Each equation given below describes a parabola. Which statement best compares their graphs?
x=5y²
x=3y²
A. Both parabolas open upward, and x = 5y² is wider than x = 3y².
B. Both parabolas open to the right, and x = 5y² is wider than x = 3y².
C. Both parabolas open to the right, and x = 3y² is wider than x = 5y².
D. Both parabolas open upward, and x = 3y² is wider than x = 5y².
Answer (2)
Both parabolas open to the right.
The width of a parabola x = a y 2 is inversely proportional to a .
Since 3 < 5 , the parabola x = 3 y 2 is wider than x = 5 y 2 .
Therefore, the correct answer is C: Both parabolas open to the right, and x = 3y² is wider than x = 5y². C
Explanation
Understanding the Problem We are given two equations of parabolas: x = 5 y 2 and x = 3 y 2 . We need to determine which statement best compares their graphs.
General Form of the Parabola The general form of a parabola opening to the right is x = a y 2 , where a is a constant. If 0"> a > 0 , the parabola opens to the right. The larger the value of a , the narrower the parabola, and the smaller the value of a , the wider the parabola.
Direction of the Parabolas In the given equations, x = 5 y 2 and x = 3 y 2 , both have the form x = a y 2 with 0"> a > 0 . Therefore, both parabolas open to the right.
Width of the Parabolas Comparing the coefficients, we have a = 5 for the first parabola and a = 3 for the second parabola. Since 3 < 5 , the parabola x = 3 y 2 is wider than the parabola x = 5 y 2 .
Conclusion Therefore, both parabolas open to the right, and x = 3 y 2 is wider than x = 5 y 2 .
Examples
Parabolas are useful in designing reflective surfaces, such as those found in satellite dishes and solar ovens. The shape of the parabola focuses incoming rays to a single point, which is why these shapes are so effective. Understanding how the equation of a parabola affects its width and direction helps engineers design more efficient and effective devices.
Both parabolas x = 5 y 2 and x = 3 y 2 open to the right, with the parabola x = 3 y 2 being wider than x = 5 y 2 . Therefore, the correct answer is option C. The width of a parabola is inversely proportional to the coefficient in front of y 2 .
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