Consider the following function:
r(x)=(x-4)^2-4
Find two points on the graph of the parabola other than the vertex and $x$-intercepts.
Answer (2)
Find the vertex of the parabola: ( 4 , − 4 ) .
Find the x-intercepts by setting r ( x ) = 0 : ( 2 , 0 ) and ( 6 , 0 ) .
Choose two other x-values, such as x = 0 and x = 1 .
Calculate the corresponding y-values: ( 0 , 12 ) and ( 1 , 5 ) . The two points are ( 0 , 12 ) and ( 1 , 5 ) .
Explanation
Understanding the Problem We are given the quadratic function r ( x ) = ( x − 4 ) 2 − 4 and asked to find two points on the graph of the parabola, other than the vertex and the x-intercepts.
Finding the Vertex First, let's find the vertex of the parabola. The vertex form of a parabola is r ( x ) = a ( x − h ) 2 + k , where ( h , k ) is the vertex. In our case, h = 4 and k = − 4 , so the vertex is ( 4 , − 4 ) .
Finding the x-intercepts Next, let's find the x-intercepts by setting r ( x ) = 0 :
( x − 4 ) 2 − 4 = 0
( x − 4 ) 2 = 4
x − 4 = ± 2
x = 4 ± 2
So, x = 4 + 2 = 6 or x = 4 − 2 = 2 . The x-intercepts are ( 2 , 0 ) and ( 6 , 0 ) .
Choosing x-values Now, we need to find two other points on the parabola. We can choose any two values for x other than 2 , 4 , and 6 . Let's choose x = 0 and x = 1 .
Calculating r(0) If x = 0 , then: r ( 0 ) = ( 0 − 4 ) 2 − 4 = ( − 4 ) 2 − 4 = 16 − 4 = 12
So, the point is ( 0 , 12 ) .
Calculating r(1) If x = 1 , then: r ( 1 ) = ( 1 − 4 ) 2 − 4 = ( − 3 ) 2 − 4 = 9 − 4 = 5
So, the point is ( 1 , 5 ) .
Final Answer Therefore, two points on the graph of the parabola r ( x ) = ( x − 4 ) 2 − 4 , other than the vertex and x-intercepts, are ( 0 , 12 ) and ( 1 , 5 ) .
Examples
Understanding quadratic functions is crucial in various real-world applications. For instance, engineers use parabolas to design suspension bridges and antennas. Architects apply quadratic equations to model arches and other curved structures. In business, quadratic functions can help model profit and cost curves, allowing for optimization of production and pricing strategies. By finding key points like the vertex and intercepts, we can analyze and predict the behavior of these systems.
The vertex of the parabola is at (4, -4) with x-intercepts at (2, 0) and (6, 0). Two additional points on the graph are (0, 12) and (1, 5). Hence, the required points are (0, 12) and (1, 5).
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