Graph this function: [tex]f(x)=x^2-4 x-5[/tex]

Step 1: Identify [tex]a[/tex] and [tex]b[/tex].
[tex]a=[/tex] $\square$ [tex]b=[/tex] $\square$


Check

Identify the coefficients: a = 1 and b = − 4 .
Calculate y values for different x values to complete the table.
Determine the vertex and intercepts of the parabola.
Graph the function using the vertex, intercepts, and table values.

Explanation

Identifying Coefficients and Objective The given function is f ( x ) = x 2 − 4 x − 5 . We need to identify the coefficients a and b in the quadratic expression of the form a x 2 + b x + c . Then, we will complete the table by choosing several values for x and calculating the corresponding y values.

Identifying a and b In the quadratic function f ( x ) = x 2 − 4 x − 5 , we can identify the coefficients as follows:


a = 1 (coefficient of x 2 )
b = − 4 (coefficient of x )
c = − 5 (constant term)

Creating a Table of Values Now, let's create a table of values for x and y = f ( x ) . We'll choose some values for x around the vertex of the parabola. The x-coordinate of the vertex is given by x v ​ = − b / ( 2 a ) . In our case, x v ​ = − ( − 4 ) / ( 2 ∗ 1 ) = 4/2 = 2 . So, we'll choose x values around 2.

Calculating y Values Let's calculate the y values for x = 0 , 1 , 2 , 3 , 4 , 5 :


For x = 0 : f ( 0 ) = ( 0 ) 2 − 4 ( 0 ) − 5 = − 5 For x = 1 : f ( 1 ) = ( 1 ) 2 − 4 ( 1 ) − 5 = 1 − 4 − 5 = − 8 For x = 2 : f ( 2 ) = ( 2 ) 2 − 4 ( 2 ) − 5 = 4 − 8 − 5 = − 9 For x = 3 : f ( 3 ) = ( 3 ) 2 − 4 ( 3 ) − 5 = 9 − 12 − 5 = − 8 For x = 4 : f ( 4 ) = ( 4 ) 2 − 4 ( 4 ) − 5 = 16 − 16 − 5 = − 5 For x = 5 : $f(5) = (5)^2 - 4(5) - 5 = 25 - 20 - 5 = 0

Completed Table So, the table is:




x
y



0
-5


1
-8


2
-9


3
-8


4
-5


5
0



Finding Vertex and Intercepts The vertex of the parabola is at ( 2 , − 9 ) . The x-intercepts are the points where f ( x ) = 0 . We already found one at x = 5 . To find the other, we can factor the quadratic: x 2 − 4 x − 5 = ( x − 5 ) ( x + 1 ) . So, the x-intercepts are x = 5 and x = − 1 . The y-intercept is the point where x = 0 , which we found to be f ( 0 ) = − 5 .

Graphing the Function Now we have all the information needed to graph the function. We have the vertex, intercepts, and several points.


Examples
Understanding quadratic functions is crucial in various fields, such as physics and engineering. For instance, the trajectory of a projectile, like a ball thrown in the air, can be modeled by a quadratic function. By analyzing the function's graph, we can determine the maximum height the ball reaches and the distance it travels before hitting the ground. This knowledge is essential for designing efficient sports equipment or predicting the behavior of objects in motion.

Answered by GinnyAnswer | 2025-07-08