Graph the function. [tex]$h(x)=x^2+2 x$[/tex]

Find the vertex of the parabola: x = − b / ( 2 a ) = − 1 , h ( − 1 ) = − 1 , so the vertex is ( − 1 , − 1 ) .
Find the x-intercepts by solving x 2 + 2 x = 0 , which gives x = 0 and x = − 2 .
Find the y-intercept by evaluating h ( 0 ) = 0 .
Sketch the parabola using the vertex, x-intercepts, and y-intercept. The final answer is the graph of the parabola with these key features.

Explanation

Understanding the Problem We are asked to graph the function h ( x ) = x 2 + 2 x . This is a quadratic function, and its graph is a parabola. To graph it, we need to find the vertex, x-intercepts, and y-intercept.

Finding the Vertex First, let's find the vertex of the parabola. The x-coordinate of the vertex is given by x = − b / ( 2 a ) , where a = 1 and b = 2 in our equation. So, x = − 2/ ( 2 ∗ 1 ) = − 1 . To find the y-coordinate of the vertex, we plug this x-value back into the equation: h ( − 1 ) = ( − 1 ) 2 + 2 ∗ ( − 1 ) = 1 − 2 = − 1 . Therefore, the vertex is at ( − 1 , − 1 ) .

Finding the X-Intercepts Next, let's find the x-intercepts. These are the points where h ( x ) = 0 . So we need to solve the equation x 2 + 2 x = 0 . We can factor out an x : x ( x + 2 ) = 0 . This gives us two solutions: x = 0 and x = − 2 . So the x-intercepts are at ( 0 , 0 ) and ( − 2 , 0 ) .

Finding the Y-Intercept Now, let's find the y-intercept. This is the point where x = 0 . So we evaluate h ( 0 ) = ( 0 ) 2 + 2 ∗ ( 0 ) = 0 . Therefore, the y-intercept is at ( 0 , 0 ) .

Sketching the Parabola Now we have the vertex at ( − 1 , − 1 ) , the x-intercepts at ( 0 , 0 ) and ( − 2 , 0 ) , and the y-intercept at ( 0 , 0 ) . We can plot these points and sketch the parabola. The axis of symmetry is a vertical line passing through the vertex, which is x = − 1 . The parabola opens upwards since the coefficient of x 2 is positive ( a = 1 ).

Final Answer The graph of the function h ( x ) = x 2 + 2 x is a parabola with vertex at ( − 1 , − 1 ) , x-intercepts at ( 0 , 0 ) and ( − 2 , 0 ) , and y-intercept at ( 0 , 0 ) .


Examples
Understanding quadratic functions like h ( x ) = x 2 + 2 x is crucial in various real-world applications. For instance, if you're launching a projectile, the height of the projectile over time can be modeled by a quadratic function. The vertex of the parabola would represent the maximum height the projectile reaches. Similarly, in business, the profit function can sometimes be modeled as a quadratic, where the vertex indicates the point of maximum profit. By analyzing the graph, you can determine key aspects such as the break-even points (x-intercepts) and the optimal conditions for maximizing the outcome.

Answered by GinnyAnswer | 2025-07-08

To graph the function h ( x ) = x 2 + 2 x , find the vertex at ( − 1 , − 1 ) , the x-intercepts at ( 0 , 0 ) and ( − 2 , 0 ) , and the y-intercept at ( 0 , 0 ) . The parabola opens upwards with its axis of symmetry at x = − 1 . Plotting these points will yield the graph of the function.
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Answered by Anonymous | 2025-07-28