ALGEBRA 2 CHAPTER 5 NOTES Section 5-1

Transformation of Functions Objectives:

• Transform quadratic functions. • Describe the effects of changes in the coefficients of y = a(x – h)2 + k.

A quadratic function is a function that can be written in the form of : In a quadratic function, the variable is always Notice that the graph of the parent function

f(x) = x2 is a Graph g(x) = –x2 + 6x – 8 by using a table.

g(x) = (x – 2)2 + 4 g(x) = x2 – 5

g(x) =(3x)2

This lowest or highest point is the The of a quadratic function is where a, h, and k are constants. Use the description to write the quadratic function in vertex form.

The parent function f(x) = x2 is vertically stretched by a factor of and then translated 2 units left and 5 units down to create g. The parent function f(x) = x2 is reflected across the x-axis and translated 5 units left and 1 unit up to create g.

ALGEBRA 2 CHAPTER 5 NOTES Section 5-2 Quadratics in Standard Form

Objectives:

The is the line through the vertex of a parabola that divides the parabola into two congruent Identify the axis of symmetry for the graph of

The of a quadratic function is where a ≠ 0. c is the a is the When a is positive, the parabola is When the a negative, the parabola is Consider the function f(x) = 2x2 – 4x + 5 a. Determine whether the graph opens upward or downward. b. Find the axis of symmetry. c. Find the vertex. d. Find the y-intercept.

( ) . f x x2

3 1

Consider the function f(x) = –x2 – 2x + 3. a. Determine whether the graph opens upward or downward. b. Find the axis of symmetry. c. Find the vertex. d. Find the y-intercept.

The of any quadratic function is all real numbers. The of a quadratic function depends on its vertex and the direction that the parabola opens. The value is the value at the vertex. It is the ordered pair that represents the vertex. Find the minimum or maximum value of f(x) = –3x2 + 2x – 4. Then state the domain and range of the function.

ALGEBRA 2 CHAPTER 5 NOTES Section 5-3 Solving Quadratics

Objectives: • Solve quadratic equations by graphing or factoring.

Determine a quadratic function from its roots • Functions have

• Equations have Quadratic functions can have _____ zeros, as shown. These zeros are always symmetric about the axis of symmetry.

Find the zeros of g(x) = –x2 – 2x + 3 by using a graph and a table. Find the zeros of f(x) = x2 – 6x + 8 by using a graph and table. Find the zeros of the function by factoring. f(x) = x2 – 4x – 12 g(x) = x2 – 8x g(x) = 3x2 + 18x

Quadratic expressions with two terms are Quadratic expressions with three terms are

Find the roots of the equation by factoring. 4x2 = 25 x2 – 4x = –4 18x2 = 48x – 32 If you know the zeros of a function, you can work backward to write a rule for the function Write a quadratic function in standard form with zeros 4 and –7. Write a quadratic function in standard form with zeros 5 and –5.

ALGEBRA 2 CHAPTER 5 NOTES Section 5-4 Completing the Square

Objectives: • Solve quadratic equations by completing the square.

• Write quadratic equations in vertex form. • Square Root Property:

• Solve the equation. 4x2 + 11 = 59 4x2 – 20 = 5 Solve the equation. x2 + 8x + 16 = 49 x2 + 12x + 36 = 28 Completing the square. x2 = 12x – 20 18x + 3x2 = 45 Write the function in vertex form, and identify its vertex f(x) = x2 + 16x – 12 g(x) = 3x2 – 18x + 7

ALGEBRA 2 CHAPTER 5 NOTES Section 5-5 Complex Numbers

Objectives: • Define and use imaginary and complex numbers. • Solve quadratic equations with complex roots.

You can see in the graph of f(x) = x2 + 1 that f has real zeros. If you solve the corresponding equation 0 = x2 + 1, you find that x = ,which has real solutions.

The is defined as . You can use the imaginary unit to write the square root of any Express the number in terms of i

Solve the equation x2 = -144 5x2 + 90 = 0 x2 + 48 = 0 9x2 + 25 = 0 A is a number that can be written in the form _________ where a and b are real numbers and i = .

Every complex number has a ___________________a and an ___________________b

Find the values of x and y that make the equation true . 2x – 6i = –8 + (20y)i 4x + 10i = 2 – (4y)i The solutions and are related. These solutions are a pair. The of any complex number a + bi is the complex number a – bi. 8 + 5i 6i

ALGEBRA 2 CHAPTER 5 NOTES Section 5-6 Quadratic Formula

Objectives: Solve quadratic equations using the Quadratic Formula.

Classify roots using the discriminant.

Find the zeros of the following using the Quadratic Formula. f(x)= 2x2 – 16x + 27 f(x) = x2 + 3x – 7 The is part of the Quadratic Formula that you can use to determine the number of _________________________of a quadratic equation.

Make sure the equation is in ______________________________________ before you evaluate the discriminant Find the type and number of solutions for the equation. x2 + 36 = 12x x2 + 40 = 12x x2 – 4x = –4

ALGEBRA 2 CHAPTER 5 NOTES Section 5-7 Solving Quadratic Inequalities

Objectives: Solve quadratic inequalities by using tables and graphs.

Solve quadratic inequalities by using algebra.

Quadratic inequality in two variables y ax2 + bx + c y ax2 + bx + c y ax2 + bx + c y ax2 + bx + c

Graph y ≥ x2 – 7x + 10. y < –3x2 – 6x – 7

For ____________statements, both of the conditions must be true. For ______statements, at least one of the conditions must be true. Solve the inequality x2 + 8x + 20 ≥ 5 x2 + 8x + 20 < 5 x2 – x + 5 < 7 x2 – 24 ≤ 5x

ALGEBRA 2 CHAPTER 5 NOTES Section 5-8 Curve Fitting

Objectives: Use quadratic functions to model data.

Use quadratic models to analyze and predict. Second Differences: For a set of ordered parts , a quadratic function has constant nonzero __________________________differences.

Determine whether the data set could represent a quadratic function. Explain.

x 1 3 5 7 9

y –1 1 7 17 31

x 3 4 5 6 7

y 1 3 9 27 81

Just as two points define a linear function, ______________________________________ points define a quadratic function. You can find three coefficients a, b, and c, of by using a system of three equations, one for each point. Write a quadratic function that fits the points (1, –5), (3, 5) and (4, 16).

A is a quadratic function that represents a real data set. Models are useful for making estimates. You can apply a similar statistical method to make a quadratic model for a given data set using The shows how well a quadratic function model fits the data. The closer , the better the fit. The table shows the cost of circular plastic wading pools based on the pool’s diameter. Find a quadratic model for the cost of the pool, given its diameter. Use the model to estimate the cost of the pool with a diameter of 8 ft.

Diameter (ft) 4 5 6 7

Cost $19.95 $20.25 $25.00 $34.95

The tables shows approximate run times for 16 mm films, given the diameter of the film on the reel. Find a quadratic model for the reel length given the diameter of the film. Use the model to estimate the reel length for an 8-inch-diameter film.

Film Run Times (16 mm)

Diameter (in)

Reel Length (ft)

Run Time (min)

5 200 5.55

7 400 11.12

9.25 600 16.67

10.5 800 22.22

12.25 1200 33.33

13.75 1600 44.25

ALGEBRA 2 CHAPTER 5 NOTES Section 5-9 Operations with Complex Numbers

Objectives: Perform operations with complex numbers.

The is a set of coordinate axes in which the horizontal axis represents real numbers and the vertical axis represents numbers.

2 – 3i –1 + 4i 4 + i –i

Find each absolute value. │a + bi│= |3 + 5i| |–7i| Add: (4 + 2i) + (–6 – 7i)

Find (3 – i) + (2 + 3i) by graphing.

Multiply. Write the result in the form a + bi. –2i(2 – 4i) –5i)(6i) (3 + 6i)(4 – i) (2 + 9i)(2 – 9i)

Simplify –6i14. Simplify i63 Simplify i42 You must rationalize any denominator that contains an imaginary unit.