Find the vertex of the parabola.
[tex]y=2 x^2-16 x+34[/tex]
A. (-2, -4)
B. (-4, -2)
C. (4, 2)
D. (2, 4)
Answer (2)
The vertex of the parabola described by the equation y = 2 x 2 − 16 x + 34 is ( 4 , 2 ) , which can be found by completing the square and converting the equation into vertex form. Therefore, the answer is Option C: (4, 2).
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Factor out the coefficient of x 2 : y = 2 ( x 2 − 8 x ) + 34 .
Complete the square: y = 2 ( x 2 − 8 x + 16 − 16 ) + 34 = 2 (( x − 4 ) 2 − 16 ) + 34 .
Distribute and simplify: y = 2 ( x − 4 ) 2 − 32 + 34 = 2 ( x − 4 ) 2 + 2 .
Identify the vertex from the vertex form: The vertex is ( 4 , 2 ) .
Explanation
Problem Analysis We are given the equation of a parabola y = 2 x 2 − 16 x + 34 and asked to find its vertex. The vertex form of a parabola is given by y = a ( x − h ) 2 + k , where ( h , k ) represents the vertex of the parabola. To find the vertex, we need to convert the given equation into vertex form by completing the square.
Factoring First, factor out the coefficient of the x 2 term (which is 2) from the first two terms of the equation: y = 2 ( x 2 − 8 x ) + 34
Completing the Square Next, complete the square for the expression inside the parenthesis. To do this, take half of the coefficient of the x term (which is -8), square it, and add and subtract it inside the parenthesis. Half of -8 is -4, and ( − 4 ) 2 = 16 . So we have: y = 2 ( x 2 − 8 x + 16 − 16 ) + 34
Rewriting the Expression Rewrite the expression inside the parenthesis as a squared term: y = 2 (( x − 4 ) 2 − 16 ) + 34
Distributing Distribute the 2 to both terms inside the parenthesis: y = 2 ( x − 4 ) 2 − 32 + 34
Finding the Vertex Finally, simplify the equation: y = 2 ( x − 4 ) 2 + 2 Now the equation is in vertex form, y = a ( x − h ) 2 + k , where a = 2 , h = 4 , and k = 2 . Therefore, the vertex of the parabola is ( h , k ) = ( 4 , 2 ) .
Final Answer The vertex of the parabola is ( 4 , 2 ) . Comparing this to the given options, we see that option c. ( 4 , 2 ) is the correct answer.
Examples
Understanding parabolas is crucial in various real-world applications, such as designing satellite dishes or understanding projectile motion. For instance, when designing a satellite dish, engineers use the properties of a parabola to focus incoming signals at a single point, the focus, where the receiver is placed. The vertex of the parabola plays a key role in determining the optimal placement and orientation of the dish to maximize signal reception. Similarly, in physics, the trajectory of a projectile (like a ball thrown in the air) follows a parabolic path, and understanding the vertex helps determine the maximum height reached by the projectile.
