Problem 40 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the line satisfying the given conditions. Passing through \((-1,0)\) with slope \(-4\)

Step-by-Step Solution

Answer

The equation is \( y = -4x - 4 \). Use the points \((-1, 0)\) and \((0, -4)\) to sketch the line.

Step 1: Identify the slope-intercept form of a line

The slope-intercept form of a line is given by the equation \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Substitute the given slope

The slope \(m\) is given as \(-4\). Substitute \(m = -4\) into the slope-intercept equation to get \(y = -4x + b\).

Step 3: Use the given point to find the y-intercept

The line passes through the point \((-1,0)\). Substitute \(x = -1\) and \(y = 0\) into the equation \(y = -4x + b\): \[ 0 = -4(-1) + b \] Solving for \(b\), we get: \[ b = 0 - 4 = -4 \] Thus, the y-intercept is \(-4\).

Step 4: Write the final equation of the line

Now, substituting \(b = -4\) into the equation \(y = -4x + b\), we get the final equation of the line: \[ y = -4x - 4 \]

Step 5: Sketch the graph

To sketch the graph, plot the point \((-1, 0)\) and the y-intercept \((0, -4)\). Since the slope is \(-4\), for every 1 unit you move to the right (positive direction along the x-axis), move 4 units down (negative direction along the y-axis). Draw a line through these points to complete the graph.

Key Concepts

Slope-Intercept Form

One of the most common ways to express a linear equation is the slope-intercept form. It is written as:

• \[ y = mx + b \]

Here, m represents the slope of the line. The slope indicates how steep the line is and the direction it slants. A positive slope means the line goes upwards from left to right, while a negative slope means it goes downwards from left to right.

The second term, b, is the y-intercept. This is the point where the line crosses the y-axis, and it tells you the value of y when x is zero.

By understanding the slope-intercept form, you can easily graph any linear equation. You just need to know the slope and the y-intercept. With these, you can quickly sketch the line.

Finding Y-Intercept

To find the y-intercept, we look at our equation in slope-intercept form:

• \[ y = mx + b \]

We need to solve for b, the y-intercept. Often, a given point on the line helps us find b. For instance, if the line passes through

• \((-1, 0)\)

we substitute x = -1 and y = 0 into the equation:\[ y = -4x + b \]
This substitution allows us to solve for b:
\[ 0 = -4(-1) + b \]
Simplifying, we get:

• \[ b = 0 - 4 \text{ or } \b = -4 \]

So, the y-intercept is

• \(-4\)

This means the line crosses the y-axis at

• \( (0, -4) \).

Understanding how to identify and use points like these is crucial in plotting lines on a graph.

Plotting Points

Plotting points is the final step in graphing a linear equation. With the slope and y-intercept identified, choose the points to draw our line. For the example

• \[ y = -4x - 4 \]

we already know it crosses the y-axis at

• \((0, -4)\)

and it passes through another given point,

• \((-1, 0)\).

The slope,

• \(-4\),

is also crucial in plotting. This means:

• For every 1 unit moved to the right, move 4 units down:

Start at the y-intercept

• \((0, -4)\)

From here, move 1 unit to the right (positive x-direction), then 4 units down (negative y-direction). This step gives another point

• \((1, -8)\).

Once you have these points:

• Plot them on a graph, join them with a straight line.

The line you draw represents the linear equation. Combining the understanding of slope, y-intercept, and plotting points, you can graph any linear equation easily.