Problem 4 - Calculus Single Variable

Question

Sketch the line that passes through point (-2,5) and that rises 7 units for every 2 units of left-to-right motion.

Step-by-Step Solution

Answer

The line through (-2,5) with a slope of \(\frac{7}{2}\) is \(y = \frac{7}{2}x + 12\).

Step 1: Determine the slope

The slope of a line is a measure of how steep the line is. In this exercise, the line rises 7 units for every 2 units of horizontal motion. This means the slope \( m = \frac{7}{2} \), as slope is calculated by rise over run.

Step 2: Write the point-slope form of the line

We utilize the point-slope form of a line equation, \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a point the line passes through and \( m \) is the slope. Here, the line passes through \((-2, 5)\), so \(x_1 = -2\) and \(y_1 = 5\). Substituting these values, we have: \( y - 5 = \frac{7}{2}(x + 2) \).

Step 3: Simplify the equation

To make the equation easier to graph, convert it into the slope-intercept form \( y = mx + b \). Start by distributing: \( y - 5 = \frac{7}{2}x + 7 \). Add 5 to both sides to solve for \( y \): \( y = \frac{7}{2}x + 12 \).

Step 4: Sketch the line

Using the slope-intercept form \( y = \frac{7}{2}x + 12 \), identify the y-intercept \( b = 12 \), where the line crosses the y-axis. From the point \((0, 12)\), use the slope \( \frac{7}{2} \) to find another point by moving 2 units right and 7 units up, ending at \((2, 19)\). Connect these points to draw the line. Note, the original point \((-2, 5)\) should also be on this line.

Key Concepts

Point-Slope Form

The point-slope form is a powerful way to write the equation of a line when you know a point on the line and the slope. It's particularly useful in situations where you only have partial information about the line, like in this exercise where the line goes through a specific point and has a known slope.
Effectively, the point-slope form is written as: \[ y - y_1 = m(x - x_1) \] Where

• \((x_1, y_1)\) is a given point on the line
• \(m\) is the slope of the line.

By plugging in these values into the formula, you make the relationship between your known point and the slope explicit. This is particularly valuable for graphing or converting to other forms like slope-intercept form. In this exercise, knowing the point \((-2, 5)\) and the slope \(\frac{7}{2}\) allows us to directly determine the equation in point-slope form: \( y - 5 = \frac{7}{2}(x + 2) \). This representation makes it easy to transform the equation into different forms as needed.

Line Through a Point

Understanding how to find a line through a point is crucial for working with linear equations. In mathematics, a line can be precisely described if you have a point that it passes through and its slope. Imagine you want to draw a line on a graph. It leaves the starting point, like the one at \((-2, 5)\), and follows the direction determined by the slope.
The slope, which is the rate of change as you move along the line, helps establish how steep the line is. With a slope of \(\frac{7}{2}\), this means for every 2 units you move horizontally (either to the left or right), you rise or fall 7 units vertically.
Creating the equation of a line using these tidbits of information not only anchors the line in place but also sets its direction, like tracing a path through a forest by always moving in the same direction as a compass.

Graphing Linear Equations

Graphing linear equations reveals the visual representation of mathematical relationships. A linear equation typically forms a straight line when plotted on a graph. Here's how to sketch a line using the slope-intercept form:

• First, identify the y-intercept from the line equation, where the line crosses the y-axis. In our example, the line given as \( y = \frac{7}{2}x + 12 \) means the y-intercept is 12.
• Begin by plotting this point (0, 12) on the graph.
• Next, use the slope \(\frac{7}{2}\). Starting from your first point, move 2 units right (horizontal movement) and 7 units up (vertical rise) to determine another point, like (2, 19).
• With these two points plotted, draw a straight line through them.
• Ensure previously given points on the line, like (-2, 5), also align perfectly with it.

By following these steps, you'll accurately represent the equation on a graph, offering a visual insight into how linear equations interact with variables.