Graph $f(x)=(x+1)(x-5)$.
Graph the function $f(x)=(x+1)(x-5)$. Use the drop-down menus to complete the steps needed to graph the function.
1. Identify the $x$-intercepts: $(-1,0)$ and $(3,0)$
2. Find the midpoint between the intercepts
3. Find the vertex $\square$
4. Find the y-intercept $\square$
5. Plot another point, then draw the graph.
Answer (1)
Find the x-intercepts by setting f ( x ) = 0 and solving for x : x = − 1 , 5 .
Determine the vertex by finding the midpoint of the x-intercepts, x = 2 , and substituting it into the function to find the y-coordinate: f ( 2 ) = − 9 .
Calculate the y-intercept by setting x = 0 : f ( 0 ) = − 5 .
Plot the x-intercepts, vertex, and y-intercept to graph the parabola: See graph .
Explanation
Understanding the Problem We are given the function f ( x ) = ( x + 1 ) ( x − 5 ) and asked to graph it. We are given the x-intercepts as ( − 1 , 0 ) and ( 5 , 0 ) . We need to find the midpoint between the x-intercepts, the vertex, the y-intercept, and plot another point to draw the graph.
Identifying the x-intercepts The x-intercepts are ( − 1 , 0 ) and ( 5 , 0 ) . Note that the problem states the x-intercepts are ( − 1 , 0 ) and ( 3 , 0 ) , but this is incorrect. The x-intercepts are the points where f ( x ) = 0 , so ( x + 1 ) ( x − 5 ) = 0 . This occurs when x + 1 = 0 or x − 5 = 0 , which means x = − 1 or x = 5 . Thus, the x-intercepts are ( − 1 , 0 ) and ( 5 , 0 ) .
Finding the Midpoint To find the midpoint between the x-intercepts, we average the x-coordinates: x m = 2 − 1 + 5 = 2 4 = 2 .
Finding the Vertex The x-coordinate of the vertex is the midpoint between the x-intercepts, which is x = 2 . To find the y-coordinate of the vertex, we substitute x = 2 into the function: 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 ( 0 ) = ( 0 + 1 ) ( 0 − 5 ) = ( 1 ) ( − 5 ) = − 5 . So, the y-intercept is ( 0 , − 5 ) .
Finding Another Point Let's choose another point, say x = 1 . Then f ( 1 ) = ( 1 + 1 ) ( 1 − 5 ) = ( 2 ) ( − 4 ) = − 8 . So, another point on the graph is ( 1 , − 8 ) .
Plotting the Points and Drawing the Graph Now we plot the x-intercepts ( − 1 , 0 ) and ( 5 , 0 ) , the vertex ( 2 , − 9 ) , the y-intercept ( 0 , − 5 ) , and the additional point ( 1 , − 8 ) . We draw a smooth curve through these points to graph the function. The graph is a parabola opening upwards.
Summary of Key Points The x-intercepts are ( − 1 , 0 ) and ( 5 , 0 ) . The midpoint between the x-intercepts is x = 2 . The vertex is ( 2 , − 9 ) . The y-intercept is ( 0 , − 5 ) . Another point on the graph is ( 1 , − 8 ) .
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. Similarly, in physics, the trajectory of a projectile, such as a ball thrown in the air, follows a parabolic path. By analyzing the x-intercepts, vertex, and y-intercept, we can determine key characteristics like the range, maximum height, and initial position of the projectile. This knowledge is invaluable in fields like sports science and aerospace engineering.
