Exponents and their effects on lines?

In summary, an exponent in a formula, such as f(x)=2x^4, will cause the line on a graph to bend because the rate of change is no longer constant. This is because an exponent, other than 1, will result in a curve rather than a straight line. The rate of change is dependent on the interval on which it is evaluated, causing the line to bend.
  • #1
galatians
2
0
Why is it that in a formula: f(x)=2x^4...), why is it that the exponent actually BENDS the line that the fomula makes when it is graphed? I know about the high and low point in algebra two (that's what I'm taking), but i just want to know WHY does an exponent BEND the line?

CD
 
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  • #2
because x^4 grows faster at x2 than at x1, if x2 > x1.

in contrast, x^1 grows at same rate everywhere. you might try to read this without waiting for them to teach you that, if you really need to know.
 
Last edited:
  • #3
galatians said:
Why is it that in a formula: f(x)=2x^4...), why is it that the exponent actually BENDS the line that the fomula makes when it is graphed? I know about the high and low point in algebra two (that's what I'm taking), but i just want to know WHY does an exponent BEND the line?

CD

Your question is a bit like asking : "Why is 2 times 2 more than twice as large as 1 times 1.
 
  • #4
Please do not confuse a line with a curve. An equation of two variables, both to the first power, represents a line. If either or both variables are raised to other powers than 1, then this represents a curve.
 
  • #5
A line is only the special case where a polynomial is of degree 1, which is of the form mx + b Any exponent different than 1 will not give a straight line as the rate of change cannot possibly be constant (the geometrical and analytical definition of a straight line). For instance, for the equation y = x, you have

x y
1 1
2 2
3 3

Here, the difference in y between two consecutive x is always constant, it's equal to 1. For y = x^2 we have,

x y
1 1
2 4
3 9

Here, 9 - 4 is not equal to 4 -1, so the rate of change is dependant on the interval on which you evaluate it.
 
  • #6
Ohhh...okay. so, like the slope formula, y=mx+b, if the x is squared or has a greater degree than 1, then it's like the 'rise and run' of the thing becomes different, Like, as you showed, instead of it being a rise (y) of 1, 2,3 and a run (x) or 1,2,3; it is now a rise (y) or 1,4,9 and a run (x) of 1,2,3. If it's graphed, then the line actually begins to curve, because its rise and run are no longer constant 1,2,3. is that right? i think it is..
 
  • #7
Yes it is. :smile:
 

1. What is an exponent?

An exponent is a mathematical notation that indicates the number of times a base number is multiplied by itself. It is written as a superscript number to the right of the base number. For example, in 23, the base number is 2 and the exponent is 3, indicating that 2 is multiplied by itself 3 times.

2. How do exponents affect lines on a graph?

Exponents can affect lines on a graph in different ways depending on the value of the exponent. For positive exponents, the line will increase at a faster rate as the x-values increase. For negative exponents, the line will decrease at a faster rate as the x-values increase. For a zero exponent, the line will be a horizontal line at y=1, and for a fraction exponent, the line will curve upwards or downwards.

3. What is the difference between linear and exponential growth?

Linear growth is when a quantity increases or decreases by a fixed amount over a given interval. This is represented by a straight line on a graph. Exponential growth, on the other hand, is when a quantity increases or decreases by a fixed percentage over a given interval. This is represented by a curved line on a graph. Exponents play a crucial role in determining the rate of growth in exponential functions.

4. How do you simplify expressions with exponents?

To simplify expressions with exponents, you can use the laws of exponents. These laws state that when multiplying exponents with the same base, you add the exponents. When dividing exponents with the same base, you subtract the exponents. And when raising an exponent to another exponent, you multiply the exponents. You can also simplify expressions by using the power rule, which states that when raising a power to another power, you multiply the exponents.

5. How are exponents used in real-life situations?

Exponents are used in real-life situations to represent quantities that grow or decay at a constant rate. For example, they can be used to model population growth, financial investments, and radioactive decay. They are also used in scientific notation to represent very large or very small numbers in a more concise way. Exponents are also used in computer programming and engineering to perform calculations and optimize processes.

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