Problem 29 - College Algebra with Modeling and Visualization

Question

State the slope of the graph of \(f\). Interpret this slope. $$ f(x)=9-x $$

Step-by-Step Solution

Answer

The slope is -1, indicating the graph declines by 1 unit for every increase of 1 unit in \(x\).

Step 1: Identify the Function Type

The function given, \(f(x) = 9 - x\), is a linear function. Linear functions are of the form \(y = mx + b\), where \(m\) represents the slope and \(b\) the y-intercept.

Step 2: Identify the Slope from the Equation

In the equation \(f(x) = 9 - x\), we can rewrite it in slope-intercept form as \(f(x) = -x + 9\). Here, \(m = -1\) is the coefficient of \(x\), and represents the slope of the graph.

Step 3: Interpret the Slope

The slope of \(-1\) means that for every 1 unit increase in \(x\), the value of \(f(x)\) decreases by 1 unit. This indicates the graph is descending with a constant rate.

Key Concepts

Slope

The slope of a linear function is a crucial concept to grasp. It represents the "steepness" or incline of the line. In mathematical terms, slope is defined as the change in the vertical direction (often noted as 'rise') compared to the change in the horizontal direction (referred to as 'run'). The formula to calculate slope is:

• Slope (m) = \( \frac{\text{rise}}{\text{run}} \)

The slope tells us how much a function's output (y-value) changes for a one-unit change in input (x-value). In the function we've been given, \(f(x) = 9 - x\), when it's rewritten as \(f(x) = -x + 9\), the slope is \(-1\). This indicates that for every single unit increase in x, y will decrease by 1 unit.
In practical terms, understanding slope helps us interpret trends. A positive slope means a positive correlation between x and y – as x increases, so does y. Conversely, a negative slope, like in our example, signifies a negative correlation – as x increases, y decreases.

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most useful forms for quickly interpreting linear functions. It is written as:

• y = mx + b

where \(m\) represents the slope and \(b\) signifies the y-intercept, or the point at which the line crosses the y-axis.
This form is especially helpful because it directly provides the slope of the line and the starting point when x equals zero. In our function \(f(x) = 9 - x\), when rearranged into the slope-intercept form as \(-x + 9\), \(m = -1\) (the slope) and \(b = 9\) (the y-intercept).
Knowing the slope-intercept form allows students to swiftly identify the rate at which y changes with respect to x and where the line begins on the y-axis, making graphing linear functions much easier.

Graph Interpretation

Interpreting a graph of a linear function involves understanding several key components, such as slope, intercepts, and the overall direction of the line. For the given function \(f(x) = 9 - x\), with its slope of \(-1\), we see a line that steadily decreases as we move from left to right.

• The negative slope (-1) tells us that the line descends.
• The y-intercept (9) indicates that the line starts at the point where y equals 9.

This means that if you were to graph this function, you would start at (0, 9) on the y-axis. Then, for a move of one unit to the right in the x-direction, you would move down by one unit in the y-direction, forming a line that heads downward.
Understanding these elements allows us to visualize how the function behaves and make predictions about values of y for given values of x. This ability is immensely beneficial in various real-world applications, from predicting financial trends to physics calculations.