Graph the equation $y=-x^2+8x-12$ on the accompanying set of axes. You must plot 5 points including the roots and the vertex. Using the graph, determine the roots of the equation $-x^2+8x-12=0$.
Click to plot points. Click points to delete them.
Answer (1)
Find the roots of the quadratic equation by factoring: x = 2 and x = 6 .
Determine the vertex of the parabola using the formula x v = − b / ( 2 a ) and substituting to find y v , resulting in the vertex ( 4 , 4 ) .
Calculate two additional points on the parabola, such as ( 1 , − 5 ) and ( 7 , − 5 ) .
Plot the roots, vertex, and additional points to graph the parabola and confirm the roots are x = 2 and x = 6 .
The roots of the equation − x 2 + 8 x − 12 = 0 are 2 , 6 .
Explanation
Understanding the Problem We are asked to graph the quadratic equation y = − x 2 + 8 x − 12 , identify the roots and vertex, and plot 5 points on the graph.
Finding the Roots First, let's find the roots of the equation by setting y = 0 : − x 2 + 8 x − 12 = 0 Multiplying by -1, we get: x 2 − 8 x + 12 = 0 Factoring the quadratic, we have: ( x − 2 ) ( x − 6 ) = 0 Thus, the roots are x = 2 and x = 6 . These are the x-intercepts of the graph.
Finding the Vertex Next, let's find the vertex of the parabola. The x-coordinate of the vertex is given by x v = − b / ( 2 a ) , where a = − 1 and b = 8 . x v = − 8/ ( 2 ∗ − 1 ) = 4 Now, substitute x v = 4 into the equation to find the y-coordinate of the vertex: y v = − ( 4 ) 2 + 8 ( 4 ) − 12 = − 16 + 32 − 12 = 4 So, the vertex is at ( 4 , 4 ) .
Finding Additional Points Now, let's find two additional points to plot. We can choose x = 1 and x = 7 . For x = 1 : y = − ( 1 ) 2 + 8 ( 1 ) − 12 = − 1 + 8 − 12 = − 5 So, the point is ( 1 , − 5 ) . For x = 7 : y = − ( 7 ) 2 + 8 ( 7 ) − 12 = − 49 + 56 − 12 = − 5 So, the point is ( 7 , − 5 ) .
Graphing and Determining the Roots We have the following 5 points: Roots: ( 2 , 0 ) and ( 6 , 0 ) Vertex: ( 4 , 4 ) Additional points: ( 1 , − 5 ) and ( 7 , − 5 ) Now we can plot these points and graph the parabola. From the graph, we can confirm that the roots of the equation − x 2 + 8 x − 12 = 0 are x = 2 and x = 6 .
Examples
Understanding quadratic equations and their graphs is essential in many real-world applications. For instance, engineers use parabolas to design arches in bridges, architects use them to model the curves in buildings, and physicists use them to analyze projectile motion. By finding the roots and vertex of a quadratic equation, we can determine key features such as maximum height, range, and optimal launch angles in these scenarios. This knowledge helps in creating efficient and safe designs.
