Graph the equation.
[tex]$y=-\frac{3}{2} x^2-6 x$[/tex]
Answer (1)
Find the x-intercepts by setting y = 0 and solving for x , resulting in x = 0 and x = − 4 .
Determine the x-coordinate of the vertex using the formula x = − 2 a b , which gives x = − 2 .
Calculate the y-coordinate of the vertex by substituting x = − 2 into the equation, yielding y = 6 .
Sketch the parabola using the x-intercepts ( 0 , 0 ) and ( − 4 , 0 ) , and the vertex ( − 2 , 6 ) . The final answer is the graph of this parabola.
Explanation
Analyze the Equation We are asked to graph the equation y = − 2 3 x 2 − 6 x . This is a quadratic equation, which represents a parabola. Since the coefficient of the x 2 term is negative, the parabola opens downward.
Find the x-intercepts To graph the parabola, we first find the x-intercepts by setting y = 0 and solving for x : 0 = − 2 3 x 2 − 6 x 0 = x ( − 2 3 x − 6 ) So, x = 0 or − 2 3 x − 6 = 0 . Solving the second equation: − 2 3 x = 6 x = 6 ⋅ − 3 2 = − 4 Thus, the x-intercepts are x = 0 and x = − 4 .
Find the x-coordinate of the vertex Next, we find the vertex of the parabola. The x-coordinate of the vertex is given by x = − 2 a b , where a = − 2 3 and b = − 6 . x = − 2 ( − 2 3 ) − 6 = − − 3 − 6 = − 2 The x-coordinate of the vertex is -2.
Find the y-coordinate of the vertex Now, we find the y-coordinate of the vertex by substituting x = − 2 into the equation: y = − 2 3 ( − 2 ) 2 − 6 ( − 2 ) = − 2 3 ( 4 ) + 12 = − 6 + 12 = 6 The y-coordinate of the vertex is 6. So, the vertex is at the point ( − 2 , 6 ) .
Sketch the Parabola We now have the x-intercepts at ( 0 , 0 ) and ( − 4 , 0 ) , and the vertex at ( − 2 , 6 ) . We can sketch the parabola using these points. The parabola opens downward, passing through the x-intercepts and having its maximum point at the vertex.
Final Answer The graph of the equation y = − 2 3 x 2 − 6 x is a parabola that opens downward, with x-intercepts at x = 0 and x = − 4 , and a vertex at ( − 2 , 6 ) .
Examples
Understanding quadratic equations and their graphs is essential in various fields, such as physics and engineering. For example, the trajectory of a projectile under the influence of gravity follows a parabolic path. By analyzing the equation of the parabola, we can determine the maximum height reached by the projectile and its range. Similarly, in engineering, parabolic shapes are used in designing antennas and reflectors to focus signals or energy efficiently. The vertex of the parabola represents the optimal point for focusing the signal or energy.
