Problem 93 - Algebra 1

Question

Use a table of values to graph the equation. Label the \(x \text { -intercept and the } y \text { -intercept. (Review } 4.2,4.3)\) $$y=-9+3 x$$

Step-by-Step Solution

Answer

The x-intercept is at (3, 0) and the y-intercept is at (0, -9). Plotting the points from the value table and drawing a straight line through them produces the graph of the equation \(y = -9 + 3x\).

Step 1: Create a Table

Firstly, create a table of values for \(x\) and \(y\). Choose arbitrary values for \(x\), and calculate corresponding \(y\) values by substituting \(x\) into the equation \(y = -9 + 3x\). For example, if \(x = -1, 0, 1, 2, 3\), the corresponding \(y\) values would be -12, -9, -6, -3, 0.

Step 2: Plot Points on The Graph

Once the table of values has been created, plot the pair of x and y coordinates on a graph. Chart the xy-coordinates (-1, -12), (0, -9), (1, -6), (2, -3) and (3, 0). Connect the points with a straight line, since we're graphing a linear equation.

Step 3: Determine and Label the X and Y Intercepts

The x-intercept is found when \(y = 0\) in the equation \(y = -9 + 3x\). If we set y to 0 and solve for x, we find that x=3. The y-intercept is found when \(x = 0\) in the equation. If we substitute x=0 into the equation, we find that y=-9. So, the x-intercept is (3,0) and the y-intercept is (0,-9).

Key Concepts

Table of Values

When dealing with linear equations, a table of values is an excellent tool for understanding the relationship between variables. It's essentially a structured way to organize pairs of numbers that satisfy an equation, in this case, the linear equation y = -9 + 3x.

Creating a table begins with selecting particular x-values, which could be any numbers but often include zero to find the y-intercept easily. After choosing x-values, you then apply the equation to find the corresponding y-values. For instance, if you select x to be -1, you replace x with -1 in the equation, giving you y = -9 + 3*(-1) = -12. Repeating this process for different x-values builds a complete table of values, revealing how y changes in response to x. This method not only simplifies graphing but also enhances your understanding of the function's behavior.

X-Intercept and Y-Intercept

The x-intercept and y-intercept are crucial points on any graph because they show where the line crosses the axes. The x-intercept occurs where the graph crosses the x-axis, which means at this point, the y-value is zero. To find the x-intercept from the equation y = -9 + 3x, you set y equal to zero and solve for x, resulting in the point (3,0).

Conversely, the y-intercept is where the graph crosses the y-axis, where x is zero. By substituting x with zero in the equation, you discover that the y-intercept is (0,-9). These intercepts are important because they are easy to find and provide two points through which the line will pass, thus helping in plotting the linear graph accurately.

Plotting Points on a Graph

With your table of values in hand, plotting points on a graph becomes a straightforward task. Each pair of x and y values corresponds to a point on the graph, with x representing the horizontal position and y the vertical position. For instance, if you have the coordinates (-1, -12), you move left from the origin (the point where the x and y axes cross) to -1 on the x-axis, and then down to -12 on the y-axis, marking the spot on the graph.

After plotting all your points, connecting them will reveal the shape of the function. If it's a linear function, like y = -9 + 3x, you should see a straight line. The beauty of linear equations is that you only need two points to draw the complete line, but having more points can ensure accuracy. It's especially important to include the intercepts, as they anchor the line at known locations on the axes.