Problem 27 - Algebra Form and Function

Question

Find a possible formula for the linear function \(h(x)\) if \(h(-30)=80\) and \(h(40)=-60\)

Step-by-Step Solution

Answer

Question: Given the points (-30, 80) and (40, -60) on a line, find the formula for the linear function h(x). Answer: The formula for the linear function is h(x) = -2x + 20.

Step 1: Find the slope

To find the slope, we can use the formula \(m = \frac{y2 - y1}{x2 - x1}\), using the given points \((-30, 80)\) and \((40, -60)\). Substitute the values into the formula: \(m = \frac{-60 - 80}{40 - (-30)}\) Now simply calculate the value of \(m\): \(m = \frac{-140}{70} = -2\)

Step 2: Use the point-slope form to find the equation of the line

Now that we have the slope \(m\), we can use the point-slope form of a linear equation: \(y - y1 = m(x - x1)\). Choose one of the given points (we'll use \((-30, 80)\)) and substitute the values into the equation: \(y - 80 = -2(x - (-30))\)

Step 3: Convert the equation to the \(h(x) = mx + b\) form

Now we will simplify the equation and put it into the desired form for a linear function: \(y - 80 = -2(x + 30)\) \(y - 80 = -2x - 60\) Now, add \(80\) to both sides to isolate \(y\): \(y = -2x + 20\)

Step 4: Write the equation as \(h(x)\)

Finally, write the equation using \(h(x)\) notation, which effectively replaces \(y\): \(h(x) = -2x + 20\) So, the formula for the linear function is \(h(x) = -2x + 20\).

Key Concepts

Understanding the Slope

The slope is a crucial concept in linear functions. It's essentially a measure of how steep a line is on a graph.
To find the slope, you can use two points through which the line passes. The formula for slope \( m \) is given by \( m = \frac{y2 - y1}{x2 - x1} \).
The numerator \( (y2 - y1) \) represents the change in the vertical direction, while the denominator \( (x2 - x1) \) represents the change in the horizontal direction.

• A positive slope means the line is rising from left to right.
• A negative slope means it is falling.
• A slope of zero indicates a horizontal line, and an undefined slope means a vertical line.

In our exercise, the points given are \((-30, 80)\) and \((40, -60)\). By substituting these into our formula, \( m = \frac{-60 - 80}{40 - (-30)} \),we calculate the slope as \( m = -2 \).
This tells us the line decreases by two units for every unit it moves to the right.

Exploring the Point-Slope Form

Once we know the slope, we can use the point-slope form to construct the equation of the line. This form is quite handy because it directly uses one of the points and the slope. The point-slope formula is:
\( y - y_1 = m(x - x_1) \).In this formula:

• \( (x_1, y_1) \) is one of the points on the line. In our example, we picked \((-30, 80)\).
• \( m \) is the slope, which we've calculated as \(-2\).

So, our equation becomes \( y - 80 = -2(x - (-30)) \), which simplifies to \( y - 80 = -2(x + 30) \).
The point-slope form acts as a stepping-stone to writing more traditional forms of equations, like the slope-intercept form.

Crafting a Linear Equation

To express the linear equation in a more familiar form, we convert from the point-slope form to the slope-intercept form \( y = mx + b \). This keeps things neat and highlights the slope and y-intercept directly.
Let's break it down using our derived equation \( y - 80 = -2(x + 30) \):

• First, distribute the \(-2\): \( y - 80 = -2x - 60 \).
• Next, solve for \( y \) by adding \( 80 \) to both sides: \( y = -2x + 20 \).
• Finally, replace \( y \) with \( h(x) \) to write it as a linear function: \( h(x) = -2x + 20 \).

In this equation, \( -2 \) is the slope, and \( 20 \) is the point where the line crosses the y-axis, also known as the y-intercept.
This form showcases the essence of a linear function, providing a complete and easy-to-understand picture of how \( h(x) \) changes with \( x \).