1) Find the midpoint of the line with endpoints A(-3, -2) and B(4, 3).
2) Copy and complete the table for $f(x) = 7 + 4x - 3x^2$ for the interval $-3 \leq x \leq 3$.
| x | -3 | -2 | -1 | 0 | 0.5 | 1.5 | 2.5 | 3 |
|---|---|---|---|---|---|---|---|---|
| f(x) | -32 | | | 7 | 7.5 | | 1 | 8 |
3) Using 1 cm as 1 unit on the x-axis and 2 units on the y-axis, draw the graph of $y = f(x)$ for the given interval.
Answer (1)
Calculate the midpoint of the line segment using the midpoint formula: M = ( 2 x 1 + x 2 , 2 y 1 + y 2 ) .
Substitute the coordinates A(-3, -2) and B(4, 3) into the midpoint formula to find the midpoint (0.5, 0.5).
Evaluate the quadratic function f ( x ) = 7 + 4 x − 3 x 2 for each x-value in the table to complete it.
Plot the points from the completed table on a coordinate plane with the given scale to draw the graph of the function.
The midpoint is ( 0.5 , 0.5 ) and the completed table is in the solution.
Explanation
Problem Overview The problem consists of three parts: finding the midpoint of a line segment, completing a table for a quadratic function, and drawing the graph of the quadratic function.
Midpoint Calculation To find the midpoint of the line segment with endpoints A(-3, -2) and B(4, 3), we use the midpoint formula: M = ( 2 x 1 + x 2 , 2 y 1 + y 2 ) Substituting the coordinates of A and B, we get: M = ( 2 − 3 + 4 , 2 − 2 + 3 ) = ( 2 1 , 2 1 ) = ( 0.5 , 0.5 ) Thus, the midpoint is (0.5, 0.5).
Completing the Table To complete the table for the function f ( x ) = 7 + 4 x − 3 x 2 , we need to evaluate the function for the given x-values. The x-values are -3, -2, -1, 0, 0.5, 1, 1.5, 2, 2.5, and 3. We have already calculated the y-values using python: f ( − 3 ) = 7 + 4 ( − 3 ) − 3 ( − 3 ) 2 = 7 − 12 − 27 = − 32 f ( − 2 ) = 7 + 4 ( − 2 ) − 3 ( − 2 ) 2 = 7 − 8 − 12 = − 13 f ( − 1 ) = 7 + 4 ( − 1 ) − 3 ( − 1 ) 2 = 7 − 4 − 3 = 0 f ( 0 ) = 7 + 4 ( 0 ) − 3 ( 0 ) 2 = 7 f ( 0.5 ) = 7 + 4 ( 0.5 ) − 3 ( 0.5 ) 2 = 7 + 2 − 0.75 = 8.25 f ( 1 ) = 7 + 4 ( 1 ) − 3 ( 1 ) 2 = 7 + 4 − 3 = 8 f ( 1.5 ) = 7 + 4 ( 1.5 ) − 3 ( 1.5 ) 2 = 7 + 6 − 6.75 = 6.25 f ( 2 ) = 7 + 4 ( 2 ) − 3 ( 2 ) 2 = 7 + 8 − 12 = 3 f ( 2.5 ) = 7 + 4 ( 2.5 ) − 3 ( 2.5 ) 2 = 7 + 10 − 18.75 = − 1.75 f ( 3 ) = 7 + 4 ( 3 ) − 3 ( 3 ) 2 = 7 + 12 − 27 = − 8 So, the completed table is: \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline x & -3 & -2 & -1 & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 \ \hline f(x) & -32 & -13 & 0 & 7 & 8.25 & 8 & 6.25 & 3 & -1.75 & -8 \ \hline \end{array}
Graphing the Function To draw the graph of y = f ( x ) = 7 + 4 x − 3 x 2 , we plot the points from the completed table on a coordinate plane. The scale is 1 cm per unit on the x-axis and 2 units per cm on the y-axis. Plotting these points and connecting them with a smooth curve will give us the graph of the quadratic function.
Final Answer The midpoint of the line segment is (0.5, 0.5). The completed table is shown above. The graph can be plotted using the table values with the specified scale.
Examples
Understanding quadratic functions and their graphs is essential in many real-world applications. For example, engineers use quadratic equations to model the trajectory of projectiles, such as rockets or balls. By analyzing the equation, they can determine the maximum height, range, and other critical parameters. Similarly, in business, quadratic functions can model profit curves, helping companies find the optimal price point to maximize their earnings. The ability to complete tables and draw graphs allows for a visual representation and analysis of these functions, making it easier to understand and predict outcomes.
