What is the vertex of f(x)=5×2 +20x−16 ?
The Correct Answer and Explanation is:
To find the vertex of the quadratic function ( f(x) = 5x^2 + 20x – 16 ), we can use the vertex formula derived from the standard form of a quadratic equation, ( f(x) = ax^2 + bx + c ). In this function, we have:
- ( a = 5 )
- ( b = 20 )
- ( c = -16 )
The vertex ( (h, k) ) of a parabola represented by the equation ( f(x) = ax^2 + bx + c ) can be found using the formula for the x-coordinate of the vertex:
[
h = -\frac{b}{2a}
]
Step 1: Calculate the x-coordinate of the vertex
Substituting the values of ( a ) and ( b ):
[
h = -\frac{20}{2 \times 5} = -\frac{20}{10} = -2
]
Step 2: Calculate the y-coordinate of the vertex
To find the y-coordinate ( k ), we substitute ( h = -2 ) back into the original function:
[
k = f(-2) = 5(-2)^2 + 20(-2) – 16
]
Calculating each term:
- ( 5(-2)^2 = 5 \times 4 = 20 )
- ( 20(-2) = -40 )
- Therefore,
[
k = 20 – 40 – 16 = 20 – 40 – 16 = -36
]
Conclusion
Thus, the vertex of the quadratic function ( f(x) = 5x^2 + 20x – 16 ) is at the point ( (-2, -36) ).
Significance of the Vertex
The vertex of a parabola is a crucial point as it represents the minimum or maximum value of the function. In this case, since ( a > 0 ) (the coefficient of ( x^2 ) is positive), the parabola opens upwards, indicating that the vertex represents the minimum point of the function. This point can also provide valuable information regarding the graph of the function, including its axis of symmetry, which in this case is the vertical line ( x = -2 ). Understanding the vertex allows for a better interpretation of the quadratic function and its behavior in various applications, including optimization problems in mathematics and real-life situations.