Problem 26 - Algebra 1: Concepts and Skills

Question

Graph the equation. $$y=x+3$$

Step-by-Step Solution

Answer

The graph is a straight line that intersects the y-axis at the point (0, 3). It slopes upward to the right since the slope is positive.

Step 1: Identify the y-intercept

The y-intercept is the value of 'y' when 'x' is zero. In this equation (\( y = x + 3 \)), the y-intercept is 3.

Step 2: Identify the slope

The slope is the coefficient of 'x'. In this equation (\( y = x + 3 \)), the slope is 1.

Step 3: Plot the y-intercept

Begin at the origin of the graph (where x = 0 and y = 0). From there, move up to where y = 3 and put a point. This point is the y-intercept.

Step 4: Draw the Line

From the y-intercept, as the slope is 1, move up one unit and to the right one unit and make another point. Draw the line through the two points.

Key Concepts

Y-Intercept

The y-intercept is a vital part of graphing linear equations. In any equation of the form \( y = mx + b \), the y-intercept is the value of \( y \) when \( x \) is 0. Essentially, it is where the line crosses the y-axis.
In our equation, \( y = x + 3 \), the y-intercept is 3. This means when \( x = 0 \), \( y = 3 \). Visually, on a graph, you start by placing a point on the y-axis directly at this value, which is 3 in this case. This helps in setting a reference point for the entire line.

• Finding the y-intercept is straightforward—just look for the constant term in the equation without an \( x \).
• This point never changes unless the equation of the line changes.
• Remember, it solely depends on the constant present in the equation.

Understanding the y-intercept grounds your graphing process and is the initial step in plotting a line.

Slope

The slope is a key factor in defining how steep or flat a line appears and dictates its direction. Think of the slope as the line’s rate of change. It tells you how much \( y \) changes with respect to a one-unit increase in \( x \).
In our equation, \( y = x + 3 \), the slope is 1. That means for every step you take horizontally along the x-axis, you also move up by one unit vertically on the y-axis. This movement direction can help you visualize and draw the line.

• Positive slopes, like our 1, suggest the line moves upwards from left to right.
• If the slope were negative, the line would tilt downwards as you move right.
• The steeper the line, the larger the magnitude of the slope.

Grasping how the slope functions allow you to draw accurate lines and understand their implications better.

Plotting Points

Once you've identified both the y-intercept and the slope, it's time to plot these on a graph. Plotting points involve marking positions on a 2-dimensional plane using coordinates.
Here's a simple way to start:

• Use the y-intercept as your starting point. In this case, plot a point at (0, 3) on the y-axis.
• Using the slope, determine your next point. Since the slope here is 1, you move 1 unit right (an increase in x) and 1 unit up (an increase in y) to mark another point.
• Connect these points with a straight edge to extend the line in both directions.

Plotting requires precision in marking your points accurately, as each subsequent point derives from the relationship between x and y defined by the slope.
With these steps, you'll be drawing lines and exploring linear relationships like a pro!