Graph the equation $y=x^2-14 x+48$ 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 $x^2-14 x+48=0$.
Answer (1)
Factor the quadratic equation x 2 − 14 x + 48 = 0 to find the roots x = 6 and x = 8 .
Determine the vertex of the parabola y = x 2 − 14 x + 48 using the formula x = − b / ( 2 a ) and substituting the x-value into the equation to find the y-value, resulting in the vertex ( 7 , − 1 ) .
Find two additional points on the parabola by substituting x = 0 and x = 10 into the equation, resulting in the points ( 0 , 48 ) and ( 10 , 8 ) .
Plot the roots, vertex, and additional points on the graph, and identify the roots of the equation as the x-coordinates where the graph intersects the x-axis: 6 , 8 .
Explanation
Problem Analysis We are asked to graph the quadratic equation y = x 2 − 14 x + 48 , plot 5 points including the roots and the vertex, and determine the roots of the equation x 2 − 14 x + 48 = 0 using the graph. First, we need to find the roots and the vertex of the parabola.
Finding the Roots To find the roots of the quadratic equation x 2 − 14 x + 48 = 0 , we can factor the quadratic expression. We are looking for two numbers that multiply to 48 and add up to -14. These numbers are -6 and -8. Therefore, we can factor the equation as ( x − 6 ) ( x − 8 ) = 0 . Setting each factor equal to zero gives us the roots x = 6 and x = 8 .
Finding the Vertex To find the vertex of the parabola y = x 2 − 14 x + 48 , we can use the formula for the x-coordinate of the vertex, which is x = − b / ( 2 a ) , where a = 1 and b = − 14 . So, x = − ( − 14 ) / ( 2 ∗ 1 ) = 14/2 = 7 . Now, we substitute x = 7 into the equation to find the y-coordinate of the vertex: y = ( 7 ) 2 − 14 ( 7 ) + 48 = 49 − 98 + 48 = − 1 . Therefore, the vertex is at ( 7 , − 1 ) .
Finding Additional Points Now we need two additional points. Let's choose x = 0 and x = 10 . When x = 0 , y = ( 0 ) 2 − 14 ( 0 ) + 48 = 48 . So, the point is ( 0 , 48 ) . When x = 10 , y = ( 10 ) 2 − 14 ( 10 ) + 48 = 100 − 140 + 48 = 8 . So, the point is ( 10 , 8 ) .
Graphing and Determining the Roots We have the following 5 points: ( 6 , 0 ) , ( 8 , 0 ) , ( 7 , − 1 ) , ( 0 , 48 ) , and ( 10 , 8 ) . Plotting these points on the graph and drawing a smooth curve through them gives us the parabola. The roots of the equation x 2 − 14 x + 48 = 0 are the x-coordinates of the points where the graph intersects the x-axis, which are x = 6 and x = 8 .
Final Answer The roots of the equation x 2 − 14 x + 48 = 0 are x = 6 and x = 8 .
Examples
Understanding quadratic equations and their roots is crucial in various fields. For example, engineers use quadratic equations to model the trajectory of projectiles, such as designing a catapult or analyzing the path of a ball thrown in the air. The roots of the equation help determine when the projectile will hit the ground, which is essential for accurate targeting and safety. By finding the vertex, they can also determine the maximum height the projectile will reach.
