Problem 8 - The Calculus with Analytic Geometry

Question

In Exercises 7 through 28 , draw a sketch of the graph of the equation. $$ y=4 x-3 $$

Step-by-Step Solution

Answer

Plot the y-intercept (0, -3), use the slope to find another point (1, 1), and draw a line through these points.

Step 1 - Identify the type of equation

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

Step 2 - Determine the slope and y-intercept

From the equation \(y = 4x - 3\), compare it with the form \(y = mx + b\). Here, the slope \(m\) is 4 and the y-intercept \(b\) is -3.

Step 3 - Plot the y-intercept

Start by plotting the y-intercept, which is the point where the line crosses the y-axis. For \(y = 4x - 3\), the y-intercept is -3. Plot the point (0, -3) on the graph.

Step 4 - Use the slope to find another point

The slope \(m\) indicates the rise over run. For \(m = 4\), rise 4 units up and run 1 unit to the right. Starting from the y-intercept (0, -3), move up 4 units to (0, 1) and then 1 unit to the right to (1, 1). Plot this point (1, 1).

Step 5 - Draw the line

Connect the points (0, -3) and (1, 1) with a straight line. Extend the line in both directions, ensuring it passes through the plotted points. This is the graph of the equation \(y = 4x - 3\).

Key Concepts

slope-intercept form

The slope-intercept form of a linear equation is a way to write the equation of a line so that you can instantly know its slope and y-intercept. The general form is \(y = mx + b\), where

• x is the independent variable (often representing horizontal distance).
• y is the dependent variable (often representing vertical distance).
• m is the slope of the line.
• b is the y-intercept, the point where the line crosses the y-axis.

This form makes it easy to graph the equation because it directly tells you two things: how steep the line is (slope) and where it starts (y-intercept). Understanding the slope and y-intercept helps break down the problem into simpler steps you can follow.

slope

Slope is a measure of the steepness or incline of a line. It tells you how much y changes for a given change in x. In the slope-intercept form \(y = mx + b\), slope is represented by m. You can think of it as 'rise over run'.
For example, if the slope \(m = 2\), it means that for every unit you move right along the x-axis, the y-coordinate goes up by 2 units. If the slope is negative, the line will slope downwards. In our equation \(y = 4x - 3\), the slope \(m\) is 4. This means:

• For every 1 unit you move right horizontally, you move up 4 units vertically.

The slope is crucial for determining the direction and angle of your line on the graph.

y-intercept

The y-intercept is the point where the line crosses the y-axis. This occurs when x = 0. In the slope-intercept form \(y = mx + b\), b represents the y-intercept. For our equation \(y = 4x - 3\), the y-intercept is -3. This means the line crosses the y-axis at the point (0, -3). Plotting this point on the graph gives you a starting point for drawing the line. Without this point, you wouldn't know where the line begins its journey. Remember, this is the vertical intercept of the line, which provides an easy-to-find location for initiating your graphing process.

plotting points

Plotting points is essential for accurately drawing lines on a graph. To graph the equation \(y = 4x - 3\), start with the y-intercept. In this case, plot the point (0, -3) since b = -3. Next, use the slope to find another point. With a slope of 4, move up 4 units and 1 unit to the right from (0, -3) to get to (1, 1). Plot this second point (1, 1).
With these two points, draw a straight line passing through them. This line is the graph of the equation. By plotting accurately, you ensure your graph reflects the true behavior of the equation. Be precise with your points, and your line will reliably show the relationship between x and y as described by your equation.