Problem 28 - Algebra Form and Function

Question

Find a possible formula for the linear function \(f(x)\) if \(f(20)=70\) and \(f(70)=10\)

Step-by-Step Solution

Answer

Answer: The linear function is f(x) = -6/5x + 94.

Step 1: Find the slope

Using the coordinates \((x_1, y_1) = (20, 70)\) and \((x_2, y_2) = (70, 10)\), we calculate the slope (m) using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Plugging in the coordinates, we get \(m = \frac{10 - 70}{70 - 20} = \frac{-60}{50} = -\frac{6}{5}\).

Step 2: Determine the y-intercept

We can use the point-slope form, \(y - y_1 = m(x - x_1)\), and one of the points to express the linear function. Using the point (20, 70), we have: \(y - 70 = -\frac{6}{5}(x - 20)\) Now we need to solve for y to find the y-intercept (b) in the equation \(y = mx + b\). \(y = -\frac{6}{5}(x - 20) + 70\)

Step 3: Simplify the equation

To obtain the equation in the slope-intercept form, we simplify the equation above: \(y = -\frac{6}{5}x + \frac{6}{5} \times 20 + 70\) \(y = -\frac{6}{5}x + 24 + 70\) \(y = -\frac{6}{5}x + 94\)

Step 4: Write the linear function

Now that we have the slope and the y-intercept, we can write the linear function f(x) as: \(f(x) = -\frac{6}{5}x + 94\)

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of describing a straight line using a simple equation. It is expressed as \(y = mx + b\). Here, \(m\) represents the slope, and \(b\) is the y-intercept.
You can think of the y-intercept as the point where the line crosses the y-axis. It tells you the value of \(y\) when \(x\) is zero.
The slope-intercept form is beloved by many because it makes it super easy to graph linear functions. Just plot the y-intercept and use the slope to find other points on the line.

• Slope (\(m\)): This tells you how steep the line is and the direction of the line (uphill or downhill).
• Y-intercept (\(b\)): This shows where the line crosses the y-axis. It's a simple ending point when \(x = 0\).

In the context of our exercise, the function is expressed as \(f(x) = -\frac{6}{5}x + 94\). Here, the slope is \(-\frac{6}{5}\) and the y-intercept is \(94\). Though it might seem complicated at first, this form helps in quickly understanding the core behavior of linear equations.

Point-Slope Form

The point-slope form is another useful way to express linear functions. It comes in handy when you know a point on the line and the slope of that line. The formula to express this is \(y - y_1 = m(x - x_1)\).
Here, \((x_1, y_1)\) represents any point on the line, and \(m\) is the slope. This form helps you when you know specific points and you need to calculate other features, like the y-intercept.
In our solved exercise, we used the point \((20, 70)\) and a slope \(-\frac{6}{5}\) to express the linear function. By plugging these values into the formula, we found:

• \(y - 70 = -\frac{6}{5}(x - 20)\)

This form is beneficial because it can be easily rearranged into slope-intercept form. Plus, when only a point and a slope are available, it's often the simplest starting point for finding the full linear equation.

Calculating Slope

Calculating the slope is a crucial part of understanding linear functions. The slope basically tells you how the function behaves as \(x\) changes. It's calculated as the change in \(y\) divided by the change in \(x\). This is expressed through the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
The slope is important for understanding how steep a line is. A positive slope means the line goes upwards as you move from left to right; a negative slope means it goes downwards.

• Positive Slope: Indicates an upward trend of the line.
• Negative Slope: Indicates a downward trend of the line.

In the given exercise, we calculated the slope using the points \((20, 70)\) and \((70, 10)\). By substituting these values into the formula, we found \(m = \frac{10 - 70}{70 - 20} = -\frac{6}{5}\).
Understanding how to calculate the slope helps in plotting linear graphs and solving real-world problems involving rates of change.