Problem 30 - College Algebra with Corequisite Support

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 the equation of a line that passes through the points \((-1, 4)\) and \((5, 1)\). This line can be expressed in the form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Calculate the slope

The slope \(m\) of the line through points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Plugging in the points \((-1, 4)\) and \((5, 1)\), we find the slope: \[ m = \frac{1 - 4}{5 - (-1)} = \frac{-3}{6} = -\frac{1}{2} \]

Step 3: Use one point to solve for the y-intercept

We’ll use the slope \(m = -\frac{1}{2}\) and one of the given points to find the y-intercept \(b\). Let's use point \((-1, 4)\): \[ 4 = -\frac{1}{2}(-1) + b \]Simplify to find \(b\):\[ 4 = \frac{1}{2} + b \] Subtract \(\frac{1}{2}\) from both sides: \[ 4 - \frac{1}{2} = b \] \[ b = \, \frac{7}{2}\]

Step 4: Write the final linear equation

Now, substitute the slope \(m = -\frac{1}{2}\) and the y-intercept \(b = \frac{7}{2}\) into the equation \(f(x) = mx + b\): \[ f(x) = -\frac{1}{2}x + \frac{7}{2} \]This is the equation of the line that satisfies the given conditions.

Key Concepts

Slope Calculation

Calculating the slope of a line is essential in understanding how steep the line is. The slope, often represented as \( m \), indicates the direction and incline of a line along the coordinate plane. It tells us how much the \( y \)-value of a line changes for a corresponding change in the \( x \)-value.

For any two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line, the slope is calculated using the formula:

• \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

In this exercise, we calculated the slope using the points

• \((-1, 4)\)
• \((5, 1)\)

By substituting these points into our formula, we found that \( m = -\frac{1}{2} \). This negative value means the line is decreasing as we move from left to right across the graph.

Y-Intercept

The y-intercept is a crucial component of linear equations. It is represented by the \( b \) in the equation \( f(x) = mx + b \).

The y-intercept refers to the point where the line crosses the y-axis. At this point, the value of \( x \) is zero. To find it, one can plug a known point and the calculated slope into the line equation.

In our solution, we used the point \((-1, 4)\) along with the slope \(-\frac{1}{2}\):

• Equation: \( 4 = -\frac{1}{2}(-1) + b \)

Simplifying gives us \( b = \frac{7}{2} \). Thus, our line crosses the y-axis at \( y = \frac{7}{2} \). Understanding the y-intercept helps visualize the starting point of a line when we think of graphs.

Coordinate Geometry

Coordinate geometry combines algebra and geometry, allowing us to solve problems by graphically representing equations. It is a powerful tool in mathematics, revealing relationships between algebraic expressions and geometric figures.

By plotting points on a graph and analyzing lines, we can solve numerous mathematical problems. In this case, the concepts of slope and y-intercept come under the umbrella of coordinate geometry.

• The slope indicates how each point on the line relates to its neighbors.
• The y-intercept tells us where the line begins its journey on the graph when considering the horizontal axis.

Thus, coordinate geometry helps us understand how lines are organized within the plane and predict their behavior. Understanding these basics paves the way for more advanced studies, making it easier to tackle complex coordinate geometry problems in the future.