Problem 79 - Algebra and Trigonometry

Question

Find a function whose graph is the given curve. The line segment joining the points \((-2,1)\) and \((4,-6)\)

Step-by-Step Solution

Answer

The function is \( y = -\frac{7}{6}x - \frac{4}{3} \).

Step 1: Understand the Problem

We are asked to find a function that represents the line between the two points \((-2, 1)\) and \((4, -6)\). A line can be represented by a linear equation, often in the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.

Step 2: Calculate the Slope (m)

The slope \(m\) of the line can be calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For our points, this becomes \(m = \frac{-6 - 1}{4 - (-2)} = \frac{-7}{6}\). Thus, the slope of our line is \(-\frac{7}{6}\).

Step 3: Write the Equation in Point-Slope Form

Using the slope \(-\frac{7}{6}\) and starting from either of the points, let's choose \((-2, 1)\), use the point-slope form \(y - y_1 = m(x - x_1)\). Plug in the values: \(y - 1 = -\frac{7}{6}(x + 2)\).

Step 4: Simplify to Slope-Intercept Form

Distribute and simplify the equation: \(y - 1 = -\frac{7}{6}x - \frac{14}{6}\). Simplifying gives \(y = -\frac{7}{6}x - \frac{7}{3} + 1\). Combine the constants: \(y = -\frac{7}{6}x -\frac{7}{3} + \frac{3}{3}\), which simplifies to \(y = -\frac{7}{6}x -\frac{4}{3}\).

Step 5: Verify the Equation

Check if both points satisfy the equation \(y = -\frac{7}{6}x -\frac{4}{3}\). For \(x = -2\), \(y = -\frac{7}{6}(-2) - \frac{4}{3} = 1\), and for \(x = 4\), \(y = -\frac{7}{6}(4) - \frac{4}{3} = -6\). Thus, both points satisfy the equation, confirming its correctness.

Key Concepts

Slope Calculation

In order to determine the equation of a line, it is essential first to calculate the slope. The slope tells us how steep the line is and its direction. We calculate the slope using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where

• \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates from two distinct points on the line.
• \( m \) is the slope.

For the given points \((-2, 1)\) and \((4, -6)\), plug in the values:\[ m = \frac{-6 - 1}{4 - (-2)} = \frac{-7}{6} \]This negative slope of \(-\frac{7}{6}\) indicates that the line decreases as it moves from left to right.

Point-Slope Form

Once the slope is calculated, a linear equation can initially be represented using the point-slope form. This is particularly useful when you know a point on the line and the slope. The point-slope formula is:\[ y - y_1 = m(x - x_1) \]

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

For example, using the slope \(-\frac{7}{6}\) and the point \((-2, 1)\), we have:\[ y - 1 = -\frac{7}{6}(x + 2) \]This step is critical as we nail down the specific line showing both direction and starting point.

Slope-Intercept Form

To simplify the equation further, you can convert it into the slope-intercept form, which is easier to use for graphing. The slope-intercept form is:\[ y = mx + c \]where

• \( m \) is again the slope
• \( c \) is the y-intercept where the line crosses the y-axis.

Let's simplify the equation from the previous form:\[ y - 1 = -\frac{7}{6}x - \frac{14}{6} \]After distributing and simplifying, it becomes:\[ y = -\frac{7}{6}x - \frac{4}{3} \]This form is very popular for quickly identifying how the line behaves in a graph.

Graphing a Line

When you need to graph a line from its equation, understanding the slope-intercept form is handy. With the equation \[ y = -\frac{7}{6}x - \frac{4}{3} \]

• The line crosses the y-axis at \( -\frac{4}{3} \).
• The slope \(-\frac{7}{6}\) tells us the line goes down 7 units for every 6 units it goes right.

To graph:

• Start at the y-intercept, \(-\frac{4}{3}\).
• From there, apply the slope. Go down 7 units and right 6 units to locate another point.
• Draw a line through these points extending in both directions.

Graphing visually represents the relationship outlined by the equation and can help check its accuracy.