Graph the function.
[tex]$h(x)=-\frac{1}{5} x^2+2 x$[/tex]
Answer (1)
Find the x-coordinate of the vertex using x = − 2 a b , which gives x = 5 .
Calculate the y-coordinate of the vertex by plugging x = 5 into the function, resulting in y = 5 .
Determine the x-intercepts by setting h ( x ) = 0 and solving for x , yielding x = 0 and x = 10 .
Find the y-intercept by setting x = 0 , which gives y = 0 .
The graph is a parabola with vertex ( 5 , 5 ) , x-intercepts at 0 and 10 , and y-intercept at 0 .
Explanation
Analyze the problem We are asked to graph the function h ( x ) = − 5 1 x 2 + 2 x . This is a quadratic function, so its graph is a parabola. Since the coefficient of the x 2 term is negative, the parabola opens downward. To graph the parabola, we need to find the vertex, x-intercepts, and y-intercept.
Find the vertex x-coordinate The x-coordinate of the vertex is given by the formula x = − 2 a b , where a = − 5 1 and b = 2 . Plugging in these values, we get x = − 2 ( − 5 1 ) 2 = − − 5 2 2 = 5.
Find the vertex y-coordinate To find the y-coordinate of the vertex, we plug the x-coordinate into the function: h ( 5 ) = − 5 1 ( 5 ) 2 + 2 ( 5 ) = − 5 1 ( 25 ) + 10 = − 5 + 10 = 5. So, the vertex of the parabola is ( 5 , 5 ) .
Find the x-intercepts To find the x-intercepts, we set h ( x ) = 0 and solve for x :
− 5 1 x 2 + 2 x = 0 x ( − 5 1 x + 2 ) = 0 So, x = 0 or − 5 1 x + 2 = 0 . Solving for x in the second equation, we get − 5 1 x = − 2 x = 10. Thus, the x-intercepts are x = 0 and x = 10 .
Find the y-intercept To find the y-intercept, we set x = 0 and evaluate h ( 0 ) :
h ( 0 ) = − 5 1 ( 0 ) 2 + 2 ( 0 ) = 0. So, the y-intercept is y = 0 .
Plot the points and sketch the parabola Now we plot the vertex ( 5 , 5 ) , the x-intercepts ( 0 , 0 ) and ( 10 , 0 ) , and the y-intercept ( 0 , 0 ) on a coordinate plane. Then, we sketch the parabola through these points, remembering that it opens downward.
Final Answer The graph of the function h ( x ) = − 5 1 x 2 + 2 x is a parabola that opens downward, with vertex at ( 5 , 5 ) , x-intercepts at x = 0 and x = 10 , and y-intercept at y = 0 .
Examples
Understanding quadratic functions and their graphs is crucial in many real-world applications. For instance, when designing a bridge, engineers use parabolas to model the arch's shape, ensuring structural stability and efficient load distribution. Similarly, in sports, the trajectory of a ball thrown or kicked can be modeled using a parabola, helping athletes optimize their performance. By analyzing the vertex and intercepts of the parabola, one can determine the maximum height and range of the projectile, which is essential for strategic planning and execution.
