Problem 38 - Contemporary Precalculus

Question

Graph the equation. Label all intercepts. $$2 x-3 y=12$$

Step-by-Step Solution

Answer

Answer: The x-intercept is (6, 0) and the y-intercept is (0, -4).

Step 1: Find the x-intercept

Set y to 0 and solve for x: $$2x - 3(0) = 12$$ $$2x = 12$$ $$x = 6$$ So the x-intercept is (6, 0).

Step 2: Find the y-intercept

Set x to 0 and solve for y: $$2(0) -3y = 12$$ $$-3y = 12$$ $$y=-4$$ So the y-intercept is (0, -4).

Step 3: Label intercepts on the graph

Plot the x-intercept, (6, 0), and the y-intercept, (0, -4), on the Cartesian plane. Be sure to label each intercept accordingly.

Step 4: Draw the line

Now that we have both intercepts, draw a straight line that passes through (6, 0) and (0, -4). The line represents the function $$2x - 3y = 12$$. We have successfully graphed the equation and labeled the intercepts.

Key Concepts

X-Intercept

Understanding the x-intercept is essential when graphing linear equations. The x-intercept is the point where the graph of an equation crosses the x-axis on the Cartesian plane. In simpler terms, it's where the output, or y-value, is zero. To find this point, you set the y-variable to zero and solve the equation for x.

Consider the equation from our exercise, 2x - 3y = 12. By substituting y with zero, you isolate x to find that x = 6. Therefore, the graph of this equation crosses the x-axis at the point (6,0). This is an essential step in sketching the overall graph of the function.

Y-Intercept

Conversely, the y-intercept is where the graph intersects the y-axis, hence where the input, or x-value, is zero. It represents the starting point of the linear function on the graph when looked at from left to right. For the same equation, 2x - 3y = 12, setting x to zero allows us to solve for y, giving us the y-intercept at (0, -4). Graphically, this tells us that if we follow the y-axis down to -4, that's where our line will begin or pass through.

Cartesian Plane

The Cartesian plane is a two-dimensional surface defined by two perpendicular axes: the horizontal x-axis, and the vertical y-axis. The intersection of these axes is known as the origin, which has coordinates (0,0). Points on this plane are represented as pairs (x, y).

When graphing, we plot points such as the x-intercept and y-intercept, then draw lines or curves to represent equations like our linear function. The Cartesian plane is a fundamental element in graphing equations as it provides a visual representation of the relationship between variables.

Linear Functions

A linear function is an algebraic equation that forms a straight line when graphed on the Cartesian plane. It's represented in the form y = mx + b, where m is the slope of the line, and b is the y-intercept. Linear functions illustrate a constant rate of change between the x and y variables.

Our exercise equation, 2x - 3y = 12, is slightly different in appearance but can be rewritten in the slope-intercept form. When graphed, it gives a clear, straight line that cuts across the plane, passing through the identified x-intercept and y-intercept. Linear functions like this are the foundation for much more complex mathematical concepts and real-world applications, such as predicting trends and modeling relationships.