Problem 27 - College Algebra and Calculus: An Applied Approach

Question

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. \(y=2 x-1\)

Step-by-Step Solution

Answer

The \( x \) - intercept is \( 0.5 \) and the \( y \) - intercept is \( -1 \)

Step 1: Find the \( x \) - intercept

Set \( y = 0 \) to find out where the graph intersects the \( x \) - axis. Solve the equation \( 0 = 2x - 1 \) for \( x \). In this case, by adding 1 to both sides and then dividing by 2, you will find that when \( y = 0 \), \( x = 0.5 \). So the graph intersects the \( x \) - axis at \( x = 0.5 \).

Step 2: Find the \( y \) - intercept

Set \( x = 0 \) to find out where the graph intersects the \( y \) - axis. Replace \( x \) with 0 in the original equation to get: \( y = 2(0) - 1 \). With simple calculations, one can see that when \( x = 0 \), \( y = -1 \). So the graph intersects the \( y \) - axis at \( y = -1 \)

Step 3: Summary of the \( x \) - and \( y \) - intercepts

From Steps 1 and 2, it can be concluded that the \( x \) - intercept is \( 0.5 \) and the \( y \) - intercept is \( -1 \)

Key Concepts

X-Intercept Calculation

Understanding the x-intercept of a graph is essential for students studying algebra. The x-intercept is the point where a graph crosses the x-axis, which means it's the point where the value of y is zero. To find this point for the linear equation y = 2x - 1, we set the y variable to zero and solve for x. Here's how it's done:

y = 2x - 1
0 = 2x - 1
2x = 1
x = 0.5

This calculation reveals that the x-intercept of the graph of y = 2x - 1 is (0.5, 0). When you're graphing the line, you'd simply place a point at 0.5 along the x-axis to mark where the line will cross it. It's important to accurately plot this point, as it's one of the two key points needed to draw the entire line.

Y-Intercept Calculation

Conversely, the y-intercept is found where the graph intersects the y-axis, which occurs when x equals zero. To find the y-intercept of the given equation, we insert zero in place of x and calculate the value of y:

y = 2(0) - 1
y = -1

This simple substitution shows that the graph of the equation crosses the y-axis at (0, -1). To graph this point, you would place a dot at -1 on the y-axis. This y-intercept is crucial because it indicates the starting point of the line on the graph, often referred to as the 'initial value' in the context of the equation.

Linear Equation Graphing

Graphing linear equations like y = 2x - 1 involves using the intercepts calculated earlier for a clear visual representation of the equation. Start by plotting both the x-intercept (0.5, 0) and the y-intercept (0, -1) on a coordinate plane. Once these points are marked, draw a straight line through them, extending it in both directions. This line represents all the possible solutions to the equation and is the visual essence of graphing linear equations.

Remember, the slope, which in this case is 2, indicates how steep the line is. It reflects the change in y (rise) for every unit change in x (run). For y = 2x - 1, the line rises 2 units for every 1 unit it moves to the right. By connecting the intercepts with consideration of the slope, you accurately depict the behavior of the linear equation on a two-dimensional plane.