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§ 1 - 2 Quadratic Functions

The student will learn about:

Quadratic function equations,

quadratic graphs, and

applications.

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IntroductionA quadratic is an equation of the form

f (x) = a x2 + b x + c , and graphs as a parabola.

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Life has many paths!

There are several ways in which we may study functions.

• Algebraically – as we have been doing.

• Calculus – throughout the upcoming weeks.

• Graphing Calculator – as we have been doing.

We need to be careful of the last method. Many instructors do not permit the use of graphing calculators. For that reason I expect you to be conversant in all three methods!

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Definition

The graph of a quadratic function is called a parabola.

Def: A function f is a quadratic function if f (x) = a x2 + b x + c, where a 0, and a, b, and c are real numbers. The domain of a quadratic function is the set of all real numbers.

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Example 1

f (x) = x2 + x – 2.

1. Find the x intercepts algebraically

Sketch the graph of f (x) in a Cartesian coordinate system.

To find the x intercept, let f (x) = 0 and solve for x. Factor or use the quadratic formula.

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Solving a Quadratic Equation

f (x) = x2 + x – 2.

Solutions may be found algebraically.

Let f (x) = 0 and solve for x :

The solutions of a quadratic (the x-intercepts) are also called roots or zeros.

a. Factor : 0 = (x + 2)(x – 1) and x = 1 and – 2.

OR

b. Use the quadratic formula – Next slide please!

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Solving a Quadratic Equation

f (x) = x2 + x – 2.

b. Use the quadratic formula : a = 1, b = 1, c = -2.

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811x

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Example 1 - Continued

f (x) = x2 + x – 2.

2. Find the y-intercept algebraically

Sketch the graph of f (x) in a Cartesian coordinate system.

To find the y-intercept let x = 0 and solve for f (x). The solution will be c.

3. Using the x-intercepts of 1 and -2 and the y-intercept of -2 we can now sketch this parabola.

A graphing calculator might help!

Knowing the vertex might help!

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2. Enter the function Y1 = x 2 + x – 2 and then press ZOOM and 6 for a standard window.

x-intercepts using CALC and zero giving 1 and -2

Graphing a Quadratic Function: Calculator

1. Turn the calculator on and press the y = button. If something is there press clear.

3. Find the x and y intercepts using the calculator.

y-intercept using CALC and value giving -2OR use table!

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Properties of Quadratics

Many useful properties of the quadratic function can be found by transforming the general equation f (x) = a x2 + b x + c into the form f (x) = a (x – h)2 + k. This second form is called the standard form (or vertex form) and can be derived from the general form by completing the square.

There is a simple (non algebraic) way to do this.

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Example 2 * f (x) = -2x2 + 4x + 6 f (x) = ax2 + bx + c = a(x – h)2 + k

2. Solve f(x) by completing the square:

1. The intercepts can be calculated as previously shown.

y = -2x2 + 4x + 6

Hence a = - 2, h = 1 and k = 8.

(-1, 0), (3, 0), and (0, 6)

y = -2 (x2 - 2x) + 6y = -2 (x2 - 2x + 1) + 6y = -2 (x –1)2 + 8

+ 2

There is a simple way to do this.

? ? ?

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3. The graph is a parabola with:

With intercepts: (-1, 0), (3, 0), and (0, 6) and

• vertex at (h, k), and

a = -2, h = 1 and k = 8.

• axis of symmetry: x = h, and• a minimum if a > 0 or• a maximum if a < 0.

Example 2 Continued * f (x) = -2x2 + 4x + 6 f (x) = ax2 + bx + c = a(x – h)2 + k

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With intercepts: (-1, 0), (3, 0), and (0, 6) and a = -2, h = 1 and k = 8.

4. The graph of f is a transformation of the graph of g (x) = x2 .

• reflected in the x axis since a is negative• stretched double since |a| = 2 > 1• shifted one to the right since h = 1

• shifted eight up since k = 8.

Example 2 Continued * f (x) = -2x2 + 4x + 6 f (x) = ax2 + bx + c = a(x – h)2 + k

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Example 2 Continuedf (x) = -2x2 + 4x + 6

With intercepts: (-1, 0), (3, 0), and (0, 6) and a = -2, h = 1 and k = 8 and• reflected in the x axis

• stretched double

• shifted one to the right• shifted eight up.

There is a simple way to do this.

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Reviewf (x) = ax2 + bx + c OR y = a (x – h)2 + k

x intercepts (x, 0)

opens up if a > 0

y intercept (0, y)

stretch if |a| > 1

squish if |a| < 1

opens down if a < 0

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Reviewf (x) = ax2 + bx + c OR y = a (x – h)2 + k

Has a minimum if a > 0

Domain is all Reals

Has a maximum if a < 0

Range is [k, ) if a > 0

Range is (- , k] if a < 0

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Reviewf (x) = ax2 + bx + c OR y = a (x – h)2 + k

vertex at (h, k)

MAX or min of y = k

symmetry axis x = h

note h = -b/2a

note k = c – b2/4a

MAX or min at x = h

There is a simple way to do this.

So, what is it?

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The simple way!f (x) = ax2 + bx + c OR y = a (x – h)2 + k

From example 2, f (x) = - 2x 2 + 4x + 6 we need to get y = a (x – h) 2 + k.But we already know from the standard form that a = - 2,

The equation must be: y = - 2 (x – 1) 2 + 8. TA DA!

But we already know from the standard form that a = - 2.

Press CALC and 3 for a minimum. Giving h = 1 and k = 8.

I love my calculator!

Graph it!

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Summary.

• We learned about quadratics and the different forms for quadratic equations.

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ASSIGNMENT

§1.2; Page 7; 1 – 11, odd.