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 .
Find the y-coordinate of the vertex by plugging x = 5 into h ( x ) , which gives h ( 5 ) = 5 .
Find the x-intercepts by setting h ( x ) = 0 and solving for x , which gives x = 0 and x = 10 .
The graph is a parabola opening downward with vertex ( 5 , 5 ) and x-intercepts ( 0 , 0 ) and ( 10 , 0 ) .
Explanation
Understanding the Function We are given the quadratic function h ( x ) = − 5 1 x 2 + 2 x . Our goal is to graph this function, which is a parabola.
Finding the Vertex To graph the parabola, we need to find the vertex and the x-intercepts. The vertex is the highest point on the parabola since the coefficient of the x 2 term is negative. The x-coordinate of the vertex can be found using the formula x = − 2 a b , where a = − 5 1 and b = 2 .
Calculating the x-coordinate of the Vertex Plugging in the values of a and b , we get: x = − 2 ( − 5 1 ) 2 = − − 5 2 2 = 5 So, the x-coordinate of the vertex is 5.
Calculating the y-coordinate of the Vertex Now, we need to find the y-coordinate of the vertex by plugging x = 5 into the function h ( x ) : h ( 5 ) = − 5 1 ( 5 ) 2 + 2 ( 5 ) = − 5 1 ( 25 ) + 10 = − 5 + 10 = 5 So, the y-coordinate of the vertex is 5. Therefore, the vertex of the parabola is ( 5 , 5 ) .
Finding the x-intercepts Next, we need to find the x-intercepts by setting h ( x ) = 0 and solving for x : − 5 1 x 2 + 2 x = 0 We can factor out an x from the equation: x ( − 5 1 x + 2 ) = 0 This gives us two possible solutions for x :
x = 0
− 5 1 x + 2 = 0 ⇒ − 5 1 x = − 2 ⇒ x = 10 So, the x-intercepts are x = 0 and x = 10 .
Sketching the Parabola Now we have the vertex ( 5 , 5 ) and the x-intercepts ( 0 , 0 ) and ( 10 , 0 ) . We can plot these points on a coordinate plane and sketch the parabola. The parabola opens downward, passes through the x-intercepts, and has its highest point at the vertex.
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 ) and x-intercepts at ( 0 , 0 ) and ( 10 , 0 ) .
Examples
Understanding quadratic functions like h ( x ) = − 5 1 x 2 + 2 x is crucial in various real-world applications. For instance, if you're launching a projectile, this function could model its trajectory, helping you determine the maximum height it reaches (the vertex) and how far it travels before landing (the x-intercepts). Similarly, in business, it could represent the profit curve of a product, showing the optimal price point for maximum profit. By analyzing the graph, you can make informed decisions about the projectile's launch angle or the product's pricing strategy.
