(U15) A is the point $(-3, -2)$ and B is $(4, 3)$.
1. Find the midpoint of the line segment AB.
2. Complete the table for $y = 7 + 4x - 3x^2$ for the interval $-3 \leq x \leq 3$.
| x | -3 | -2 | -1 | 0 | 0.5 | 1 | 1.5 | 2.5 | 3 |
| --- | ----- | -- | -- | --- | --- | --- | --- | --- | --- |
| y | -32 | | | 7.5 | 8.5 | | 1 | 1 | 8 |
3. Using 2 cm as 1 unit on the x-axis and 1 cm as 1 unit on the y-axis, draw the graph of y for the given intervals.
Answer (1)
Find the other endpoint using the midpoint formula: x = 11 , y = 8 .
Calculate the missing y-values for the quadratic function y = 7 + 4 x − 3 x 2 .
Complete the table with the calculated y-values.
Plot the points from the completed table and draw the graph of the quadratic function. The other endpoint is ( 11 , 8 ) .
Explanation
Problem Analysis We are given a line segment with one endpoint A ( − 3 , − 2 ) and midpoint ( 4 , 3 ) . We need to find the other endpoint. We also need to complete a table for the quadratic function y = 7 + 4 x − 3 x 2 and then draw its graph.
Midpoint Formula Let the other endpoint be B ( x , y ) . The midpoint formula states that the midpoint of a line segment with endpoints ( x 1 , y 1 ) and ( x 2 , y 2 ) is ( 2 x 1 + x 2 , 2 y 1 + y 2 ) .
Setting up Equations Using the midpoint formula, we have ( 4 , 3 ) = ( 2 − 3 + x , 2 − 2 + y ) . This gives us two equations: 4 = 2 − 3 + x and 3 = 2 − 2 + y .
Solving for the Endpoint Solving for x , we have 4 = 2 − 3 + x ⇒ 8 = − 3 + x ⇒ x = 11 . Solving for y , we have 3 = 2 − 2 + y ⇒ 6 = − 2 + y ⇒ y = 8 . Thus, the other endpoint is B ( 11 , 8 ) .
Completing the Table Now, we need to complete the table for y = 7 + 4 x − 3 x 2 . We already have the values for x = − 3 , 0 , 0.5 , 3 . We need to calculate the values for x = − 2 , − 1 , 1 , 1.5 , 2 , 2.5 .
Calculating Table Values For x = − 2 , y = 7 + 4 ( − 2 ) − 3 ( − 2 ) 2 = 7 − 8 − 12 = − 13 . For x = − 1 , y = 7 + 4 ( − 1 ) − 3 ( − 1 ) 2 = 7 − 4 − 3 = 0 . For x = 1 , y = 7 + 4 ( 1 ) − 3 ( 1 ) 2 = 7 + 4 − 3 = 8 . For x = 1.5 , y = 7 + 4 ( 1.5 ) − 3 ( 1.5 ) 2 = 7 + 6 − 6.75 = 6.25 . For x = 2 , y = 7 + 4 ( 2 ) − 3 ( 2 ) 2 = 7 + 8 − 12 = 3 . For x = 2.5 , y = 7 + 4 ( 2.5 ) − 3 ( 2.5 ) 2 = 7 + 10 − 18.75 = − 1.75 .
Completed Table The completed table is:
x
-3
-2
-1
0
0.5
1
1.5
2
2.5
3
y
-32
-13
0
7
8.5
8
6.25
3
-1.75
-8
Drawing the Graph Finally, we need to draw the graph of y = 7 + 4 x − 3 x 2 using the completed table. The x-axis and y-axis both use a scale of 1 cm per unit. Plot the points from the table and draw a smooth curve through them.
Final Answer The other endpoint is ( 11 , 8 ) , and the completed table is shown above. The graph can be plotted using these values.
Examples
Understanding quadratic functions and line segments is crucial in various fields. For instance, engineers use quadratic equations to model projectile motion, optimizing trajectories for rockets or designing parabolic reflectors for antennas. Similarly, the midpoint formula is used in computer graphics to calculate the center of objects for scaling or rotation. These mathematical concepts provide a foundation for solving real-world problems in science and technology.
