Problem 80 - 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: Determine the Slope

To find the slope of the line (denoted as \( m \)), use the formula: \( m = \frac{y_2-y_1}{x_2-x_1} \). For the points \( (-3,-2) \) and \( (6,3) \), substitute the coordinates: \( m = \frac{3 - (-2)}{6 - (-3)} = \frac{5}{9} \). So, the slope of the line is \( \frac{5}{9} \).

Step 2: Use the Point-Slope Form Equation

The equation of a line using the point-slope formula is \( y - y_1 = m(x - x_1) \). Choose one of the given points, let's use \( (-3, -2) \). Substituting the values, we get: \( y + 2 = \frac{5}{9}(x + 3) \).

Step 3: Simplify the Equation

To express the equation in the slope-intercept form (\( y = mx + b \)), simplify \( y + 2 = \frac{5}{9}(x + 3) \). Distribute \( \frac{5}{9} \): \( y + 2 = \frac{5}{9}x + \frac{15}{9} \). Simplifying further: \( y = \frac{5}{9}x + \frac{15}{9} - 2 \).

Step 4: Express in Simplified Form

Convert \( -2 \) to a fraction with a denominator of 9: \( -2 = \frac{-18}{9} \). Now combine terms: \( y = \frac{5}{9}x + \frac{15}{9} - \frac{18}{9} \). This simplifies to \( y = \frac{5}{9}x - \frac{3}{9} \), or \( y = \frac{5}{9}x - \frac{1}{3} \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a straightforward way to write the equation of a straight line. It allows you to see the slope and the y-intercept at a glance. This form is given by the equation \( y = mx + b \), where:

• \( m \) represents the slope of the line.
• \( b \) is the y-intercept, which is the value of y when x equals zero.

Once you have the slope and y-intercept of a line, you can easily formulate its equation using this form. In our case, after simplifying the point-slope equation, we arrived at \( y = \frac{5}{9}x - \frac{1}{3} \). Here, \( \frac{5}{9} \) is the slope, and \( -\frac{1}{3} \) is the y-intercept. This form makes it very clear how the line behaves and where it crosses the y-axis.

Point-Slope Formula

The point-slope formula is a powerful tool when you know a point on a line and the line's slope. Its equation is: \( y - y_1 = m(x - x_1) \), where:

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

You start by choosing any point on the line, though either could be used. For our line between points \((-3, -2)\) and \((6, 3)\), we opted for \((-3, -2)\). By knowing the slope as \( \frac{5}{9} \), we can substitute these values into the formula, leading to \( y + 2 = \frac{5}{9}(x + 3) \). This step bridges the gap between the raw data given (points) and the neat equation in slope-intercept form.

Slope Calculation

Calculating the slope is the essential first step in forming the equation of a line. The slope tells us how steep the line is, or how much y increases as x increases. The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

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

For our specific points \((-3, -2)\) and \((6, 3)\), the slope calculation becomes \( m = \frac{3 - (-2)}{6 - (-3)} = \frac{5}{9} \). Think of the slope as the 'rise' over 'run' metric, illustrating how much y 'rises' for each unit x 'runs' to the right. This important value is central to deriving further iterations of line equations, such as in point-slope or slope-intercept forms.