Problem 43 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the given equation. Label the intercepts. $$y=0.75 x-4.8$$

Step-by-Step Solution

Answer

The y-intercept is \((0, -4.8)\) and the x-intercept is \((6.4, 0)\). Sketch the line passing through these points.

Step 1: Identify the slope and y-intercept

The given equation is in slope-intercept form: \(y = mx + b\). Here, \(m = 0.75\) and \(b = -4.8\). So, the slope is 0.75 and the y-intercept is -4.8.

Step 2: Plot the y-intercept

Start by plotting the y-intercept point on the graph at \((0, -4.8)\). This is the point where the line crosses the y-axis.

Step 3: Use the slope to find a second point

The slope of 0.75 means rising 0.75 units for every 1 unit you move to the right. From \((0, -4.8)\), move 1 unit right to \((1, -4.8 + 0.75) = (1, -4.05)\). Plot this second point at \((1, -4.05)\).

Step 4: Draw the line

Draw a straight line through the two points \((0, -4.8)\) and \((1, -4.05)\). Extend the line across the graph.

Step 5: Determine the x-intercept

To find the x-intercept, set \(y = 0\) and solve for \(x\): \[ 0 = 0.75x - 4.8 \Rightarrow 0.75x = 4.8 \Rightarrow x = \frac{4.8}{0.75} = 6.4 \]. So the x-intercept is at \((6.4, 0)\). Plot this point and label it.

Step 6: Label other points and finalize the graph

Label the intercepts that are plotted. The y-intercept \((0, -4.8)\) and the x-intercept \((6.4, 0)\). Ensure all points and the line are accurately drawn and labeled.

Key Concepts

slope-intercept form

The slope-intercept form of a linear equation is one of the most popular ways to express a linear function. It's written as \(y = mx + b\), where

• \(m\) represents the slope of the line
• \(b\) represents the y-intercept of the line

This format makes it straightforward to read off the slope and the y-intercept directly from the equation. For example, in the equation \(y = 0.75x - 4.8\), the slope \(m\) is 0.75, and the y-intercept \(b\) is -4.8. This tells us that the line will cross the y-axis at -4.8 and will rise 0.75 units for every 1 unit it moves to the right.

slope

The slope of a line shows how steep the line is. It is calculated as the change in the y-coordinate divided by the change in the x-coordinate (\text{rise \rightarrow run}). In our example equation, \(y = 0.75 x - 4.8\), the slope is 0.75. This means for each unit you move to the right along the x-axis, you'll move up 0.75 units along the y-axis.
To visualize this, start from a point you know on the graph, such as the y-intercept. From \text(0, -4.8\text), if you move 1 unit right (in the positive x-direction), you will move 0.75 units up (in the positive y-direction), reaching the point \text(1, -4.05\text). Plotting these points and understanding how to ‘rise’ and ‘run’ according to the slope is crucial for accurate graphing.

intercepts

Intercepts are the points where a line crosses the axes. They help us understand key points on the graph.
To find the y-intercept, look at the constant term, \(b\), in the slope-intercept form equation \(y = mx + b\). This will be the y-value when x = 0. For example, in \(y = 0.75x - 4.8\), the y-intercept is -4.8. Plot this point at \text(0, -4.8\text) on the graph.
To find the x-intercept, set y to 0 in the equation and solve for x. For \(y = 0.75x - 4.8\):
\text{ 0 = 0.75x - 4.8\text}
\text{ 0.75x = 4.8\text}
\text{ x = \frac{4.8}{0.75}\text}
This gives x = 6.4. Plot this point at \text(6.4, 0\text) on the graph. Label these intercepts clearly on your graph to provide reference points.

plotting points

Plotting points accurately is essential for drawing a precise graph of a linear equation. Start with easy-to-identify points such as the intercepts.
For \(y = 0.75x - 4.8\), plot the y-intercept \text(0, -4.8\text) first, as it is a key reference point where the line crosses the y-axis. Then use the slope to find another point. From \text(0, -4.8\text), if the slope is 0.75, moving one unit right along the x-axis means you move 0.75 units up on the y-axis, giving the point \text(1, -4.05\text).
Plot this second point, then draw a straight line through these points to graph the equation. Extending this line will cover more points modeled by the equation. Double-check that your plotted points align with the determined slope and intercepts for an accurate graph. Label all critical points and your graph is complete!