Problem 33 - Algebra 2

Question

Find the \(x\)-intercept and the \(y\)-intercept of the graph of each equation. Then graph the equation. \(y=4 x-2\)

Step-by-Step Solution

Answer

The y-intercept is \((0, -2)\) and the x-intercept is \(\left(\frac{1}{2}, 0\right)\).

Step 1: Find the y-intercept

The y-intercept is the point where the graph crosses the y-axis, which means at this point, the value of x is 0. Substitute x with 0 in the equation to find the y-intercept.Substitute: \(y = 4(0) - 2 \)Simplify: \(y = -2\)Thus, the y-intercept is at the point \((0, -2)\).

Step 2: Find the x-intercept

The x-intercept is the point where the graph crosses the x-axis, which means at this point, the value of y is 0. Substitute y with 0 in the equation to find the x-intercept.Substitute: \(0 = 4x - 2\)Solve for x: \ \(4x = 2\) \ \(x = \frac{2}{4} = \frac{1}{2} \)Thus, the x-intercept is at the point \(\left(\frac{1}{2}, 0\right)\).

Step 3: Graph the equation

To graph the equation, plot the points found from the intercepts on a coordinate plane. The points are \((0, -2)\) and \(\left(\frac{1}{2}, 0\right)\). Plot these and use a ruler to draw a straight line through them, extending the line in both directions.This line represents the graph of the equation \(y = 4x - 2\).

Key Concepts

Understanding X-Intercept

The x-intercept is a key point on a graph where the line crosses the x-axis. At this point, the y-value in the equation is zero. To find the x-intercept for the equation, set the equation equal to zero and solve for x. This means replacing y with 0 in the equation. For example, in the equation \(y = 4x - 2\), replace y with 0 and solve, which gives:

• \(0 = 4x - 2\)
• Rearrange to find \(4x = 2\)
• Simplify to get \(x = \frac{1}{2}\)

This means the x-intercept is at the point \(\left(\frac{1}{2}, 0\right)\). Finding the x-intercept is important for graphing because it gives you one specific point a line goes through.

Understanding Y-Intercept

The y-intercept is where the graph crosses the y-axis and, at this point, the x-value is zero. To find the y-intercept, simply substitute x with 0 in the linear equation and solve for y. Using the same equation \(y = 4x - 2\), putting x as 0 gives:

• \(y = 4(0) - 2\)
• Resulting in \(y = -2\)

This tells us the y-intercept is at point \((0, -2)\). The y-intercept allows you to start plotting the graph since it is a fixed point on the graph line.

Graphing Lines with Intercepts

Graphing lines using intercepts is a straightforward approach. Once you have both intercepts, plotting a line becomes simple:

• Begin by plotting the y-intercept on the graph. Place a point at \((0, -2)\).
• Next, plot the x-intercept at \(\left(\frac{1}{2}, 0\right)\).
• Use a ruler to connect these two points with a straight line. Make sure to extend the line beyond these points in both directions.

This line is the visual representation of the linear equation \(y = 4x - 2\). Graphing lines this way gives a clear picture of how the equation behaves visually on the coordinate plane.