Problem 54 - Algebra 2

Question

Find the \(x\)-intercept and the \(y\)-intercept of the graph of each equation. Then graph the equation. \(y=3-2 x\)

Step-by-Step Solution

Answer

The y-intercept is (0, 3) and the x-intercept is (1.5, 0). Graph the line through these points.

Step 1: Find the y-intercept

The y-intercept is the point where the graph intersects the y-axis, which happens when \(x = 0\). To find the y-intercept, substitute \(x = 0\) into the equation: \[y = 3 - 2(0)\]\[y = 3\]So, the y-intercept is \((0, 3)\).

Step 2: Find the x-intercept

The x-intercept is the point where the graph intersects the x-axis, which happens when \(y = 0\). To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\):\[0 = 3 - 2x\]Add \(2x\) to both sides:\[2x = 3\]Divide both sides by 2:\[x = \frac{3}{2}\]So, the x-intercept is \(\left(\frac{3}{2}, 0\right)\).

Step 3: Graph the equation

To graph the equation, use the intercepts found:- The y-intercept is \((0, 3)\).- The x-intercept is \(\left(\frac{3}{2}, 0\right)\).Plot these points on the coordinate plane and draw a straight line through them. This line represents the graph of the equation \(y = 3 - 2x\).

Key Concepts

Intercepts

Intercepts are key points where a graph meets the axes on a coordinate plane. They reveal important features of a linear equation.

• **Y-Intercept**: This is where the graph intersects the y-axis. It occurs when the value of x is zero. In the exercise, setting x to zero in the equation helps to discover the y-intercept, which is (0, 3).


• **X-Intercept**: This point is where the graph crosses the x-axis. It is found by setting y to zero in the equation. In the given example, solving the equation when y is zero, finds the x-intercept at \(\left(\frac{3}{2}, 0\right)\).

Finding intercepts is a fundamental step in graphing, as they provide precise points to help accurately draw the graph.

Graphing

Graphing a linear equation connects algebra with geometry. It visually represents equations on a coordinate plane. Graphing is made simpler by knowing the intercepts.

• First, plot the y-intercept on the y-axis at (0, 3).


• Next, mark the x-intercept on the x-axis at \(\left(\frac{3}{2}, 0\right)\).


• After both intercepts are plotted, draw a straight line through these points. This line reflects the linear equation's graph.

Remember, a straight line is characteristic of any linear equation. The line continues indefinitely in both directions beyond points.

Coordinate Plane

The coordinate plane is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis). It is essential for graphing equations, visualizing how equations behave, and understanding intercepts.

• **Quadrants**: The plane is divided into four quadrants by the x and y axes. Each quadrant signifies a different combination of positive and negative values of x and y.


• **Origin**: The point (0, 0) is called the origin, where the x and y axes intersect.


• **Plotting Points**: Coordinates are written as (x, y) and are used to specify locations on the plane.

Understanding the coordinate plane is crucial for graphing, as it provides the framework necessary to interpret intercepts and draw accurate graphs.