Problem 77 - A Graphical Approach to Precalculus with Limits

Question

Sketch by hand the line that passes through the points \((1,-2)\) and \((3,2)\).

Step-by-Step Solution

Answer

The line passing through the points can be sketched using the equation \(y = 2x - 4\).

Step 1: Calculate the Slope

To sketch a line through two points \((1, -2)\) and \((3, 2)\), first calculate the slope (m) using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Plugging in the points, we have \(m = \frac{2 - (-2)}{3 - 1} = \frac{4}{2} = 2\). So, the slope of the line is 2.

Step 2: Choose the Point-Slope Formula

Use the point-slope form of the equation to write the equation of the line. The formula is \(y - y_1 = m(x - x_1)\). We use the point \((1, -2)\) with the slope \(m = 2\).

Step 3: Write the Equation of the Line

Substitute the point \((1, -2)\) and slope \(m = 2\) into the point-slope formula: \(y + 2 = 2(x - 1)\). Simplify this to get the standard form: \(y = 2x - 4\).

Step 4: Plot the Points

On a coordinate plane, plot the given points \((1, -2)\) and \((3, 2)\). These points will guide the sketching of the line.

Step 5: Draw the Line

Using a ruler, draw a straight line that passes through the two plotted points \((1, -2)\) and \((3, 2)\). Ensure that the line is straight and extends in both directions.

Key Concepts

Slope Calculation

Understanding how to calculate the slope of a line is a vital skill in algebra. The **slope** of a line measures its steepness and direction. To find the slope between two points, \((x_1, y_1)\) and \((x_2, y_2)\), use the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

\(m\) represents the slope.
It's a simple subtraction between the \(y\) coordinates, divided by the subtraction of the \(x\) coordinates. With the points given, \((1, -2)\) and \((3, 2)\), we calculate:

• \( m = \frac{2 - (-2)}{3 - 1} = \frac{4}{2} = 2 \)

The result \( m = 2 \) indicates a line rising up two units for each unit it goes to the right.

Point-Slope Form

The **point-slope form** of a linear equation is another way to represent a line using its slope and a point on the line. The formula is:

• \( y - y_1 = m(x - x_1) \)

The point-slope form is particularly useful when you know a point on the line and the slope. Simply plug in these values to form an equation.
For example, with the point \((1, -2)\) and a slope \( m = 2 \), it translates to:

• \( y + 2 = 2(x - 1) \)

This gives a dynamic way to express the equation, which can be adjusted for different points or slopes.

Standard Form

Converting a line's equation from point-slope form to **standard form** is often beneficial for calculations. Standard form looks like this:

• \( Ax + By = C \)

Where \(A\), \(B\), and \(C\) are integers, and \(A\) should be positive.
From our earlier point-slope equation, \( y + 2 = 2(x - 1) \), simplify to the following steps:

• Distribute the \(2\): \( y + 2 = 2x - 2 \)
• Rearrange: \( y = 2x - 4 \)

This is now a linear equation in standard form. Although not in the perfect \(Ax + By = C\) form, it effectively helps identify critical parts like slope and y-intercept.

Coordinate Plane

A **coordinate plane** is a two-dimensional surface on which we can plot points, lines, and curves to visualize algebraic equations. The coordinate plane is formed by two perpendicular number lines intersecting at the origin (0,0):

• The horizontal axis is the x-axis.
• The vertical axis is the y-axis.

Each point on the plane is defined by an ordered pair, such as \((x, y)\).
The line passing through the points \((1, -2)\) and \((3, 2)\) can be sketched by plotting these points on the graph. Each point identifies a specific position on the plane. Connecting these points with a straight line provides a visual representation of the equation \( y = 2x - 4 \). It effectively shows the slope and the path through multiple pairs that satisfy the equation.