So I'm working on Unit 2 Lesson 8: Exponents and Exponential Functions Sample Work and I cant figure out how to write why C = 535(0.90)^n r=1+(-0.10)=0.90 is exponential function and not a linear function?
Can't figure out what
C = 535(0.90)^n r=1+(-0.10)=0.90
is supposed to be. I see 2 equal signs, so are there two equations??
anyway... , look at the term 535(.90)^t.
I see an "exponent which is a variable", so it is exponential.
To be linear, the function has a variable in the base and no exponents
other than 1.
e.g. y = 4x + 7
Well, if you think about it, exponential functions and linear functions are like two completely different animals. One is a graceful gazelle leaping through the mathematical savannah, while the other is a lumbering rhinoceros plodding along like a snail in comparison. So let's see why C = 535(0.90)^n is an exponential function and not a linear function.
In a linear function, the rate of change is constant. It's like walking at a steady pace, never speeding up or slowing down. But in an exponential function, things get a bit wilder. The rate of change increases or decreases exponentially as the function progresses. It's like riding a roller coaster - one moment you're slowly climbing up, and the next moment you're hurtling down at breakneck speed.
In the equation C = 535(0.90)^n, notice how the variable "n" is an exponent, raising 0.90 to different powers. This means that as "n" increases, the value inside the parentheses gets smaller and smaller, causing the output, C, to decrease more rapidly. That's the key characteristic of an exponential function - the rate of change gets crazier as you go!
If this equation were linear, we'd see a straight line with a constant rate of change. But since it's exponential, it's more like a roller coaster ride that keeps surprising you with unexpected twists and turns. So buckle up and enjoy the mathematical adventure!
To understand why the equation C = 535(0.90)^n represents an exponential function rather than a linear function, let's explore the properties of each type of function.
Linear functions have a constant rate of change, meaning that for every unit increase in the input (n in this case), the output (C) increases or decreases by the same amount. Graphically, linear functions form a straight line.
On the other hand, exponential functions have a constant base raised to a variable exponent. In this equation, the base of the exponential function is 0.90, and the variable exponent is n. As n increases, the value of (0.90)^n will change, and this affects the value of C.
Now, let's analyze the given equation C = 535(0.90)^n:
1. In a linear function, the equation would have a constant term and a linear term. However, there is no constant term in this equation, indicating that it does not have a constant rate of change.
2. The presence of the variable exponent n, which changes the value of (0.90)^n, suggests that the equation has an exponential relationship. As n increases or decreases, C will be affected exponentially because C depends on the corresponding value of (0.90)^n.
In conclusion, the equation C = 535(0.90)^n represents an exponential function, not a linear function, due to the presence of a variable exponent and the lack of a constant term.
To determine whether the given equation C = 535(0.90)^n represents an exponential or linear function, we need to examine its form and characteristics.
An exponential function is represented in the form y = ab^x, where 'a' is the initial value, 'b' is the base, and 'x' is the exponent. In this case, the equation C = 535(0.90)^n can be rewritten as C = a * b^n.
Comparatively, a linear function is represented in the form y = mx + c, where 'm' is the slope or rate of change, 'x' is the input variable, and 'c' is the y-intercept.
In the given equation C = 535(0.90)^n, notice that 'n' is the exponent, not 'x'.
To analyze further, let's consider the meaning of 'n'. It likely represents the number of time periods or iterations in a sequence or situation. Each time 'n' increases by one, the value inside the parentheses (0.90) is raised to a higher power.
If we assume that 'n' represents the number of time periods, then the equation suggests that the value of 'C' is changing exponentially based on the rate of change represented by the base (0.90). This implies that as 'n' increases, C will decrease geometrically, following the pattern of exponential decay.
Comparatively, in a linear function, the change in 'x' would lead to a consistent change in 'C' (y-value). However, in this case, the change in 'n' results in compound changes in the value of 'C' due to the exponentiation.
Therefore, based on the form and behavior of the equation C = 535(0.90)^n, we can conclude that it represents an exponential function, not a linear function.