Given the equations, identify which way the parabola opens by matching an equation on the left with a term on the right.
| $y^2=6 x$ | $\square$ | left |
| $x^2=3 y$ | $\square$ | up |
| $y^2=-2 x$ | $\square$ | down |
| $x^2=-10 y$ | $\square$ | right |
Answer (1)
y 2 = 6 x opens to the right.
x 2 = 3 y opens upwards.
y 2 = − 2 x opens to the left.
x 2 = − 10 y opens downwards.
The final answer is matching each equation with the direction in which the parabola opens.
Explanation
Understanding the Problem We are given four equations of parabolas and asked to match each with the direction in which it opens. The standard forms of parabolas are:
y 2 = 4 a x opens to the right.
y 2 = − 4 a x opens to the left.
x 2 = 4 a y opens upwards.
x 2 = − 4 a y opens downwards.
Matching the Equations
y 2 = 6 x : This equation is in the form y 2 = 4 a x , where 4 a = 6 . Since the coefficient of x is positive, the parabola opens to the right.
x 2 = 3 y : This equation is in the form x 2 = 4 a y , where 4 a = 3 . Since the coefficient of y is positive, the parabola opens upwards.
y 2 = − 2 x : This equation is in the form y 2 = − 4 a x , where − 4 a = − 2 . Since the coefficient of x is negative, the parabola opens to the left.
x 2 = − 10 y : This equation is in the form x 2 = − 4 a y , where − 4 a = − 10 . Since the coefficient of y is negative, the parabola opens downwards.
Final Answer Therefore, the matches are:
y 2 = 6 x opens to the right.
x 2 = 3 y opens upwards.
y 2 = − 2 x opens to the left.
x 2 = − 10 y opens downwards.
Examples
Understanding the direction a parabola opens is crucial in various real-world applications. For instance, satellite dishes and radar antennas are designed with parabolic shapes to focus incoming signals to a single point. The orientation of the parabola determines the direction from which the signals are received most effectively. Similarly, in architecture, parabolic arches are used for their structural strength, and knowing the direction of the parabola helps in designing stable and aesthetically pleasing structures. By recognizing the relationship between the equation and the direction of a parabola, engineers and designers can optimize the performance and appearance of these applications.
