Problem 76 - Algebra and Trigonometry

Question

Find a function whose graph is the given curve. The line segment joining the points \((-3,-2)\) and \((6,3)\)

Step-by-Step Solution

Answer

The function is \( y = \frac{5}{9}x - \frac{1}{3} \).

Step 1: Identify the Given Points

We are given two points on the line, \((-3,-2)\) and \((6,3)\). We need to find the equation of the line that joins these two points.

Step 2: Calculate the Slope

To find the slope, use the formula \(m = \frac{y_2-y_1}{x_2-x_1}\). Substitute \((x_1, y_1) = (-3, -2)\) and \((x_2, y_2) = (6, 3)\). Calculate: \[ m = \frac{3 - (-2)}{6 - (-3)} = \frac{3 + 2}{6 + 3} = \frac{5}{9} \]. Thus, the slope \(m\) is \(\frac{5}{9}\).

Step 3: Use the Point-Slope Formula

With the slope \(m = \frac{5}{9}\) and one of the points, use the point-slope form of a line: \( y - y_1 = m (x - x_1) \). Here choose \((x_1, y_1) = (-3, -2)\). Plug in the values: \[ y + 2 = \frac{5}{9} (x + 3) \].

Step 4: Simplify the Equation

Distribute the slope on the right side of the equation:\[ y + 2 = \frac{5}{9}x + \frac{15}{9} \]. Subtract 2 from both sides to solve for \(y\):\[ y = \frac{5}{9}x + \frac{15}{9} - 2 \]. Convert 2 into a fraction with a denominator of 9:\[ y = \frac{5}{9}x + \frac{15}{9} - \frac{18}{9} \].Simplify further:\[ y = \frac{5}{9}x - \frac{3}{9} \]\[ y = \frac{5}{9}x - \frac{1}{3} \].

Step 5: Write the Final Answer

The function that describes the line joining the given points is:\[ y = \frac{5}{9}x - \frac{1}{3} \].

Key Concepts

Slope Calculation

To understand how to calculate the slope of a line, let's start by exploring what the slope represents. The slope measures the steepness and direction of a line. It tells us how much the line rises or falls as we move from left to right on a graph. The formula to calculate the slope, denoted as \( m \), between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[ m = \frac{y_2-y_1}{x_2-x_1} \]This formula essentially captures how much the \( y \)-coordinate (vertical change) changes for a given change in the \( x \)-coordinate (horizontal change).

• In our example, the points are \((-3, -2)\) and \((6, 3)\).
• Calculate the difference in \( y \)-values, which is \(3 - (-2)\), and the difference in \( x \)-values, which is \(6 - (-3)\).
• The result is a slope of \( \frac{5}{9} \), indicating a gentle rise as you move from left to right.

Understanding the slope is crucial because it helps define the behavior of a line and is a foundational component in forming linear equations.

Point-Slope Form

The point-slope form is a method for writing the equation of a line when you know the slope and one point on the line. The equation of the point-slope form is given by:\[ y - y_1 = m(x - x_1) \]Here, \((x_1, y_1)\) is a known point on the line, and \( m \) is the slope.

• In our example, we chose the point \((-3, -2)\) and the slope \( \frac{5}{9} \).
• Plugging these into the formula results in: \[ y + 2 = \frac{5}{9}(x + 3) \].

Using the point-slope form is helpful because it directly integrates the slope and a specific point, making it easy to transition into other forms like the slope-intercept form. Additionally, this form is flexible, allowing you to choose either given point for finding the same line equation.

Linear Equation

A linear equation represents a straight line on a graph, and it can be written in several forms, with the slope-intercept form being the most common:\[ y = mx + b \]- Here, \( m \) is the slope, and \( b \) is the \( y \)-intercept, which is the point where the line crosses the \( y \)-axis.- The standard form of a linear equation is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants.In our example, after using the point-slope form, we simplified to:\[ y = \frac{5}{9}x - \frac{1}{3} \]

• This equation is in the slope-intercept form, where \( \frac{5}{9} \) is the slope, and \( -\frac{1}{3} \) is the \( y \)-intercept.
• Understanding how to manipulate and convert between different linear equation forms is crucial in algebra, as it provides multiple perspectives on the behavior of a line.