This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Complete the table for the relation y = 2x + 3 - x^2. We substitute each given x value into the equation y = 2x + 3 - x^2 to find the corresponding y values. • For x = -2: y = 2(-2) + 3 - (-2)^2 = -4 + 3 - 4 = -5 • For x = 1: y = 2(1) + 3 - (1)^2 = 2 + 3 - 1 = 4 • For x = 1.5: y = 2(1.5) + 3 - (1.5)^2 = 3 + 3 - 2.25 = 3.75 • For x = 2: y = 2(2) + 3 - (2)^2 = 4 + 3 - 4 = 3 • For x = 3: y = 2(3) + 3 - (3)^2 = 6 + 3 - 9 = 0 • For x = 3.5: y = 2(3.5) + 3 - (3.5)^2 = 7 + 3 - 12.25 = -2.25 The completed table is: |c|c|c|c|c|c|c|c|c|c|c| x & -2 & -1 & 0 & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 & 4 \\ y & -5 & 0 & 3 & 4 & 3.75 & 3 & 1.75 & 0 & -2.25 & -5 \\ Step 2: Draw the graph of y = 2x + 3 - x^2 (Part b). To draw the graph, plot the points from the completed table in Step 1 on a graph paper. Use a scale of 2cm to 1 unit on both the x-axis and y-axis. Connect the points with a smooth curve to form the parabola. The coordinates to plot are: (-2, -5), (-1, 0), (0, 3), (1, 4), (1.5, 3.75), (2, 3), (2.5, 1.75), (3, 0), (3.5, -2.25), (4, -5). Step 3: Draw the graph of y - x = 0 (Part c). The equation y - x = 0 simplifies to y = x. This is a straight line. To draw this line on the same axes, plot at least two points and connect them. For example: • For x = -2: y = -2. Point: (-2, -2) • For x = 0: y = 0. Point: (0, 0) • For x = 4: y = 4. Point: (4, 4) Draw a straight line passing through these points. Step 4: Use the graphs to solve the equation 3 + 2x = x^2 + x (Part d, i). Rearrange the given equation to match the forms of the graphs: $3 + 2x = x^2 +