Name: Brady MasonDate: $\qquad$2) Given $p(t) = 3t + 5$, Find $p(t-3)$.A) $3t - 7$ B) $3t - 4$ C) $-9t + 5$ D) $-3t + 5$

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Understanding Polynomial Functions

Polynomial functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, calculus, and engineering. A polynomial function is a function that can be written in the form of a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power. In this article, we will focus on substituting values into a polynomial function, specifically the function p(t) = 3t + 5.

The Function p(t) = 3t + 5

The given function is p(t) = 3t + 5. This function takes a value of t as input and returns a value of p(t) as output. The function is a linear function, meaning that it has a constant rate of change. The coefficient of t is 3, which means that for every unit increase in t, the value of p(t) increases by 3 units.

Substituting Values into the Function

To find p(t-3), we need to substitute t-3 into the function p(t) = 3t + 5. This means that we will replace every instance of t in the function with t-3.

Step 1: Replace t with t-3

We will start by replacing t with t-3 in the function p(t) = 3t + 5.

p(t-3) = 3(t-3) + 5

Step 2: Distribute the 3

Next, we will distribute the 3 to the terms inside the parentheses.

p(t-3) = 3t - 9 + 5

Step 3: Combine Like Terms

Finally, we will combine the like terms -9 and 5.

p(t-3) = 3t - 4

Conclusion

In conclusion, to find p(t-3), we need to substitute t-3 into the function p(t) = 3t + 5. We replaced t with t-3, distributed the 3, and combined the like terms to get the final answer p(t-3) = 3t - 4.

Answer

The correct answer is B) 3t - 4.

Tips and Tricks

  • When substituting values into a polynomial function, make sure to replace every instance of the variable with the new value.
  • When distributing a coefficient to terms inside parentheses, make sure to multiply each term by the coefficient.
  • When combining like terms, make sure to add or subtract the coefficients of the like terms.

Real-World Applications

Polynomial functions have many real-world applications, including:

  • Modeling population growth
  • Describing the motion of objects
  • Analyzing data
  • Solving optimization problems

Practice Problems

Try substituting values into the following polynomial functions:

  • p(t) = 2t + 1, find p(t-2)
  • p(t) = t^2 + 3t - 4, find p(t+1)
  • p(t) = 4t - 2, find p(t-1)

Conclusion

Frequently Asked Questions

In this article, we will answer some frequently asked questions about substituting values into polynomial functions.

Q: What is a polynomial function?

A polynomial function is a function that can be written in the form of a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power.

Q: How do I substitute values into a polynomial function?

To substitute values into a polynomial function, you need to replace every instance of the variable with the new value. For example, if you have the function p(t) = 3t + 5 and you want to find p(t-3), you would replace t with t-3.

Q: What is the difference between a linear function and a polynomial function?

A linear function is a function that can be written in the form of y = mx + b, where m is the slope and b is the y-intercept. A polynomial function is a function that can be written in the form of a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power.

Q: How do I distribute a coefficient to terms inside parentheses?

To distribute a coefficient to terms inside parentheses, you need to multiply each term by the coefficient. For example, if you have the function p(t) = 3(t-3) + 5, you would distribute the 3 to the terms inside the parentheses to get p(t) = 3t - 9 + 5.

Q: How do I combine like terms?

To combine like terms, you need to add or subtract the coefficients of the like terms. For example, if you have the function p(t) = 3t - 9 + 5, you would combine the like terms -9 and 5 to get p(t) = 3t - 4.

Q: What are some real-world applications of polynomial functions?

Polynomial functions have many real-world applications, including:

  • Modeling population growth
  • Describing the motion of objects
  • Analyzing data
  • Solving optimization problems

Q: How do I solve a polynomial equation?

To solve a polynomial equation, you need to set the equation equal to zero and then factor the equation. For example, if you have the equation 2x^2 + 5x - 3 = 0, you would factor the equation to get (2x - 1)(x + 3) = 0.

Q: What is the difference between a quadratic function and a polynomial function?

A quadratic function is a polynomial function of degree two, meaning that it has a squared variable term. A polynomial function can have any degree, meaning that it can have any number of terms with variables raised to any power.

Q: How do I graph a polynomial function?

To graph a polynomial function, you need to find the x-intercepts and the y-intercept of the function. You can also use a graphing calculator or a computer program to graph the function.

Conclusion

In this article, we answered some frequently asked questions about substituting values into polynomial functions. We discussed the definition of a polynomial function, how to substitute values into a polynomial function, and how to distribute a coefficient to terms inside parentheses. We also discussed some real-world applications of polynomial functions and how to solve a polynomial equation.