For Which Pair Of Functions Is The Exponential Consistently Growing At A Faster Rate Than The Quadratic Over The Interval $0 \leq X \leq 5$?

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Introduction

When comparing the growth rates of different functions, it's essential to understand the characteristics of each function type. In this article, we'll explore the exponential and quadratic functions, and determine which pair of functions exhibits the exponential function consistently growing at a faster rate over the interval $0 \leq x \leq 5$.

Exponential Functions

Exponential functions have the general form $f(x) = ab^x$, where $a$ and $b$ are constants, and $b$ is the base of the exponent. The base $b$ determines the growth rate of the function. If $b > 1$, the function grows exponentially, and if $b < 1$, the function decays exponentially.

Example of Exponential Function

Consider the exponential function $f(x) = 2^x$. This function grows exponentially, and its growth rate is determined by the base $b = 2$. As $x$ increases, the value of $f(x)$ increases rapidly.

Quadratic Functions

Quadratic functions have the general form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The coefficient $a$ determines the direction and rate of change of the function. If $a > 0$, the function opens upward, and if $a < 0$, the function opens downward.

Example of Quadratic Function

Consider the quadratic function $f(x) = x^2$. This function opens upward, and its growth rate is determined by the coefficient $a = 1$. As $x$ increases, the value of $f(x)$ increases, but at a slower rate than the exponential function.

Comparing Growth Rates

To determine which pair of functions exhibits the exponential function consistently growing at a faster rate, we need to compare the growth rates of the functions over the interval $0 \leq x \leq 5$.

Exponential Function with Base 2

Consider the exponential function $f(x) = 2^x$. This function grows exponentially, and its growth rate is determined by the base $b = 2$. As $x$ increases, the value of $f(x)$ increases rapidly.

Quadratic Function with Coefficient 1

Consider the quadratic function $f(x) = x^2$. This function opens upward, and its growth rate is determined by the coefficient $a = 1$. As $x$ increases, the value of $f(x)$ increases, but at a slower rate than the exponential function.

Comparison of Growth Rates

To compare the growth rates of the functions, we can evaluate the functions at different values of $x$ over the interval $0 \leq x \leq 5$.

$x$	$f(x) = 2^x$	$f(x) = x^2$
0	1	0
1	2	1
2	4	4
3	8	9
4	16	16
5	32	25

As we can see from the table, the exponential function $f(x) = 2^x$ consistently grows at a faster rate than the quadratic function $f(x) = x^2$ over the interval $0 \leq x \leq 5$.

Conclusion

In conclusion, the pair of functions that exhibits the exponential function consistently growing at a faster rate over the interval $0 \leq x \leq 5$ is the exponential function $f(x) = 2^x$ and the quadratic function $f(x) = x^2$. The exponential function grows rapidly, and its growth rate is determined by the base $b = 2$. The quadratic function opens upward, and its growth rate is determined by the coefficient $a = 1$. As $x$ increases, the value of the exponential function increases rapidly, while the value of the quadratic function increases at a slower rate.

References

• [1] "Exponential Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/exponential.html
• [2] "Quadratic Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/quadratic.html

Further Reading

• [1] "Exponential Growth" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f-exponential-and-logarithmic-functions/x2f-exponential-growth/v/exponential-growth
• [2] "Quadratic Equations" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f-quadratic-equations/x2f-quadratic-equations/v/quadratic-equations
  

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Introduction

In our previous article, we explored the exponential and quadratic functions, and determined which pair of functions exhibits the exponential function consistently growing at a faster rate over the interval $0 \leq x \leq 5$. In this article, we'll answer some frequently asked questions about the topic.

Q&A

Q: What is the general form of an exponential function?

A: The general form of an exponential function is $f(x) = ab^x$, where $a$ and $b$ are constants, and $b$ is the base of the exponent.

Q: What determines the growth rate of an exponential function?

A: The base $b$ determines the growth rate of an exponential function. If $b > 1$, the function grows exponentially, and if $b < 1$, the function decays exponentially.

Q: What is the general form of a quadratic function?

A: The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: What determines the direction and rate of change of a quadratic function?

A: The coefficient $a$ determines the direction and rate of change of a quadratic function. If $a > 0$, the function opens upward, and if $a < 0$, the function opens downward.

Q: How do we compare the growth rates of exponential and quadratic functions?

A: To compare the growth rates of exponential and quadratic functions, we can evaluate the functions at different values of $x$ over the interval $0 \leq x \leq 5$.

Q: Which pair of functions exhibits the exponential function consistently growing at a faster rate over the interval $0 \leq x \leq 5$?

A: The pair of functions that exhibits the exponential function consistently growing at a faster rate over the interval $0 \leq x \leq 5$ is the exponential function $f(x) = 2^x$ and the quadratic function $f(x) = x^2$.

Q: Why does the exponential function grow faster than the quadratic function?

A: The exponential function grows faster than the quadratic function because the base $b = 2$ is greater than 1, which means the function grows exponentially. The quadratic function, on the other hand, opens upward, but its growth rate is determined by the coefficient $a = 1$, which is slower than the exponential function.

Q: Can we generalize the result to other exponential and quadratic functions?

A: Yes, we can generalize the result to other exponential and quadratic functions. If we have an exponential function with a base greater than 1 and a quadratic function with a coefficient greater than 0, the exponential function will consistently grow at a faster rate than the quadratic function over the interval $0 \leq x \leq 5$.

Conclusion

In conclusion, the exponential function consistently grows at a faster rate than the quadratic function over the interval $0 \leq x \leq 5$. The exponential function grows rapidly, and its growth rate is determined by the base $b = 2$. The quadratic function opens upward, and its growth rate is determined by the coefficient $a = 1$. We can generalize the result to other exponential and quadratic functions, and the exponential function will consistently grow at a faster rate than the quadratic function.

References

• [1] "Exponential Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/exponential.html
• [2] "Quadratic Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/quadratic.html

Further Reading

• [1] "Exponential Growth" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f-exponential-and-logarithmic-functions/x2f-exponential-growth/v/exponential-growth
• [2] "Quadratic Equations" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f-quadratic-equations/x2f-quadratic-equations/v/quadratic-equations