algebra ii chapter 4: part 2 - mrs. gruver's classroom...

Alg II Chapter 4 Notes Part 2.notebook

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Algebra IIChapter 4:

Quadratic Functions and Factoring

Part 2

Chapter 4 Lesson 6Perform Operations with Complex NumbersVocabulary


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Example 1: Solve a Quadratic Equation

Reminder: All rules for simplifying radicals still apply!

Example 2: Add and Subtract Complex Numbers

Make sure to give answers in standard form!


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Example 3: Use Addition of Complex Numbers in Real Life

Example 4: Multiply Complex NumbersReminder: i * i = ‐1, which means you should never have i2 in a simplified answer!Make sure to give answers in standard form!


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Example 5: Divide Complex NumbersYou need a real number in the denominator of a fracon so that you can give the answer in standard form. To do this, mulply the numerator and denominator by the conjugate of the denominator. This is similar to the process of "raonalizing the denominator". Reminder: i * i = ‐1, which means you should never have i2 in a simplified answer!

Example 6: Plot Complex Numbers

The x axis becomes the "real" axis. The y axis becomes the "imaginary" axis.

real

imaginary


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Example 7: Find Absolute Values of Complex Numbers

Chapter 4 Lesson 8Use the Quadratic Formula and the DiscriminantVocabulary


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Three Scenarios When Solving Quadratics:1. Two Real Solutions

*The parabola will cross the x axis two times.

2. One Real Solution*The parabola will touch the x axis only one time.

3. No Real Solutions (Two Imaginary Solutions!!!)*The parabol will NOT touch the x axis at all!

Example 1: Solve an Equation with Two Real Solutions

Make sure the equation is equal to zero before you start!


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Example 2: Solve an Equation with One Real Solution


Example 3: Solve an Equation with Imaginary Soltutions



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Example 4: Use a Discriminant

The discriminant is only the expression under the radical in the quadratic formula!!!

Example 5: Solve a Vertical Motion Problem

s


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Chapter 4 Lesson 7Complete the SquareVocabulary

Example 1: Solve a Quadratic Equation by Finding Square Roots1. Factor first.2. Look for a perfect square binomial.3. If it is a perfect square binomial, you can solve by finding square roots.Reminder: YOU WILL GET TWO ANSWERS!!!!


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Example 2: Make a Perfect Square Trinomial

Example 3: Solve ax2 + bx + c = 0 when a = 1Steps:1. Add the opposite of the constant to both sides. Leave space to add a term!2. Make a perfect square trinomial, but be sure to add that value to BOTH sides of the equation. Reminder: 3. Write the trinomial as a binomial squared.4. Take the square root of both sides. Reminder: You will get a positive and negative result.5. Solve for x.


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Example 4: Solve ax2 + bx + c = 0 when a = 1Steps:1. Divide everything by the coefficient of the x2 term.2. Add the opposite of the constant to both sides.3. Make a perfect square trinomial, but be sure to add that value to BOTH sides of the equation. Reminder: 4. Write the trinomial as a binomial squared.5. Take the square root of both sides. Reminder: You will get a positive and negative result.6. Solve for x.

Example 5: Standardized Test Practice

In cases like the ones below, distribute and then proceed.

Steps:1. Divide everything by the coefficient of the x2 term.2. Add the opposite of the constant to both sides.3. Make a perfect square trinomial, but be sure to add that value to BOTH sides of the equation. Reminder: 4. Write the trinomial as a binomial squared.5. Take the square root of both sides. Reminder: You will get a positive and negative result.6. Solve for x.


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Example 6: Write a Quadratic Function in Vertex FormReminder:

Vertex Form: y = a(xh)2+kSteps:1. Use the steps to complete the square.2. Solve for y.

Example 7: Find the Maximum Value of a Quadratic Function


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Chapter 4 Lesson 2Graph Quadratic Functions in Vertex or Intercept Form

Vocabulary

Example 1: Graph a Quadratic Function in Vertex FormSteps:1. Identify the constants a, h, and k. Open up or down?2. Plot the vertex (h,k) and draw the line of symmetry. Reminder: h will have the opposite sign.3. Use a table of values to get two points on the graph. Then use the line of symmetry to plot the corresponding points.4. Draw the curve through the points.

Always draw the parent function!!! State the domain and range.

Domain:

Range:


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Example 2: Use a quadratic Model in Vertex Form

Example 3: Graph a Quadratic Function in Intercept FormSteps:1. Identify and plot the x intercepts (p and q). 2. Find the x value of the vertex using .

3. Substitute the x value of the vertex to get the y value of the vertex.4. Plot the vertex.5. Draw the curve through the points.

Always draw the parent function!!! State the domain and range.

Domain:

Range:


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Example 4: Use a Quadratic Function in Intercept Form

a: Rewrite the function. y = – 0.026(x – 0)(x – 46). Find the intercepts and the distance.

b: Calculate the vertex. Hint: Isn't the y value of the vertex the maximum?

Example 5: Change from Intercept Form to Standard Form

Reminder:An example of quadratic standard form might be


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Example 6: Change from Vertex Form to Standard FormReminder:*An example of quadratic standard form might be * means

Chapter 4 Lesson 10Write Quadratic Functions and ModelsVocabulary


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Example 1: Write a Quadratic Function in Vertex Form1. If vertex is given, begin with vertex form.2. Substitute the vertex values (h, k) into the equation.3. Substitute the point values (x, y) into the equation.4. Solve for a.5. Rewrite the form using the vertex values (h, k) and the new a value.

Example 2: Write a Quadratic Function in Intercept Form1. If the x intercepts are given, begin with intercept form.2. Substitute the intercept values (p and q) into the equation.3. Substitute the point values (x, y) into the equation.4. Solve for a.5. Rewrite the form using the intercept values (p and q) and the new a value.


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Example 3: Write a Quadratic Function in Standard FormSteps:1. Substitute the coordinates of each point into y = ax2 + bx + c to get a system of three linear equations.2. Rewrite the system of three equations in Step 1 as a system of two equations by substituting.3. Solve the system consisting of revised equations. Use the elimination method.

Example 4: Solve a MultiStep Problem

To Enter a List:1. Touch STAT.2. Scroll to 1:Edit and touch E3. Enter your desired numbers.4. You can add a second column Note: When entering ordered pairdirectly across from each other!

To Find Quadrac Regression:1. Touch STAT.2. Touch the right cursor to hi3. Scroll to 5:QuadReg and touc4. Touch ENTER again.Note: The resulng numbers can

quadrac: .


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algebra ii chapter 4: part 2 - mrs. gruver's classroom...

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