Problem 39 - 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 of the line is y=4x+4.

Step 1: Understand the Problem

We need to sketch the graph of a line that passes through the point (-1,0) and has a slope of 4.

Step 2: Recall the Equation of a Line

The general equation of a line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.

Step 3: Substitute the Slope

Given the slope (m) is 4, substitute this value into the slope-intercept form equation: y = 4x + b.

Step 4: Use the Given Point

Substitute the point (-1, 0) into the equation y = 4x + b to solve for b (the y-intercept). 0 = 4(-1) + b

Step 5: Solve for y-intercept (b)

Solve the equation from the previous step: 0 = -4 + b Thus, b = 4.

Step 6: Write the Final Equation

Substitute the slope (4) and y-intercept (4) into the slope-intercept form to get the final equation: y = 4x + 4.

Step 7: Sketch the Graph

To sketch the graph, plot the y-intercept at (0, 4). From this point, use the slope to plot another point by rising 4 units up and running 1 unit to the right, giving the point (1, 8). Draw the line through these two points.

Key Concepts

slope-intercept form

The slope-intercept form of a linear equation is a super handy way to represent a straight line. It is written as \(y = mx + b\).

• \(m\) is the 'slope' of the line.
• \(b\) is the 'y-intercept', where the line crosses the y-axis.

Knowing \(m\) and \(b\) lets you quickly graph a line. Slope \(m\) tells us how steep the line is, and \(b\) tells us where the line hits the y-axis. This form is useful because it provides a clear and straightforward way to understand and visualize the relationship between \(x\) and \(y\).

plotting points

Plotting points on a graph is essential in understanding the behavior of a linear equation. Begin with the y-intercept \(b\), which is the point where the line crosses the y-axis. For example, in the equation \(y = 4x + 4\), the y-intercept is \(4\), so we plot the point \((0, 4)\).
Next, use the slope to find another point. The slope of \(4\) means 'rise over run', or going up 4 units for every 1 unit you move to the right. From \((0, 4)\), move up 4 and to the right 1, landing at \(1, 8)\). Now, you have two points, and you can draw the line that passes through them.
Always plot at least two points to ensure your line is accurate.

y-intercept

The y-intercept is a crucial part of the slope-intercept form \(y = mx + b\). The y-intercept \(b\) is where the line crosses the y-axis. This means at \(x = 0\), \(y = b\). For our example equation \(y = 4x + 4\), the y-intercept is \((0, 4)\). When graphing, start here and use the slope to find other points on the line.

It's important to correctly identify and plot the y-intercept because it anchors your line on the graph. From this starting point, you can use the slope to determine direction and steepness, ensuring your graph accurately represents the equation.