Problem 30 - College Algebra

Question

For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible. \(f(-1)=4\) and \(f(5)=1\)

Step-by-Step Solution

Answer

The linear equation is \( f(x) = -\frac{1}{2}x + \frac{7}{2} \).

Step 1: Understand the Problem

We need to find a linear equation in the form of \( f(x) = mx + b \) such that it passes through the given points: \((-1, 4)\) and \((5, 1)\). To do this, we need to find the slope \(m\) and the y-intercept \(b\).

Step 2: Calculate the Slope

The slope \(m\) can be calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \((x_1, y_1) = (-1, 4)\) and \((x_2, y_2) = (5, 1)\). Substitute these values into the formula: \[ m = \frac{1 - 4}{5 - (-1)} = \frac{-3}{6} = -\frac{1}{2} \]

Step 3: Write the Equation in Point-Slope Form

Now that we have the slope, we can use the point-slope form of a line, which is \( y - y_1 = m(x - x_1) \). Using the point \((-1, 4)\), we substitute: \[ y - 4 = -\frac{1}{2}(x + 1) \]

Step 4: Simplify to Slope-Intercept Form

Expand and simplify the equation from point-slope form to slope-intercept form \( y = mx + b \): Start with:\[ y - 4 = -\frac{1}{2}(x + 1) \]Simplify:\[ y - 4 = -\frac{1}{2}x - \frac{1}{2} \]Add 4 to both sides:\[ y = -\frac{1}{2}x + \frac{7}{2} \]Thus, the equation is \( f(x) = -\frac{1}{2}x + \frac{7}{2} \).

Step 5: Verify the Equation

To ensure the equation is correct, substitute \(x = -1\) and \(x = 5\) back into the equation \( f(x) = -\frac{1}{2}x + \frac{7}{2} \).Check for \(x = -1\): \[ f(-1) = -\frac{1}{2}(-1) + \frac{7}{2} = \frac{1}{2} + \frac{7}{2} = 4 \]Check for \(x = 5\): \[ f(5) = -\frac{1}{2}(5) + \frac{7}{2} = -\frac{5}{2} + \frac{7}{2} = 1 \]Both points verify the equation is correct.

Key Concepts

Slope

The slope of a line is a measure of its steepness and direction. It tells you how much the line rises or falls as you move from left to right. The slope is often represented by the letter \( m \) and is calculated using two points on the line. Specifically, the formula for calculating slope is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \((x_1, y_1)\) and \((x_2, y_2)\) represent the coordinates of two points on the line.
In the exercise provided, you are given the points \((-1, 4)\) and \((5, 1)\). By substituting these into the slope formula, you determine that \( m = -\frac{1}{2} \).
Key points to remember about slope:

• A positive slope means the line rises from left to right.
• A negative slope means the line falls from left to right.
• A zero slope indicates a horizontal line.
• An undefined slope indicates a vertical line.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis. It is represented as \( b \) in the equation of a line. In the slope-intercept form of a linear equation, \( y = mx + b \), the y-intercept is the constant \( b \).
To find the y-intercept, you can use the slope and one of the points through which the line passes. With the equation rearranged as \( y - y_1 = m(x - x_1) \), you can solve for \( b \) by isolating \( y \).
Let's take the example in the solution:

• Given \( m = -\frac{1}{2} \) and the point \((-1,4)\), the equation in point-slope form becomes \( y - 4 = -\frac{1}{2}(x + 1) \).
• Rearranging this gives us the slope-intercept form, \( y = -\frac{1}{2}x + \frac{7}{2} \), showing that the y-intercept \( b = \frac{7}{2} \).

Point-Slope Form

The point-slope form of a linear equation is particularly useful when you know the slope of a line and a point on the line. The formula is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is a point on the line.
This form is handy because it directly introduces the concept of slope into the equation while relating it to a specific point on the line. It serves as an intermediate step in finding the slope-intercept form.
In the exercise, the point-slope form starts with one point \((-1,4)\) and the slope \( m = -\frac{1}{2} \). You form the equation: \( y - 4 = -\frac{1}{2}(x + 1) \). This form allows easy conversion to slope-intercept form by using algebraic manipulation.
Remember, when writing equations in point-slope form:

• You usually have a choice of which point to use, as long as the slope is correct.
• This form is very helpful in deriving other forms of a linear equation.

Slope-Intercept Form

The slope-intercept form is one of the most recognized forms of a linear equation, denoted as \( y = mx + b \). It clearly shows the slope \( m \) and the y-intercept \( b \), making it straightforward to graph and understand the line.
In our exercise, once you have the point-slope form \( y - 4 = -\frac{1}{2}(x + 1) \), you can manipulate this using algebra to convert it to \( y = -\frac{1}{2}x + \frac{7}{2} \). This form immediately tells you:

• The slope of the line is \(-\frac{1}{2}\), meaning it declines as you move right.
• The y-intercept is \( \frac{7}{2} \), which is where the line crosses the y-axis.

Slope-intercept form is particularly helpful for:

• Quickly determining the overall direction and steepness of a line.
• Finding where a line intersects the y-axis.
• Making the task of graphing a line on the cartesian plane a lot easier.