Question

Find the intercepts and graph each equation by plotting points. Be sure to label the intercepts. $$ y=3 x-9 $$

Short answer

The y-intercept is (0, -9) and the x-intercept is (3, 0).

Step-by-step solution

Find the y-intercept

To find the y-intercept, set the value of \(x\) to zero in the equation \(y = 3x - 9\).\Substitute \(x = 0\):\[y = 3(0) - 9\]\[y = -9\]\So, the y-intercept is \( (0, -9) \). Label this point on the graph.

Find the x-intercept

To find the x-intercept, set the value of \(y\) to zero in the equation \(y = 3x - 9\).\Substitute \(y = 0\):\[0 = 3x - 9\]\Solve for \(x\) by adding 9 to both sides:\[9 = 3x\]\Divide both sides by 3 to isolate \(x\):\[x = 3\]\So, the x-intercept is \((3, 0)\). Label this point on the graph.

Plot additional points (optional)

To make the graph more accurate, choose additional values for \(x\) and solve for \(y\). For example, if \(x = 1\):\[y = 3(1) - 9\]\[y = -6\]\So, another point is \( (1, -6) \). Similarly, if \(x = 2\):\[y = 3(2) - 9\]\[y = -3\]\So, another point is \( (2, -3) \). Plot these points on the graph.

Draw the line

Draw a straight line through the points that have been plotted, including the intercepts \( (0, -9) \) and \( (3, 0) \). Extend the line in both directions and ensure that it is straight.

Label the intercepts

In the graph, clearly label the intercepts: \( (0, -9) \) as the y-intercept and \( (3, 0) \) as the x-intercept.

Key concepts

intercepts

Intercepts are the points where a line crosses the x-axis or y-axis. The x-intercept is where the line meets the x-axis, meaning y is equal to zero. To find it, set y to zero in your equation and solve for x. The y-intercept is where the line meets the y-axis, meaning x is equal to zero. To find it, set x to zero in your equation and solve for y. In the provided exercise, the x-intercept is (3, 0), and the y-intercept is (0, -9).

plotting points

Plotting points is a key step in graphing linear equations. After finding the intercepts, it's helpful to plot additional points to ensure your graph is accurate. You pick any value for x, plug it into the equation, and solve for y. For example, if we substitute x = 1 into the equation y = 3x - 9, we get y = -6, giving us the point (1, -6). Plot these points on your graph to get a clear picture of the line.

linear equations

Linear equations are equations of the first degree, meaning they produce a straight line when graphed. They generally have the form y = mx + b, where m is the slope and b is the y-intercept. In the provided example, y = 3x - 9, 3 is the slope, meaning the line rises 3 units up for every 1 unit it moves to the right. The -9 is the y-intercept, where the line crosses the y-axis. Understanding this form is critical for graphing.

graphing steps

Graphing a linear equation involves several steps for accuracy. First, find the y-intercept by setting x to zero and solving for y. Next, find the x-intercept by setting y to zero and solving for x. Then, plot these intercept points on the graph. For more accuracy, choose additional values for x, solve for y, and plot these points too. Finally, draw a straight line through all the plotted points and extend it in both directions. Make sure to clearly label the intercepts, as they help anyone reading the graph understand it better.