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Author: nsharp1
No. Questions: 3
Created: May 26, 2020

# Linear Functions and Growth Rate

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Let $f(x) = mx+b$ be a real-valued linear function. The following questions will investigate the growth of this linear function, or the value $Delta f = f(x + Delta x) - f(x)$, over equal intervals of length $Delta x$, where $Delta x > 0$.
Grade 10 Functions and Relations CCSS: HSF-LE.A.1, HSF-LE.A.1a
A.
What is the value of $Delta f = f(x + Delta x) - f(x)$, using the definition of $f(x) ?$ Simplify your answer fully.
1. $m Delta x + 2b$
2. $Delta x - 2b$
3. $m Delta x$
4. $Delta x + 2m$
Grade 10 Functions and Relations CCSS: HSF-LE.A.1, HSF-LE.A.1a
B.
Given the answer in the previous question, which of the following gives the best reasoning as to why one can conclude that a linear function grows by equal differences over equal intervals?
1. Because the resulting equation for $Delta f$ is also linear, it will increase at a constant rate.
2. Because there is no slope in the resulting equation, the value of $Delta f$ is constant.
3. Since the resulting equation for $Delta f$ has no $b$ value, it is independent of the y-intercept. As such, $Delta f$ will increase by equal amounts over equal intervals.
4. Since $Delta f$ is dependent only on the length of the interval, as long as the interval $Delta x$ is constant, $Delta f$ will be the same.
Grade 10 Functions and Relations CCSS: HSF-LE.A.1, HSF-LE.A.1a
C.
What if, for the linear function $f(x)$, $m =0 ?$
1. This means that the above reasoning is invalid.
2. For how the function is defined above, $m$ cannot equal zero.
3. It makes things easier, since the growth rate of $f(x)$ simply becomes zero (and thus constant) for all intervals.
4. It does not change anything, since $Delta f$ is not dependent on $m$.