Problem 16 - Applied Calculus

Question

For each equation, find the slope \(m\) and \(y\) -intercept \((0, b)\) (when they exist) and draw the graph. $$ y=2 x $$

Step-by-Step Solution

Answer

Slope \( m = 2 \), y-intercept \( (0, 0) \); line through \( (0, 0) \) and \( (1, 2) \).

Step 1: Identify the equation's form

The equation provided is \( y = 2x \). This equation is in the slope-intercept form, \( y = mx + b \), which makes it easy to identify the slope \( m \) and the y-intercept \( b \).

Step 2: Determine the slope and y-intercept

From the equation \( y = 2x \), we can see it corresponds to \( y = mx + b \) with \( m = 2 \) and \( b = 0 \). Therefore, the slope \( m \) is 2, and the y-intercept is the point \( (0, 0) \).

Step 3: Graph the equation

Begin by plotting the y-intercept \( (0, 0) \) on the graph. From this point, use the slope \( m = 2 \) to determine another point: for every unit increase in \( x \), \( y \) increases by 2. Therefore, moving one unit to the right from the origin, go up 2 units to point \( (1, 2) \). Connect these points with a straight line, extending in both directions.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is an efficient and straightforward way to express a line. It is written as \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept, which is the point where the line crosses the y-axis. This form is widely used because it readily reveals the key features of a line.

In the equation \( y = mx + b \), 'm' and 'b' are constants. The slope 'm' tells us how steep the line is, while 'b' gives us the starting point of the line on the y-axis. This makes it a handy form for quickly sketching a graph. You simply start at point \((0, b)\) on the y-axis, then use the slope 'm' to find other points on the line. For any linear equation, if you rewrite it in this form, identifying slope and y-intercept becomes simple.

Slope

The slope of a line is a measurement of its steepness. It is represented by the symbol \( m \) in the slope-intercept form and is calculated as the "rise over run."

What does "rise over run" mean? Simply put, it means how much the line goes up (or down) for each step it takes to the right. For example, a slope \( m = 2 \) means that for every 1 unit you move to the right, the line rises 2 units.

In our specific equation \( y = 2x \), the slope is \( m = 2 \). This positive value indicates an upward tilt to the right. If the slope were negative, the line would tilt down instead. Understanding the slope is crucial, as it helps in predicting how the line extends across the graph.

Y-Intercept

The y-intercept is where the line crosses the y-axis. This is given by the point \((0, b)\) in the slope-intercept form \( y = mx + b \).

In other words, it's the value of \( y \) when \( x \) is 0. It's like the starting point of the line on the graph. For example, in our equation \( y = 2x \), the y-intercept is \((0, 0)\). This happens because \( b = 0 \) in this particular equation.

The y-intercept is easily plotted and provides a good reference when graphing a line. Once you have this point, you can apply the slope to find other points and draw the line accurately. So, knowing the y-intercept is vital for charting any linear equation effectively.