Problem 3 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

On the same set of axes, sketch lines through point \((0,1)\) with the slopes indicated. Label the lines. (a) slope \(=0\) (b) slope \(=\frac{1}{2}\) (c) slope \(=1\) (d) slope \(=2\) (e) slope \(=-\frac{1}{2}\) (f) slope \(=-1\) (g) slope \(=-2\)

Step-by-Step Solution

Answer

Step-by-step visualization of different slopes using lines that pass through the point (0,1). Lines are labeled as \(m=0\), \(m=1/2\), \(m=1\), \(m=2\), \(m=-1/2\), \(m=-1\), and \(m=-2\) on the same set of axes.

Step 1: Plot and Label for Slope = 0

Start with the point (0,1) on the coordinate system. Since the slope is 0, implying the line is horizontal, simply draw a horizontal line through the point (0,1). Label it as \(m=0\)..

Step 2: Plot and Label for Slope = 1/2

Again, start from the same point (0,1). Now, as the slope is positive and less than 1, the line inclines upward to the right but with a slight slope. Draw and label it as \(m=1/2\).

Step 3: Plot and Label for Slope = 1

With slope = 1, the line inclines upward at a 45 degree angle to the x-axis. Draw and label it as \(m=1\).

Step 4: Plot and Label for Slope = 2

This time, the slope is steeper than the previous one. Draw a line from point (0,1) that inclines sharply upward to the right. Label it as \(m=2\).

Step 5: Plot and Label for Slope = -1/2

Here, the slope is negative and less than 1 so the line slants downward, but with a slight slope from the point (0,1). Draw and label it as \(m=-1/2\)..

Step 6: Plot and Label for Slope = -1

With a slope of -1, the line will incline downward at a 45 degree angle to the x-axis. Draw and label it as \(m=-1\).

Step 7: Plot and Label for Slope = -2

This slope is more negative than the previous one, implying a steeper decline. Draw a line from the point (0,1) to the right that inclines sharply downward. Label it as \(m=-2\)..

Key Concepts

slope-intercept form

The slope-intercept form is a widely used method for expressing the equation of a straight line. It's given by the formula \(y = mx + b\). Here, \(m\) represents the slope of the line, while \(b\) is the y-intercept, which is where the line crosses the y-axis.
This form is extremely helpful when graphing linear equations because it clearly shows both the slope and the y-intercept. As soon as you have these two values, you can easily draw the line.
For example, if you are working with the slope-intercept form \(y = 2x + 3\), it tells you that the line crosses the y-axis at (0,3) and rises 2 units for every 1 unit it moves to the right.

plotting points

Plotting points is a fundamental skill in graphing. It's about marking specific coordinates on a graph. Each point is defined by a pair of numbers (x, y), which tell you where to place the point on a coordinate grid.
To plot points, you first move along the x-axis (horizontal) to the x-coordinate, and then move vertically to reach the y-coordinate. The spot where these two movements intersect is where your point should be placed.
For instance, to plot the point (0, 1), you do not need to move along the x-axis because x=0. You just move up to 1 on the y-axis and place your point there. That's how you start drawing the lines described in linear equations.

coordinate system

The coordinate system is like a map you use to find points and draw lines on a plane. This system is made up of two axes: the x-axis running horizontally and the y-axis vertically. The point where these two axes intersect is called the origin. Its coordinates are (0,0).
Using the coordinate system allows us to visualize equations graphically and understand how numbers correspond to positions. It's a grid where each point is identified by a unique pair of numbers, (x, y).
The coordinate system is essential when graphing linear equations because it helps accurately track the slope and direction of lines, ensuring that your plotted points and drawn lines are accurate and true to the equation they're based on.

negative slopes

A negative slope in a linear equation indicates that the line decreases; it falls as it moves from left to right across the graph. This slope is represented by a negative number, and it means that for every unit increase along the x-axis, the y-value decreases by the amount of the slope.
Negative slopes create downward-slanting lines. For example, a slope of -1 signifies that you move down one unit on the y-axis for every one unit you move right on the x-axis. A steeper negative slope, like -2, results in a more pronounced downward angle.
Understanding negative slopes is key to predicting how a line should look on a graph and ensuring your drawing accurately reflects the equation's behavior across a coordinate system.