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:

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:

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:

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:

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

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