]\( y^2 = 5x \)[/tex]:
- This matches the form [tex]\( y^2 = 4ax \)[/tex] with [tex]\( a = 5/4 = 1.25 \)[/tex], which is positive. Therefore, the focus is [tex]\( (1.25, 0) \)[/tex], satisfying the positive x-coordinate condition.
After analyzing each equation, the parabolas that Lauren could be describing are:
[tex]\[ y^2 = x \][/tex]
[tex]\[ y^2 = 10x \][/tex]
[tex]\[ y^2 = 5x \][/tex]
Thus, the correct set of parabolas are options 3, 4, and 6.
