Problem 22

Question

Show that if $f(x)$ is the linear function $y=m x+b$ where $m$ and $b$ are constants, then increases in $f(x)$ are proportional to increases in $x .$ That is, suppose initially that $x=x_{0}$, and $y=y_{0}=$ $m x_{0}+b .$ Then we increase $x$ by $\Delta x$ to $x=x_{0}+\Delta x .$ Calculate the increase in $y .$ Show that the increase in $y$ depends on $\Delta x$ but does not depend on $x_{0}$. This means that the same increment in $x$ always produces the same increment in $y$, independently of the starting value of $x$. Contrast this behavior with a concave down function.

Step-by-Step Solution

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Answer

The increase in $y$ due to an increase in $x$ is $m\Delta x$, independent of $x_0$, indicating proportionality for linear functions. In contrast, concave down functions depend on $x_0$.

1Step 1: Initial Function and Values

Given the linear function $f(x) = mx + b$, where $m$ and $b$ are constants. Initially, $x = x_0$ and $y = y_0 = mx_0 + b$.

2Step 2: Apply Change in x

Increase $x$ by $\Delta x$, so the new value of $x$ becomes $x_0 + \Delta x$.

3Step 3: Calculate the New Function Value

Substitute $x = x_0 + \Delta x$ into the function to get the new value of $y$, which is $y = m(x_0 + \Delta x) + b$.

4Step 4: Simplify the New Function Value

Expand the expression: $y = mx_0 + m\Delta x + b$.

5Step 5: Determine the Increase in y

Subtract the initial value $y_0 = mx_0 + b$ from the new value $y = mx_0 + m\Delta x + b$. The increase in $y$ is $\Delta y = m\Delta x$.

6Step 6: Analyze Dependence on x_0

Notice that $\Delta y = m\Delta x$ does not include $x_0$, meaning the increase in $y$ depends only on $\Delta x$ and not on the initial value $x_0$.

7Step 7: Contrast with a Concave Down Function

For a concave down function like $y = -x^2$, the change in $y$ would depend on the specific starting point $x_0$. This is because the derivative (slope) of a concave down function varies with $x$.

Key Concepts

Proportional IncreasesFunction BehaviorConcave Down Function

Proportional Increases

In mathematics, when we talk about proportional increases, we're often dealing with a relationship where two quantities increase at a constant rate relative to each other. The equation of a straight line, such as a linear function described by $ f(x) = mx + b $, showcases this kind of relationship.

Here $ m $ represents the slope of the line, which indicates how much $ y $ increases for a unit increase in $ x $.
Whenever $ x $ is increased by a certain amount, $ \Delta x $, the increase in $ y $ is simply $ m \Delta x $.

This change in $ y $, $ \Delta y = m \Delta x $, highlights the concept of proportional increases.Because $ \Delta y $ depends purely on the constant $ m $ and the change in $ x $, $ \Delta x $, it remains unaffected by the initial value of $ x $. This means no matter where your starting point $ x_0 $ is on the line, any incremental change in $ x $ will consistently yield the same change in $ y $. This beautiful predictability is what makes linear equations a staple in understanding proportional relationships and the reason they are easy to work with.

Function Behavior

The behavior of the function gives us insights into how the relationship between the dependent variable $ y $ and independent variable $ x $ changes over the domain.For a linear function like $ f(x) = mx + b $, the function behavior is very straightforward.

The linearity implies a constant rate of change, embodied by $ m $, the slope.
This constant slope results in a graph that's a straight line, ensuring consistent interval changes throughout.

As we see, the behavior of a linear function is unique due to its simplicity:- No matter where you start on the line, the increment in $ x $ translates directly into a proportionate increment in $ y $.This is in stark contrast to other function types where slope and rate of change may vary. Thus, understanding linear functions becomes foundational in exploring how different mathematical models behave.

Concave Down Function

When exploring different types of functions, a concave down function presents a contrast to linear functions. A typical example of a concave down function could be $ y = -x^2 $.

This function type is characterized by its downward-curving shape, like an upside-down bowl.
The critical feature here is that the slope of the function, or its derivative, changes depending on the value of $ x $.

For a function like $ y = -x^2 $, the derivative is $ -2x $, indicating a slope that becomes more negative as $ x $ increases.This means if we increase $ x $ by a small amount $ \Delta x $ from a starting point $ x_0 $, the increase in $ y $ depends on the value of $ x_0 $.Such variability contrasts sharply with the constant proportional increases seen in linear functions, adding complexity to analyzing real-world phenomena with non-linear tendencies.

Problem 22

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