Find the vertex of the parabola.
[tex]f(x)=5 x^2-30 x+49[/tex]
a. (3,0)
c. (4,0)
b. (3,4)
d. (4,3)
Answer (2)
The vertex of the parabola given by the function f ( x ) = 5 x 2 − 30 x + 49 is ( 3 , 4 ) .
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Use the formula x = − 2 a b to find the x-coordinate of the vertex: x = − 2 ( 5 ) − 30 = 3 .
Plug x = 3 into the function f ( x ) = 5 x 2 − 30 x + 49 to find the y-coordinate: f ( 3 ) = 5 ( 3 ) 2 − 30 ( 3 ) + 49 = 4 .
The vertex of the parabola is ( 3 , 4 ) .
The answer is ( 3 , 4 ) .
Explanation
Understanding the Problem We are given the quadratic function f ( x ) = 5 x 2 − 30 x + 49 and asked to find the vertex of the parabola it represents. The vertex form of a parabola is given by f ( x ) = a ( x − h ) 2 + k , where ( h , k ) is the vertex. We can find the vertex by completing the square or by using the formula x = − 2 a b to find the x-coordinate of the vertex, and then plugging that value into the function to find the y-coordinate.
Finding the x-coordinate Let's use the formula x = − 2 a b to find the x-coordinate of the vertex. In our equation, a = 5 and b = − 30 . Plugging these values into the formula, we get: x = − 2 ( 5 ) − 30 = 10 30 = 3 So, the x-coordinate of the vertex is 3.
Finding the y-coordinate Now, let's plug x = 3 into the function f ( x ) = 5 x 2 − 30 x + 49 to find the y-coordinate of the vertex: f ( 3 ) = 5 ( 3 ) 2 − 30 ( 3 ) + 49 = 5 ( 9 ) − 90 + 49 = 45 − 90 + 49 = − 45 + 49 = 4 So, the y-coordinate of the vertex is 4.
Stating the Vertex Therefore, the vertex of the parabola is ( 3 , 4 ) .
Examples
Understanding the vertex of a parabola is crucial in various real-world applications. For instance, when designing a suspension bridge, engineers need to determine the lowest point of the cable's curve, which can be modeled as a parabola. Similarly, in projectile motion, the vertex represents the maximum height reached by an object, helping to predict its trajectory. Knowing the vertex allows for optimizing designs and predicting outcomes in fields ranging from engineering to physics.
