Question

Graph the function \(3 \mathrm{x}-5\).

Short answer

The short answer to the task, based on the step-by-step solution provided, involves first identifying the slope (3) and y-intercept (-5) of the line from the function \(3x - 5\). These values are then used to plot the y-intercept at point (0, -5) as well as another point using the slope (rise over run). The slope indicates that for every unit moved to the right along the x-axis, the line would go up by 3 units. Thus, from the y-intercept, moving 1 unit right and 3 units up gives us point (1, -2). These points are then connected to form a straight line representing the function \(3x - 5\). Finally, to verify the graph, select a test point not already on the line, substitute the x-value into the equation to solve for y, and cross-verify that the y-value matches with the one on the graph.

Step-by-step solution

Identify the Slope and Y-intercept

The equation of the line is in the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. Comparing this with the given equation, \(3x - 5\), we identify that the slope, \(m\), is 3 and the y-intercept, \(c\), is -5.

Plot the Y-intercept

The y-intercept is the point where the line crosses the y-axis. This is at the point (0, -5). Plot this point on the graph.

Use the Slope to Plot More Points

The slope of the line is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. Since the slope is 3, this means that for each 1 unit movement to the right along the x-axis, the line will move up by 3 units. From the y-intercept (0, -5), if we move 1 unit to the right to the point (1, -5), then move up 3 units, we would be at the point (1, -2). Plot this point.

Draw the Line

Now that you have at least two points: the y-intercept (0, -5) and the point (1, -2) from the slope, draw a straight line through these points. This line represents the graph of the function \(3x - 5\).

Verify the Graph

To ensure the graph is correct, you can choose a test point not already on your line. Substitute the x-value of this point into your equation and solve for y. If the calculated y-value corresponds with the y-value at that x-position on your line, then your graph is correct.

Key concepts

Slope and Y-Intercept

Understanding the slope and the y-intercept is crucial when graphing linear equations. These two key features will determine the unique straight line that represents the equation.

The slope, typically denoted as m, defines how steep the line is. It represents the rate at which the y value of a point changes per unit increase in the x value. For the equation given in the exercise, y = 3x - 5, the slope is 3. This means that for each step we move to the right along the x-axis, the y-value increases by 3 units.

Meanwhile, the y-intercept is where the line crosses the y-axis. It is written as 'c' in the equation y = mx + c and describes the starting point of the line when x is zero. In our equation, the y-intercept is -5, which is the point (0, -5) on the graph. This is the first point we plot on our graph and serves as the foundation upon which we build the line by using the slope.

Plotting Points

Plotting points is a methodical process that provides a visual representation of the relationship between two variables described by an equation. Starting with the y-intercept, as we did with the point (0, -5), you then use the slope to find additional points.

You can think of plotting points as laying down breadcrumbs to mark your trail. Our slope is 3, which means for every horizontal movement of 1 unit, we make a vertical movement of 3 units up. From the y-intercept, we move right to (1, -5), then up to (1, -2), and plot this second point.

It's always good to plot more than two points to ensure accuracy when drawing your line. With each point plotted, you’re piece by piece, creating a path that the line will follow. Connect these points with a ruler for precision, and you've graphed your equation.

Slope of a Line

The slope of a line is a measure of its steepness and direction. Calculated as the 'rise over run,' it determines how much the line 'rises' vertically for each unit of 'run' horizontally.

In the context of our exercise, the slope of the line is 3. Visually, this slope creates a line that ascends rapidly because for every unit we move horizontally, the rise is three times as much. The slope would be different if, for instance, our equation were y = 2x - 5, our line would rise more slowly, 2 units up for each 1 unit across.

Finally, the sign of the slope also indicates the direction of the line. A positive slope, as in our exercise, means the line tilts upwards from left to right. Conversely, a negative slope would tilt downwards. By mastering the concept of the slope, students gain an intuitive understanding of how linear equations graphically represent various rates of change.