Graph the equation $y=-x^2-4 x-3$ on the accompanying set of axes. You must plot 5 points including the roots and the vertex. Click to plot points. Click points to delete them.
Answer (1)
Find the roots by factoring the quadratic equation: x = − 3 and x = − 1 .
Determine the vertex using the formula x = − b / ( 2 a ) and substituting to find the y-coordinate: ( − 2 , 1 ) .
Choose two additional points, such as x = − 4 and x = 0 , and calculate their corresponding y-values: ( − 4 , − 3 ) and ( 0 , − 3 ) .
Plot the roots, vertex, and additional points to graph the parabola: ( − 3 , 0 ) , ( − 1 , 0 ) , ( − 2 , 1 ) , ( − 4 , − 3 ) , ( 0 , − 3 )
Explanation
Understanding the Problem We are asked to graph the quadratic equation y = − x 2 − 4 x − 3 . To do this, we need to find the roots, the vertex, and a couple of other points to accurately sketch the parabola.
Finding the Roots First, let's find the roots of the equation by setting y = 0 and solving for x . This gives us − x 2 − 4 x − 3 = 0 . We can multiply both sides by -1 to get x 2 + 4 x + 3 = 0 . Factoring this quadratic, we have ( x + 3 ) ( x + 1 ) = 0 . Thus, the roots are x = − 3 and x = − 1 . These are the points ( − 3 , 0 ) and ( − 1 , 0 ) .
Finding the Vertex Next, we need to find the vertex of the parabola. The x-coordinate of the vertex is given by x = − b / ( 2 a ) , where a = − 1 and b = − 4 . So, x = − ( − 4 ) / ( 2 ∗ ( − 1 )) = 4/ ( − 2 ) = − 2 . To find the y-coordinate of the vertex, we substitute x = − 2 into the equation: y = − ( − 2 ) 2 − 4 ( − 2 ) − 3 = − 4 + 8 − 3 = 1 . Therefore, the vertex is at the point ( − 2 , 1 ) .
Finding Additional Points Now, let's choose two additional x-values to get two more points. We'll pick x = − 4 and x = 0 . When x = − 4 , y = − ( − 4 ) 2 − 4 ( − 4 ) − 3 = − 16 + 16 − 3 = − 3 . So, we have the point ( − 4 , − 3 ) . When x = 0 , y = − ( 0 ) 2 − 4 ( 0 ) − 3 = − 3 . So, we have the point ( 0 , − 3 ) .
Plotting the Points and Graphing We have found the following five points: ( − 3 , 0 ) , ( − 1 , 0 ) , ( − 2 , 1 ) , ( − 4 , − 3 ) , and ( 0 , − 3 ) . These points can now be plotted to graph the parabola.
Examples
Understanding quadratic equations and their graphs is crucial in various fields. For example, engineers use parabolas to design bridges and arches, ensuring structural stability. In physics, projectile motion follows a parabolic path, allowing us to predict the trajectory of objects like balls or rockets. By finding roots and vertices, we can determine key parameters such as maximum height and range, optimizing designs and predictions in real-world applications.
