We are asked to write a paragraph proof about linear and exponential functions. First, we will write a proof to show that linear functions grow by equal differences over equal intervals. Next, we will show that exponential functions grow by equal factors over equal intervals. At the end we will create our proof.
Linear
Let f(x)=mx+b represent a linear function for some values of m and b. Let l be the fixed length of an interval. We want to show that the function grows by equal differences over equal intervals.
Given: f(x)=mx+b
Prove: Function f(x) grows by equal differences over equal intervals.
Let x_0 be the starting point of the interval with length l. Then, the endpoint of the interval is x_0+l.
To show that f(x) grows by equal differences, it is enough to show that the difference Δ y — which is the difference between the values at the endpoints of the interval — is constant.
Conjecture: Δ y=f(x_0+l)-f(x_0) is constant.
First, let's find the values of f(x_0) and f(x_0+l).
The difference Δ y=ml depends only on the constants m and l. This tells us that the function f(x) grows by equal differences over equal intervals, which ends the proof.
Exponential
Let f(x)=a^x represent an exponential function for some value of a, where a>0 and a≠1. Let l be the fixed length of an interval. We want to show that the function grows by equal factors over equal intervals.
Given: f(x)=a^x
Prove: Function f(x) grows by equal factors over equal intervals.
Let x_0 be the startpoint of the interval with length l. Then, the endpoint of the interval is x_0+l.
To show that f(x) grows by equal factors, it is enough to show that the ratio between the values at the endpoints of the interval, f(x_0+l)f(x_0), is constant.
Conjecture: f(x_0+l)/f(x_0) is constant.
First, let's find the values of f(x_0) and f(x_0+l).
The ratio a^l depends only on the constants a and l. This tells us that the function f(x) grows by equal factors over equal intervals, which ends the proof.
To show that f(x) grows by equal factors, it is enough to show that the ratio between the values at the endpoints of the interval, f(x_0+l)f(x_0), is constant.
Paragraph Proof for Linear Functions
Based on the complete proof for linear functions, we will write a paragraph proof.
Paragraph Proof
Let f(x)=mx+b be a linear function for some values of m and b. Let l be the fixed length of an interval with endpoints of x_0 and x_0+l. We will show that Δ y=f(x_0+l)-f(x_0) is constant. Since f(x_0)=mx_0+b and f(x_0+l)= m(x_0+l)+b= mx_0+ml+b, we get that Δ y= mx_0+ml+b-(mx_0+b)= ml. The expression ml does not depend on x_0. Therefore, we end the proof.
Paragraph Proof for Exponential Functions
Finally, we will write a paragraph proof for exponential functions.
Paragraph Proof
Let f(x)=a^x be an exponential function for some value of a, where a>0 and a≠1. Let l be the fixed length of an interval with endpoints of x_0 and x_0+l. We will show that the quotient f(x_0+l)/f(x_0) is constant. Since f(x_0)=a^(x_0) and f(x_0+l)= a^(x_0+l), we get that f(x_0+l)/f(x_0)= a^(x_0+l)/a^(x_0)=a^l. The expression a^l does not depend on x_0. This ends the proof.
