Problem 82 - Intermediate Algebra

Question

Graph each equation. $$y=-3 x+3$$

Step-by-Step Solution

Answer

The graph is a line with a slope of -3 and y-intercept at (0, 3).

Step 1: Identify the Equation Form

The given equation is \(y = -3x + 3\). This is in the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Determine the Slope and Y-Intercept

From the equation \(y = -3x + 3\), we can see that the slope \(m\) is \(-3\), and the y-intercept \(b\) is \(3\). This means the line crosses the y-axis at the point \((0, 3)\).

Step 3: Plot the Y-Intercept

Plot the y-intercept on the graph. Since \(b = 3\), place a point on the y-axis at \((0, 3)\).

Step 4: Use the Slope to Find Another Point

The slope \(m = -3\) means that for every 1 unit increase in \(x\), \(y\) decreases by 3 units. From the y-intercept point \((0, 3)\), move 1 unit to the right to \(x = 1\), then 3 units down to \(y = 0\), placing a point at \((1, 0)\).

Step 5: Draw the Line Through the Points

Draw a straight line through the points \((0, 3)\) and \((1, 0)\) extending in both directions to represent the equation \(y = -3x + 3\).

Key Concepts

Understanding the Slope-Intercept Form

The slope-intercept form of a linear equation is a crucial concept when graphing lines on a coordinate plane. It is typically written as \( y = mx + b \), where:

• \( y \) represents the dependent variable.
• \( x \) stands for the independent variable.
• \( m \) is the slope of the line, indicating its steepness and direction.
• \( b \) is the y-intercept, or the point where the line crosses the y-axis.

This form is extremely user-friendly, as it explicitly tells you the slope and y-intercept, making it easier to quickly plot the line on a coordinate plane. Let's explore slope and y-intercept further to understand how they impact the graph.

Exploring the Slope

The slope is an essential element of any linear equation. It describes how steep a line is and the direction it moves across the coordinate plane. For the equation \( y = -3x + 3 \), the slope \( m \) is \(-3\).A negative slope means the line slants downward as it moves from left to right. In simple terms, when \( x \) increases by 1, \( y \) decreases by 3. Hence, you can think of it as a rise over run where "rise" refers to a change in \( y \) and "run" refers to a change in \( x \):

• A positive slope moves upward.
• A negative slope moves downward.
• A slope of zero results in a horizontal line.
• An undefined slope results in a vertical line.

Understanding slope helps in visualizing how the line behaves as you move along the x-axis.

Grasping the Y-Intercept

The y-intercept is the point on a coordinate plane where the line crosses the y-axis. This point is extremely significant because it gives a starting place for plotting the line. In the equation \(y = -3x + 3\), the y-intercept \(b\) is \(3\).This means that when \(x\) is 0, \(y\) is 3. You can plot this exactly on the graph at the point \((0, 3)\). This point acts as the anchor for the line, and from here, the line's slope will determine its path across the plane. Always remember:

• The y-intercept is where the line intersects the y-axis.
• It offers a clear initial point to draw the line.

Knowing the y-intercept allows you to start graphing with a concrete point and apply the slope to determine the line's slope and further points.

The Coordinate Plane

A coordinate plane is a two-dimensional surface defined by two perpendicular axes: the horizontal axis (x-axis) and the vertical axis (y-axis). Each point on this plane is given by a pair of numbers known as coordinates \((x, y)\).When graphing equations like \(y = -3x + 3\), the coordinate plane becomes a stage where the equation showcases its graphical representation as a line:

• The x-axis (horizontal) and y-axis (vertical) intersect at the origin \((0, 0)\).
• Using the slope \(-3\) and the y-intercept \((0, 3)\), the line can be drawn effortlessly.
• It helps visualize relationships, making abstract concepts tangible.

Navigating the coordinate plane requires understanding how both axes work and how different points relate to the equation being graphed.