graph

Graphing linear equations is a fundamental skill in mathematics, and the equation y = 4x is a simple example of a linear equation. To graph this equation, follow these steps:

1. Plot the y-intercept. The y-intercept is the point where the graph crosses the y-axis. For the equation y = 4x, the y-intercept is (0, 0) because when x = 0, y = 0.
2. Find the slope of the line. The slope is a measure of how steep the line is. For the equation y = 4x, the slope is 4. This means that for every 1 unit increase in x, y increases by 4 units.
3. Use the slope and the y-intercept to plot additional points. Starting from the y-intercept, use the slope to plot additional points on the graph. For example, to plot the point (1, 4), start at the y-intercept (0, 0) and move up 4 units (because the slope is 4) and then to the right 1 unit.
4. Connect the points with a line. Once you have plotted a few points, you can connect them with a line to complete the graph.

Graphing linear equations is an important skill because it allows you to visualize the relationship between two variables. For example, the equation y = 4x could be used to represent the relationship between the number of hours worked and the amount of money earned. By graphing the equation, you can see how the amount of money earned increases as the number of hours worked increases.

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Graphing the equation $y = 1 + 2x^2$ involves plotting points on a coordinate plane that satisfy the equation. To achieve this, follow these steps:

1. Create a table of values by assigning different values to $x$ and calculating the corresponding $y$ values using the equation.
2. Plot these points on the coordinate plane, with $x$ values on the horizontal axis and $y$ values on the vertical axis.
3. Connect the plotted points with a smooth curve to visualize the graph of $y = 1 + 2x^2$.

This parabola opens upward because the coefficient of the squared term, $2$, is positive. Its vertex, the point where the parabola changes direction, can be found using the formula $x = -\frac{b}{2a}$, which gives $x = 0$ in this case. Plugging this value back into the equation yields $y = 1$, so the vertex is at the point $(0, 1)$.

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