Find the standard equation of a parabola that has a vertical axis and satisfies the given conditions: vertex $(3,-3), x$-intercept -2
Answer (2)
Substitute the vertex ( 3 , − 3 ) into the standard equation of a parabola with a vertical axis: ( x − 3 ) 2 = 4 p ( y + 3 ) .
Use the x-intercept ( − 2 , 0 ) to solve for p : 25 = 12 p , so p = 12 25 .
Substitute the value of p back into the equation: ( x − 3 ) 2 = 3 25 ( y + 3 ) .
Simplify the equation to get the standard form: 3 ( x − 3 ) 2 = 25 ( y + 3 ) .
Explanation
Problem Analysis We are given the vertex ( 3 , − 3 ) and the x -intercept − 2 of a parabola with a vertical axis. Our goal is to find the standard equation of this parabola.
Standard Equation The standard equation of a parabola with a vertical axis is given by ( x − h ) 2 = 4 p ( y − k ) , where ( h , k ) is the vertex of the parabola and p is the distance from the vertex to the focus.
Substituting the Vertex We are given the vertex ( 3 , − 3 ) , so we can substitute h = 3 and k = − 3 into the standard equation: ( x − 3 ) 2 = 4 p ( y − ( − 3 )) ( x − 3 ) 2 = 4 p ( y + 3 )
Solving for p We are also given that the x -intercept is − 2 , which means the point ( − 2 , 0 ) lies on the parabola. We can substitute x = − 2 and y = 0 into the equation to solve for p : ( − 2 − 3 ) 2 = 4 p ( 0 + 3 ) ( − 5 ) 2 = 12 p 25 = 12 p p = 12 25
Substituting p back into the equation Now we substitute the value of p back into the equation: ( x − 3 ) 2 = 4 ( 12 25 ) ( y + 3 ) ( x − 3 ) 2 = 3 25 ( y + 3 )
Final Equation To get rid of the fraction, we can multiply both sides of the equation by 3: 3 ( x − 3 ) 2 = 25 ( y + 3 ) This is the standard equation of the parabola.
Examples
Parabolas are commonly seen in the real world, such as the path of a projectile (like a ball being thrown) or the shape of a satellite dish. Understanding how to find the equation of a parabola given certain conditions allows us to model and predict these phenomena. For instance, if we know the vertex and another point on the path of a thrown ball, we can determine the equation of the parabola and thus predict where the ball will land. Similarly, the shape of a satellite dish is designed as a parabola to focus incoming signals to a single point, and knowing the equation helps in designing the dish for optimal performance. The equation we found, 3 ( x − 3 ) 2 = 25 ( y + 3 ) , could represent the cross-section of such a dish, with the vertex at (3, -3).
The standard equation of the parabola with a vertex at (3, -3) and an x-intercept at -2 is given by 3(x-3)² = 25(y+3). This is derived by first substituting the vertex into the standard form and then solving for p using the x-intercept. Finally, we normalized the equation to eliminate fractions.
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