Final answer:
To graph the equation y = x² + b, create a value table, calculate y-values, plot the points, and draw a smooth curve. The graph will be a parabola with its vertex at (0,b), and it will open upwards.
Step-by-step explanation:
The question asks us to graph the equation y = x² + b. To graph this equation, we first need to recognize that it represents a parabola. When plotting data pairs to create a graph, we choose various values for x and then calculate the corresponding y-values. Here are the steps you would follow:
- Create a table with selected x-values (for example, -3, -2, -1, 0, 1, 2, 3).
- Calculate the corresponding y-values using the equation y = x² + b.
- Plot the (x,y) points on a coordinate axis.
- Draw a smooth curve through the points to visualize the parabola.
If b is positive, the parabola will open upwards and its vertex will be located at (0,b). If b is negative, the parabola will still open upwards, but the vertex will be below the x-axis. If b is zero, the vertex will be at the origin (0,0). To provide a specific example, consider the equation y = x² + 2:
- Plotting points: (-3, 11), (-2, 6), (-1, 3), (0, 2), (1, 3), (2, 6), (3, 11)
- The vertex is at (0,2), indicating the parabola opens upward and is shifted 2 units above the x-axis.
Remember, the shape of the graph is a parabola, which is different from the linear equations expressed as y = b + mx where b would be the y-intercept and m would represent the slope of the line.