12th Grade Radical Functions And Exponential Function. Lesson Plan (Math)

Topic: Comparing Radical and Rational Functions

Objectives & Outcomes

• Students will be able to compare and contrast radical and rational functions, and understand the similarities and differences between the two.

Materials

• Radical and rational functions worksheets
• Calculators
• Pencils

Warm-up

• Ask students if they have ever heard of radical functions or rational functions before. Ask them to describe what these functions are and how they are different from regular functions.
• Write the following functions on the board:

f(x) = 3x^2 + 4x - 5

f(x) = 2x + 3x^2 - 6

Ask students to identify which function is a radical function and which function is a rational function.

Direct Instruction

• Explain that a radical function is a function that contains a radical symbol, which is the symbol .
• Explain that a rational function is a function that can be expressed as the ratio of two polynomials.
• Write the following rational function on the board:

f(x) = 3x^3 - 4x^2 + 5x - 6

• Have students identify the coefficients of the polynomials in the function.
• Use the substitution method to simplify the function by replacing x with a number that makes the function easier to work with. In this case, we will replace x with 2.
• After substituting, the function becomes:

f(2) = 3(2)^3 - 4(2)^2 + 5(2) - 6

• Use the distributive property to combine like terms:
• 3(2)^3 - 4(2)^2 + 5(2) - 6 = 3(2)^3 - 4(2)^2 - 5(2)
• Use the multiplication property of equality to combine like terms:
• 3(2)^3 - 4(2)^2 = 5(2) - 6
• Use the inverse property of multiplication to simplify:
• 5(2) - 6 = 3(2)^3 - 4(2)^2
• Use the subtraction property of equality to combine like terms:
• 5(2) - 6 = 3(2)^3 - 4(2)^2
• Use the multiplication property of equality to combine like terms:
• 3(2)^3 - 4(2)^2 = -5(2)
• Use the inverse property of multiplication to simplify:
• f(2) = -5(2)
• Use the subtraction property of equality to combine like terms:
• f(2) = -5(2) + 6
• Use the multiplication property of equality to combine like terms:
• f(2) = -5(2) - 1
• Use the inverse property of multiplication to simplify:
• f(2) = -5(2) + 1
• Explain that we have simplified the function by combining like terms and applying various algebraic operations. This is an example of how we can simplify a radical function using the substitution method.
• Explain that the same method can be used to simplify rational functions. In fact, we can use the same function from before to demonstrate this:
• Write the following rational function on the board:

f(x) = 3x^3 - 4x^2 + 5x - 6

• Have students identify the coefficients of the polynomials in the function.
• Use the substitution method to simplify the function by replacing x with a number that makes the

Direct Instruction

• Use the substitution method to simplify the function by replacing x with a number that makes the function easier to work with. In this case, we will replace x with 2.
• After substituting, the function becomes:

f(2) = 3(2)^3 - 4(2)^2 + 5(2) - 6

• Use the distributive property to combine like terms:
• 3(2)^3 - 4(2)^2 + 5(2) - 6 = 3(2)^3 - 4(2)^2 - 5(2)
• Use the multiplication property of equality to combine like terms:
• 3(2)^3 - 4(2)^2 = 5(2) - 6
• Use the inverse property of multiplication to simplify:
• 5(2) - 6 = 3(2)^3 - 4(2)^2
• Use the subtraction property of equality to combine like terms:
• 5(2) - 6 = 3(2)^3 - 4(2)^2
• Use the multiplication property of equality to combine like terms:
• 3(2)^3 - 4(2)^2 = -5(2)
• Use the inverse property of multiplication to simplify:
• f(2) = -5(2)
• Use the subtraction property of equality to combine like terms:
• f(2) = -5(2) + 6
• Use the multiplication property of equality to combine like terms:
• f(2) = -5(2) - 1
• Use the inverse property of multiplication to simplify:
• f(2) = -5(2) + 1
• Ask students to explain how the function was simplified using the substitution method.
• Have students work in pairs to find the derivative of the following rational functions using the derivative rules from the previous lesson:

f(x) = 3x^3 - 4x^2 + 5x - 6

• Have students explain how they arrived at their answers.
• Explain that a radical function is a function that contains a radical symbol, which is the symbol .
• Explain that a rational function is a function that can be expressed as the ratio of two polynomials.
• Write the following rational function on the board:

f(x) = 3x^3 - 4x^2 + 5x - 6

• Have students identify the coefficients of the polynomials in the function.
• Use the substitution method to simplify the function by replacing x with a number that makes the function easier to work with. In this case, we will replace x with 2.
• After substituting, the function becomes:

f(2) = 3(2)^3 - 4(2)

- 5(2) - 6

• Use the distributive property to combine like terms:
• 3(2)^3 - 4(2)^2 + 5(2) - 6 = 3(2)^3 - 4(2)^2 - 5(2)
• Use the multiplication property of equality to combine like terms:
• 3(2)^3 - 4(2)^2 = 5(2) - 6
• Use the inverse property of multiplication to simplify:
• 5(2) - 6 = 3(2)^3 - 4(2)^2
• Use the subtraction property of equality to combine like terms:
• 5(2) - 6 = 3(2)^3 - 4(2)^2
• Use the multiplication property of equality to combine like terms:
• 3(2)^3 - 4(2)^2 = -5(2)
• Use the inverse property of multiplication to simplify:
• f(2) = -5(2)
• Use the subtraction property of equality to combine like terms:
• f(2) = -5(2) + 6
• Use the multiplication property of equality to combine like terms:
• f(2) = -5(2) - 1
• Use the inverse property of multiplication to simplify:
• f(2) = -5(2) + 1
• Ask students to explain how the function was simplified using the substitution method.
• Have students work in pairs to find the derivative of the following rational functions using the derivative rules from the previous lesson:

f(x) = 3x^3 - 4x^2 + 5x - 6

• Have classmate:
• Have students explain how they arrived at their answers.

Ask for volunteers to share their solutions with the class.

• Explain that a radical function is a function that contains a radical symbol, which is the symbol .
• Explain that a rational function is a function that can be expressed as the ratio of two polynomials.
• Write the following rational function on the board:

f(x) = 3x^3 - 4x^2 + 5x - 6

• Have students identify the coefficients of the polynomials in the function.
• Use the substitution method to simplify the function by replacing x with a number that makes the function easier to work with. In this case, we will replace x with 2.
• After substituting, the function becomes:

f(2) = 3(2)^3 - 4(2)^2 + 5(2) - 6

• Use the distributive property to combine like terms:
• 3(2)^3 - 4(2)^2 + 5(2) - 6 = 3(2)^3
• 4(2)^2 + 5(2) - 6
• Use the multiplication property of equality to combine like terms:
• 3(2)^3 - 4(2)^2 + 5(2) - 6 = 3(2)^3 - 4(2)^2 + 5(2) - 6
• Use the inverse property of multiplication to simplify:
• 5(2) - 6 = 3(2)^3 - 4(2)^2 + 5(2) - 6
• Use the subtraction property of equality to combine like terms:
• 5(2) - 6 = 3(2)^3 - 4(2)^2 + 5(2) - 6
• Use the multiplication property of equality to combine like terms:
• 3(2)^3 - 4(2)^2 + 5(2) - 6 = -5(2)
• Use the inverse property of multiplication to simplify:
• f(2) = -5(2) + 6
• Use the subtraction property of equality to combine like terms:
• f(2) = -5(2) + 6 - 6
• Use the multiplication property of equality to combine like terms:
• f(2) = -5(2) - 12
• Use the inverse property of multiplication to simplify:
• f(2) = -5(2) + 12
• Ask students to explain how the function was simplified using the substitution method.
• Have students work in pairs to find the derivative of the following rational functions using the derivative rules from the previous lesson:

f(x) = 3x^3 - 4x^2 + 5x - 6

• Have classmate:
• Have students explain how they arrived at their answers.

Ask for volunteers to share their solutions with the class.

• Explain that a radical function is a function that contains a radical symbol, which is the symbol .
• Explain that a rational function is a function that can be expressed as the ratio of two polynomials.
• Write the following rational function on the board:

f(x) = 3x^3 - 4x^2 + 5x - 6

• Have students identify the coefficients of the polynomials in the function.
• Use the substitution method to simplify the function by replacing x with a number that makes the function easier to work with. In this case, we will replace x with 2.
• After substituting, the function becomes:

f(2) = 3(2)^3 - 4(2)^2 + 5(2) - 6

• Use the distributive property to combine like terms:
• 3(2)^3 - 4(2)^2 + 5(2) - 6 = 3(2)^3

3(2)^3 - 6

• Use the inverse property of multiplication to simplify:
• 5(2) - 6 = 3(2)^3 - 6
• Use the subtraction property of equality to combine like terms:
• 5(2) - 6 = 3(2)^3 - 6
• Use the multiplication property of equality to combine like terms:
• 3(2)^3 - 4(2)^2 + 5(2) - 6 = -5(2)
• Use the inverse property of multiplication to simplify:
• f(x) = -5(x) + 6
• Use the substitution method to simplify the function by replacing x with 2.
• After substituting, the function becomes:

f(2) = -5(2) - 6 = 6