$g(x)=\frac{1}{5} x^2-2 x-1$
Answer (1)
Find the vertex of the parabola using x = − 2 a b and g ( x ) . The vertex is ( 5 , − 6 ) .
Determine the x-intercepts by setting g ( x ) = 0 and using the quadratic formula. The x-intercepts are approximately -0.477 and 10.477.
Find the y-intercept by setting x = 0 . The y-intercept is ( 0 , − 1 ) .
The parabola opens upwards, and the minimum value is -6. The final answer is the analysis of the function. Vertex: ( 5 , − 6 ) , x-intercepts: − 0.477 , 10.477 , y-intercept: ( 0 , − 1 )
Explanation
Analyzing the Function We are given the quadratic function g ( x ) = 5 1 x 2 − 2 x − 1 . Our goal is to analyze this function by finding its key features: vertex, x-intercepts, y-intercept, and axis of symmetry. This will allow us to sketch the graph of the function and understand its behavior.
Finding the Vertex To find the vertex of the parabola, we first need to find the x-coordinate of the vertex, which is given by x = − 2 a b . In our case, a = 5 1 and b = − 2 . Plugging these values into the formula, we get: x = − 2 ( 5 1 ) − 2 = 5 2 2 = 2 ⋅ 2 5 = 5 Now, to find the y-coordinate of the vertex, we substitute x = 5 into the function: g ( 5 ) = 5 1 ( 5 ) 2 − 2 ( 5 ) − 1 = 5 1 ( 25 ) − 10 − 1 = 5 − 10 − 1 = − 6 So, the vertex of the parabola is ( 5 , − 6 ) .
Finding the X-Intercepts To find the x-intercepts, we set g ( x ) = 0 and solve for x :
5 1 x 2 − 2 x − 1 = 0 We can use the quadratic formula to solve for x : x = 2 a − b ± b 2 − 4 a c . In our case, a = 5 1 , b = − 2 , and c = − 1 . Plugging these values into the formula, we get: x = 2 ( 5 1 ) − ( − 2 ) ± ( − 2 ) 2 − 4 ( 5 1 ) ( − 1 ) = 5 2 2 ± 4 + 5 4 = 5 2 2 ± 5 24 = 5 2 2 ± 2 5 6 = 5 ( 1 ± 5 6 ) So, the x-intercepts are x = 5 ( 1 + 5 6 ) ≈ 10.477 and x = 5 ( 1 − 5 6 ) ≈ − 0.477 .
Finding the Y-Intercept To find the y-intercept, we set x = 0 and evaluate g ( 0 ) :
g ( 0 ) = 5 1 ( 0 ) 2 − 2 ( 0 ) − 1 = − 1 So, the y-intercept is ( 0 , − 1 ) .
Determining the Axis of Symmetry The axis of symmetry is a vertical line that passes through the vertex. Since the x-coordinate of the vertex is 5, the axis of symmetry is x = 5 .
Determining the Direction of the Parabola Since the coefficient of the x 2 term is positive ( 0"> a = 5 1 > 0 ), the parabola opens upwards. This means that the vertex represents the minimum point of the function.
Finding the Minimum Value The minimum value of the function is the y-coordinate of the vertex, which is -6.
Summary of Key Features In summary, the key features of the function g ( x ) = 5 1 x 2 − 2 x − 1 are:
Vertex: ( 5 , − 6 )
X-intercepts: approximately -0.477 and 10.477
Y-intercept: ( 0 , − 1 )
Axis of symmetry: x = 5
Opens upwards
Minimum value: -6
Examples
Understanding quadratic functions is crucial in various real-world applications. For example, engineers use quadratic equations to model the trajectory of projectiles, such as rockets or balls. By analyzing the vertex, intercepts, and axis of symmetry, they can determine the maximum height, range, and optimal launch angle. Similarly, economists use quadratic functions to model cost and revenue curves, helping businesses optimize production and pricing strategies to maximize profits. This analysis provides valuable insights into the behavior and characteristics of the function.
