Problem 42 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the given equation. Label the intercepts. $$y=0.3 x-2.1$$

Step-by-Step Solution

Answer

Plot the y-intercept -2.1, use the slope 0.3 to plot another point, draw the line, and label intercepts at (0, -2.1) and (7, 0).

Step 1: Understand the Equation

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

Step 2: Identify the Slope and Intercept

In the equation \( y = 0.3x - 2.1 \), the slope \( m \) is 0.3 and the y-intercept \( b \) is -2.1.

Step 3: Plot the Y-Intercept

Start by plotting the y-intercept. Locate -2.1 on the y-axis and place a point there. This is the point (0, -2.1).

Step 4: Use the Slope to Find Another Point

The slope of 0.3 means that for every 1 unit increase in \( x \), \( y \) increases by 0.3. Starting from (0, -2.1), move 1 unit to the right and 0.3 units up to plot the point (1, -1.8).

Step 5: Draw the Line

Using a ruler, draw a straight line through the points (0, -2.1) and (1, -1.8). This line represents the graph of \( y = 0.3x - 2.1 \).

Step 6: Find the X-Intercept

To find the x-intercept, set \( y = 0 \) and solve for \( x \). Solve \( 0 = 0.3x - 2.1 \) by isolating \( x \):\[0.3x = 2.1\] \[x = \frac{2.1}{0.3} \approx 7\]. The x-intercept is (7, 0).

Step 7: Label the Intercepts

Label the points (0, -2.1) and (7, 0) as the y-intercept and x-intercept, respectively, on the graph.

Key Concepts

Slope-Intercept Form

Slope-intercept form is a way of writing linear equations that makes it easy to graph them. The general form is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This format allows us to quickly identify how steep a line is (slope) and where it crosses the y-axis (y-intercept). Understanding this form is crucial for graphing linear equations.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis. In the equation \( y = 0.3x - 2.1 \), the y-intercept is -2.1, meaning the line crosses the y-axis at the point (0, -2.1). This point is significant because it serves as a starting point for plotting the line.
To plot the y-intercept, simply locate -2.1 on the y-axis and place a point there. This helps you anchor the graph and serves as a reference for drawing the rest of the line.

X-Intercept

The x-intercept is the point where the line crosses the x-axis. To find it, set \( y \) to 0 and solve for \( x \). In our example, we start with the equation \( 0 = 0.3x - 2.1 \). Solving for \( x \), we get:
\[ 0.3x = 2.1 \]
\[ x = \frac{2.1}{0.3} \ x \approx 7 \]
The x-intercept is (7, 0). This tells us that the line touches the x-axis at (7, 0). Label this point on your graph to make understanding easier.

Plotting Points

Plotting points is the process of graphing points on a coordinate plane to visualize a linear equation. Start by identifying key points like the intercepts. First, plot the y-intercept. In the equation \( y = 0.3x - 2.1 \), plot the point (0, -2.1).
Next, use the slope to find other points. The slope of 0.3 means for every 1 unit right, move 0.3 units up. From (0, -2.1), go 1 unit right and 0.3 units up to plot (1, -1.8).
Connect the points with a straight line. Repeat this for other points if needed. This visual representation aids in understanding the equation.