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 (1)
Identify the coefficients: a = 3 and b = − 18 from the quadratic function.
Calculate the x-coordinate of the vertex using the formula: x = − 2 a b = − 2 ( 3 ) − 18 = 3 .
Substitute x = 3 into the function to find the y-coordinate: f ( 3 ) = 3 ( 3 ) 2 − 18 ( 3 ) + 2 = − 25 .
State the vertex as an ordered pair: ( 3 , − 25 ) .
Explanation
Understanding the Problem 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. The vertex represents the minimum or maximum point of the parabola.
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 found using 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 So, the x-coordinate of the vertex is 3.
Finding the y-coordinate To find the y-coordinate of the vertex, we substitute the x-coordinate ( x = 3 ) back into the original function: f ( 3 ) = 3 ( 3 ) 2 − 18 ( 3 ) + 2 = 3 ( 9 ) − 54 + 2 = 27 − 54 + 2 = − 27 + 2 = − 25 Thus, the y-coordinate of the vertex is -25.
Stating 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 instance, if you're launching a projectile, the vertex represents the maximum height the projectile will reach. Similarly, in business, if a quadratic function models profit, the vertex indicates the point of maximum profit. Knowing how to find the vertex allows you to optimize processes and predict outcomes in these scenarios. For example, a company might use this to determine the optimal pricing strategy to maximize their revenue, where the price is 'x' and the revenue is 'f(x)'.
