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: Write the standard equation of a linear graph. The standard equation of a linear graph is given by: y = mx + c where m represents the gradient (slope) of the line and c represents the y-intercept (the point where the line crosses the y-axis). Step 2: Complete the table for gradient and y-intercept. We compare each given equation to the standard form y = mx + c to identify the gradient (m) and the y-intercept (c). For the equation y = 2x + 1: The gradient m = 2. The y-intercept c = 1. For the equation y = 2x: This can be written as y = 2x + 0. The gradient m = 2. The y-intercept c = 0. For the equation y = 2x - 1: The gradient m = 2. The y-intercept c = -1. The completed table is: | Equation | Gradient | Y-Intercept | | :--------- | :------- | :---------- | | y = 2x + 1 | 2 | 1 | | y = 2x | 2 | 0 | | y = 2x - 1 | 2 | -1 | Step 3: State the gradients of the 3 linear graphs. From the completed table, the gradients of all three linear graphs are the same. The gradients of the 3 linear graphs are 2. Step 4: Plot the 3 graphs in one Cartesian Plane. To plot each line, we can use the y-intercept and the gradient. The y-intercept is the point (0, c). From this point, use the gradient m = riserun = (2)/(1) to find another point (move 1 unit right, 2 units up). For y = 2x + 1: Plot the y-intercept (0, 1). From (0, 1), move 1 unit right and 2 units up to reach (1, 3). Draw a straight line through (0, 1) and (1, 3). (You can also find a third point, e.g., if x=-1, y=2(-1)+1 = -1, so (-1, -1) is on the line). For y = 2x: Plot the y-intercept (0, 0) (the origin). From (0, 0), move 1 unit right and 2 units up to reach (1, 2). Draw a straight line through (0, 0) and (1, 2). (You can also find a third point, e.g., if x=-1, y=2(-1) = -2, so (-1, -2) is on the line). For y = 2x - 1: Plot the y-intercept (0, -1). From (0, -1), move 1 unit right and 2 units up to reach (1, 1). Draw a straight line through (0, -1) and (1, 1). (You can also find a third point, e.g., if x=-1, y=2(-1)-1 = -3, so (-1, -3) is on the line). When plotted, these three lines will be parallel to each other because they all have the same gradient (m=2). That's 2 down. 3 left today — send the next one.