1Algebra

QUADRATIC FUNCTIONS

• Quadratic Expressions, Rectangles and Squares• Absolute Value, Square Roots and Quadratic Equations• The Graph Translation Theorem• Graphing • Completing the Square• Fitting a Quadratic Model to Data• The Quadratic Formula• Analyzing Solutions to Quadratic Equations• Solving Quadratic Equations and Inequalities

2Algebra

QUADRATIC FUNCTIONS

Quadratic – quadratus (Latin) , ‘to make square’

Standard form of a quadratic:

3Algebra

QUADRATIC FUNCTIONS

Quadratic expressions from Rectangles and Squares

Suppose a rectangular swimming pool 50 m by 20 m is to be built with a walkway around it. If the walkway is w meters wide, write the total area of the pool and walkway in standard form.

Write the area of the square with sides of length in standard form

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Binomial Square Theorem

For all real numbers x and y,

Note: When discussing this, ask students whether any real-number values of the variable give a negative value to the expression. [ The square of any real number is nonnegative].

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Challenge

Have students give quadratic expressions for the areas described below.

1. The largest possible circle inside a square whose side is x.

2. The largest possible square inside a circle whose radius is x.

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Absolute Value, Square Roots and Quadratic Equations

In Geometry•

In Algebra

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Activity

1. Evaluate each of the following.

2. Find a value of x that is a solution to .

3. Find a value of x that is not a solution to .

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Absolute Value – Square Root Theorem

For all real numbers x,

Example 1

Solve

Example 2

A square and a circle have the same area. The square has side 10. What is the radius of the circle?

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ChallengeThe Existence of Irrational Numbers

Prove that cannot be written as a simple fraction.

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Graphs and Translations

Consider the graphs of and What transformation maps the graph of the first function

onto the graph of the second?

Graph – Translation Theorem

In a relation described by a sentence in x and y, the following two processes yield the same graph:

1. replacing by and by

2. applying the translation to the graph of the original relation.

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QUADRATIC FUNCTIONS

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Example 1

Find an equation for the image of the graph of under the translation .

Corollary

The image of the parabola under the translation is the parabola with the equation

Example 2

a. Sketch the graph of

b. Give the coordinates of the vertex of the parabola

c. Tell whether the parabola opens up or down

d. Give the equation for the axis of symmetry.

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Graphing Suppose

a. Find when

b. Explain what each pair tells you about the height of the ball.

c. Graph the pairs over the domain of the function.

Note: Two natural questions about the thrown ball are related to questions about this parabola.

1. How high does the ball get? The largest possible value of h.

2. When does the ball hit the ground?

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QUADRATIC FUNCTIONS

Newton’s Formula

• is a constant measuring the acceleration due to gravity• is the initial upward velocity• is the initial height • the equation represents the height of the ball off the ground at time

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QUADRATIC FUNCTIONS

Completing the Square

Completing the square geometrically and algebraically

Theorem

To complete the square on .

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Practice

• Going back to , find the maximum height of the ball.• Rewrite the equation in vertex form. Locate the vertex of

the parabola.• Suppose

a. What is the domain of ?

b. What is the vertex of the graph?

c. What is the range of ?

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Fitting a Model to Data

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Practice

The number of handshakes needed for everyone in a group of people, to shake the hands of every other person is a quadratic function of Find three points of the function relating and Use these points to find a formula for this function.

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The Angry Blue Bird Problem

What if Blue Bird’s flight path is described by the function

Where is Blue Bird when she’s 8 feet high?

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The Quadratic Formula

If

Challenge

How was it derived?

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Practice

Solve

The 3-4-5 right triangle has sides which are consecutive integers. Are there any other right triangles with this property?

Challenge: Find a number such that 1 less than the number divided by the reciprocal of the number is equal to 1.

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QUADRATIC FUNCTIONS

How Many Real Solutions Does a Quadratic Equation Have?

Discriminant Theorem

Suppose are real numbers with

Then the equation has

i. two real solutions if

ii. one real solution if

iii. two complex conjugate solutions if .

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QUADRATIC FUNCTIONS

Practice

Determine the nature of the roots of the following equations. Then solve.

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Solving Quadratic Equations

Extracting Square Roots Factoring Completing the Square Quadratic Formula

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FactoringSteps:• Transform the quadratic equation into standard form if necessary.• Factor the quadratic expression.• Apply the zero product property by setting each factor of the quadratic

expression equal to 0.

Zero Product Property– If the product of two real numbers is zero, then either of the two

is equal to zero or both numbers are equal to zero.

• Solve each resulting equation.• Check the values of the variable obtained by substituting each in the original

equation.

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Practice

Solve the following equations.

1.

2.

3.

4.

5.

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HOW TO SOLVE?

1. find the "=0" points

2. in between the "=0" points, are intervals that are either

greater than zero (>0), or

less than zero (<0)

3. then pick a test value to find out which it is

(>0 or <0)

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Here is the plot of 

The equation equals zero at -2 and 3

The inequality "<0" is true 

between -2 and 3.

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Practice

1. Find the solution set of

2. Graph

3. A stuntman will jump off a 20 m building. A high-speed camera is ready to film him between 15 m and 10 m above the ground. When should the camera film him?