Problem 41 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the given equation. Label the intercepts. $$y=-0.2 x+1.4$$

Step-by-Step Solution

Answer

The graph is a straight line with y-intercept (0, 1.4) and x-intercept (7, 0).

Step 1: Identify the Equation Type

The given equation is in the slope-intercept form, which is written as \(y = mx + b\). Here, \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Determine the Slope and Y-Intercept

From the equation \(y = -0.2x + 1.4\), the slope \(m = -0.2\) and the y-intercept \(b = 1.4\). The y-intercept is where the graph crosses the y-axis.

Step 3: Plot the Y-Intercept

Plot the point \( (0, 1.4) \) on the y-axis, as this is the y-intercept.

Step 4: Use the Slope to Find Another Point

The slope \(m = -0.2\) indicates that for every unit increase in \(x\), \(y\) decreases by 0.2. From the y-intercept \( (0, 1.4) \), move 5 units to the right along the x-axis (to \(x = 5\)), then move down 1 unit (since \(-0.2 \times 5 = -1\)). Plot the point \( (5, 0.4) \).

Step 5: Connect the Points

Draw a straight line through the points \( (0, 1.4) \) and \( (5, 0.4) \). Extend the line in both directions.

Step 6: Label the X-Intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\):\[\begin{align*}0 &= -0.2x + 1.4\ \Rightarrow 0.2x &= 1.4\ \Rightarrow x &= 7\end{align*}\] So, the x-intercept is \( (7, 0) \). Plot and label this point.

Step 7: Final Graph

Ensure the graph is properly labeled, particularly the intercepts at points \( (0, 1.4) \) and \( (7, 0) \).

Key Concepts

slope-intercept form

The slope-intercept form is a way of expressing linear equations. It looks like this: \( y = mx + b \). In this form, m represents the slope of the line, and b represents the y-intercept.
When given an equation in this form, you can quickly identify how steep the line is by looking at the slope and where the line crosses the y-axis by looking at the y-intercept.
For example, in the equation \( y = -0.2x + 1.4 \), the slope \( m = -0.2 \), and the y-intercept \( b = 1.4 \).
This form is very user-friendly because it gives you direct information on how to graph the line.

y-intercept

The y-intercept of a line is where the graph crosses the y-axis. In simpler terms, it's the value of y when x is zero. You can find it quickly in the slope-intercept form. It’s the b in the equation \( y = mx + b \). For example, in \( y = -0.2x + 1.4 \), the y-intercept \( b = 1.4 \).
This means that the point (0, 1.4) is where the line crosses the y-axis.
To plot this, simply go to the y-value of 1.4 on the graph and mark it. This is an essential step in sketching the graph correctly.

x-intercept

The x-intercept is the point where the line crosses the x-axis. To find it, set y to 0 in the equation and solve for x.
In our example with the equation \( y = -0.2x + 1.4 \), you set \( y = 0 \) and solve like this:
\begin{align*} 0 &= -0.2x + 1.4 \Rightarrow 0.2x &= 1.4 \Rightarrow x &= 7 correlation table. For instance, in our example, we have the point (7, 0) as the x-intercept. To graph it, plot the point (7, 0) on the x-axis. This point is crucial because it helps us draw the line accurately.

slope

Slope indicates the steepness and direction of a line. In the equation \( y = mx + b \), m is the slope. It describes how much y changes for a one-unit change in x.
Positive slopes rise to the right, while negative slopes fall to the right. In our example \( y = -0.2x + 1.4 \), the slope \( m = -0.2 \) tells us that for every 1 unit increase in x, y decreases by 0.2 units.
To use the slope, start from the y-intercept (0, 1.4) and move based on the slope. Here, for a 5-unit increase in x-direction, we have a 1 unit decrease in y-direction because (-0.2 * 5 = -1). Therefore, we get the point (5, 0.4). Connecting these points gives you the direction of the line accurately.