Graph the function [tex]f(x)=(x+1)(x-5)[/tex]. Use the drop-down menus to complete the steps needed to graph the function.
1. Identify the [tex]x[/tex]-intercepts: [tex](-1,0)[/tex] and [tex](5,0)[/tex]
2. Find the midpoint between the intercepts (calculate, don't plot).
3. Find the vertex: [ ]
4. Find the [tex]y[/tex]-intercept: [ ]
5. Plot another point, then draw the graph.
Answer (1)
Find the midpoint between the x-intercepts: x m = 2 − 1 + 5 = 2 .
Calculate the y-coordinate of the vertex: f ( 2 ) = ( 2 + 1 ) ( 2 − 5 ) = − 9 . The vertex is ( 2 , − 9 ) .
Determine the y-intercept by setting x = 0 : f ( 0 ) = ( 0 + 1 ) ( 0 − 5 ) = − 5 . The y-intercept is ( 0 , − 5 ) .
Plot the x-intercepts, vertex, and y-intercept to graph the parabola: See graph .
Explanation
Understanding the Problem The problem asks us to graph the quadratic function f ( x ) = ( x + 1 ) ( x − 5 ) . We are given the x-intercepts and need to find the midpoint between them, the vertex, and the y-intercept to sketch the graph.
Finding the Midpoint To find the midpoint between the x-intercepts, we use the formula x m = 2 x 1 + x 2 , where x 1 = − 1 and x 2 = 5 . Thus, x m = 2 − 1 + 5 = 2 4 = 2 .
Finding the Vertex The x-coordinate of the vertex is the midpoint x m = 2 . To find the y-coordinate of the vertex, we substitute x m into the function f ( x ) : f ( 2 ) = ( 2 + 1 ) ( 2 − 5 ) = ( 3 ) ( − 3 ) = − 9 . Therefore, the vertex is ( 2 , − 9 ) .
Finding the y-intercept To find the y-intercept, we set x = 0 in the function f ( x ) : f ( 0 ) = ( 0 + 1 ) ( 0 − 5 ) = ( 1 ) ( − 5 ) = − 5 . Thus, the y-intercept is ( 0 , − 5 ) .
Graphing the Function Now we have the x-intercepts ( − 1 , 0 ) and ( 5 , 0 ) , the vertex ( 2 , − 9 ) , and the y-intercept ( 0 , − 5 ) . We can plot these points and draw a smooth parabola through them to graph the function.
Examples
Understanding quadratic functions like f ( x ) = ( x + 1 ) ( x − 5 ) is crucial in various real-world applications. For instance, engineers use parabolas to design arches in bridges, ensuring structural stability and efficient load distribution. Similarly, in physics, the trajectory of a projectile, such as a ball thrown in the air, follows a parabolic path, allowing us to predict its range and maximum height. By analyzing the x-intercepts, vertex, and y-intercept, we can optimize designs and predict outcomes in these scenarios.
