In the following exercise, find the coordinates of the vertex for the parabola defined by the given quadratic function.
[tex]$f(x)=3 x^2-18 x+2$[/tex]
The vertex is $\square$ (Type an ordered pair.)
Answer (2)
Find the x-coordinate of the vertex using the formula x = − 2 a b , where a = 3 and b = − 18 , resulting in x = 3 .
Substitute x = 3 into the function f ( x ) = 3 x 2 − 18 x + 2 to find the y-coordinate.
Calculate f ( 3 ) = 3 ( 3 ) 2 − 18 ( 3 ) + 2 = − 25 .
The vertex of the parabola is ( 3 , − 25 ) , so the answer is ( 3 , − 25 ) .
Explanation
Problem Analysis We are given the quadratic function f ( x ) = 3 x 2 − 18 x + 2 and asked to find the coordinates of the vertex of the parabola it defines.
Finding the x-coordinate The x-coordinate of the vertex of a parabola given by the quadratic function f ( x ) = a x 2 + b x + c is given by the formula x = − 2 a b . In our case, a = 3 and b = − 18 .
Calculating the x-coordinate Substituting the values of a and b into the formula, we get: x = − 2 ( 3 ) − 18 = 6 18 = 3
Finding the y-coordinate Now that we have the x-coordinate of the vertex, we can find the y-coordinate by substituting x = 3 into the original function: f ( 3 ) = 3 ( 3 ) 2 − 18 ( 3 ) + 2 = 3 ( 9 ) − 54 + 2 = 27 − 54 + 2 = − 25
The Vertex Therefore, the coordinates of the vertex are ( 3 , − 25 ) .
Examples
Understanding the vertex of a parabola is crucial in various real-world applications. For example, if you're launching a projectile, the vertex represents the maximum height the projectile will reach. Similarly, in business, if a quadratic function represents the profit of a company, the vertex can indicate the point at which the maximum profit is achieved. Knowing how to find the vertex allows you to optimize processes and make informed decisions in many fields.
The coordinates of the vertex for the parabola defined by the function f ( x ) = 3 x 2 − 18 x + 2 are ( 3 , − 25 ) . To find this, we calculated the x-coordinate using the formula x = − 2 a b and then substituted this value back into the function to find the y-coordinate. Thus, the vertex is located at ( 3 , − 25 ) .
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