QUADRATICS UNIT Properties of Quadratics

Quadratic Function:

Standard Form:

Axis of Symmetry: Vertex:

Maximums/Minimums:

Vertical Intercept (y-intercept)

Vertex Form:

Transformations:

Graph the following functions:

f(x) = 2(x + 1)2 – 4 f(x) = - (x – 2)2 + 3

Homework for Properties of Quadratics

Describe the transformation on the lines below, then graph each on the axis.

QUADRATICS UNIT Solving Quadratic Equations

Solving means finding the

Ways to Solve Quadratics:

1. Using a graph or table:

2. Factoring:

f(x)= x2 – 5x – 6 g(x) = 3x2 + 18x 4x2 = 25

Perfect Squares: Difference of Squares:

x2 – 4x = –4 25x2 = 9

Going Backwards! Write a quadratic function in standard form with zeros 6 and –1.

Square Root Property:

4x2 + 11 = 59

3. Completing the Square:

How to Complete the Square:

x2 + 12x + 36 = 28 3x2 – 24x = 27

Writing a Quadratic Equation in Vertex Form:

f(x) = x2 + 16x – 12 g(x) = 3x2 – 18x + 7

Homework for Solving Quadratic Equations

Find the roots or zeros.

Write a quadratic equation with the following roots:

-2 and 7 1/2 and -1 -3/4 and 1/3

Solve the following by completing the square:

Put the following equations in vertex form:

QUADRATIC FUNCTIONS UNIT Complex Numbers and Roots

What are the roots of the function to the right?

Imaginary Numbers:

Solve: 5x2 + 90 = 0

Complex Numbers:

Graphing Complex Numbers:

Operations with Complex Numbers:

Solve for both x and y: 2x – 6i = –8 + (20y)i

Addition and Subtraction:

(4 + 2i) - (–6 – 7i)

Multiplying:

–2i(2 – 4i) (–5i)(6i) (3 + 6i)(4 – i) (2 + 9i)(2 – 9i)

Homework for Complex Numbers

Solve for x:

Solve for x and y:

Graph the following:

Solve by graphing:

Perform the indicated operations:

QUADRATICS UNIT More Complex Numbers

Finding Absolute Values of Complex Numbers: │a + bi│= a2 + b2 |–7i| |3 + 5i| Complex Conjugates: For a + bi, its complex conjugate is: 6i 8 + 5i

i raised to a power i1 i2 i3 i4

i12 i41 i63 –6i14

Simplifying Complex Fractions:

Homework for More Complex Numbers

Find each Complex Conjugate:

Find the Absolute Value:

Simplify:

QUADRATICS UNIT Solving with Quadratic Formula

Find the zeros using the Quadratic Formula: f(x) = x2 + 3x – 7 f(x)= 2x2 – 16x + 27

Find the type and number of solutions for the equation. x2 – 4x = –4 2x2 + 36 = 12x A pebble is tossed from the top of a cliff. The pebble’s height is given by y(t) = –16t2 + 200, where t is the time in seconds. Its horizontal distance in feet from the base of the cliff is given by d(t) = 5t. How far will the pebble be from the base of the cliff when it hits the ground?

Homework for Solving with Quadratic Formula Find the zeros using the Quadratic Formula:

Find the type and number of solutions:

QUADRATICS UNIT Solving Quadratic Inequalities and Curve Fitting

By graphing the inequality: y ≥ x2 – 7x + 10, we can begin to look at what shading would look like:

Looking at the inequality: y < –3x2 – 6x – 7, write the solution:

We can also use our knowledge of and and or to solve much faster: x2 + 12x + 39 ≥ 12 x2 – 24 ≤ 5x or and

Curve Fitting With Quadratics Second Differences: For a set of ordered parts with equally spaced x-values,

Using systems of equations, write a quadratic function that fits the points (2, 0), (3, –2), and (5, –12). Using: f(x) = ax2 + bx + c A quadratic model is a quadratic function that represents a real data set. Models are useful for making estimates. Quadratic regression- The coefficient of determination (R2)shows how well a quadratic function model fits the data. The closer R2 is to 1, the better the fit 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.

Homework for Solving Quadratic Inequalities and Modeling Graph the Inequality:

Solve the following Inequalities:

The following models represent quadratic functions. Find the missing values.

Write a quadratic function that fits the given values.

Use a Graphics Calculator to solve the following:

Write an inequality and solve it to find the following: A boat operator wants to offer tours of San Francisco Bay. His profit P for a trip can be modeled by P(x) = –2x2 + 120x – 788, where x is the cost per ticket. What range of ticket prices will generate a profit of at least $500?