CHAPTER 9: Quadratic Equations and Functions

Notes #23

9-1: Exploring Quadratic Graphs

A. Graphing

·  A ______is a function that can be written in the form where a, b, and c are real numbers and a0.
Examples:

·  The graph of a quadratic function is a U-shaped curve called a ______. When graphed it will look like: OR

·  You can fold a parabola so that the two sides match exactly. This property is called: ______.

·  The highest or lowest point of the parabola is called the ______, which is on the axis of symmetry.

B. Identifying a Vertex

Identify the vertex of each graph. Tell whether it is a minimum or a maximum.

1.) 2.)

Vertex: ( , ) ______Vertex: ( , ) ______

3.) 4.)

Vertex: ( , ) ______Vertex: ( , ) ______


Graph each function. State the domain, the vertex (min/max point), the range, the x-intercepts, and the axis of symmetry.

5.) f(x)= x2 – 4
/
Domain: ______
Range: ______
Vertex:______
Max or min?______
x-intercepts: ______
Axis of symmetry: ______
6.) h(x) = -2x2
/
Domain: ______
Range: ______
Vertex:______
Max or min?______
x-intercepts: ______
Axis of symmetry: ______
7.) f(x) =
/
Domain: ______
Range: ______
Vertex:______
Max or min?______
x-intercepts: ______
Axis of symmetry: ______

8.) k(x) = x2 + 2x + 1
/


Domain: ______
Range: ______
Vertex:______
Max or min?______
x-intercepts: ______
Axis of symmetry: ______

9.) f(x) = x2 – x - 6
/




Domain: ______
Range: ______
Vertex:______
Max or min?______
x-intercepts: ______
Axis of symmetry: ______

C. Comparing Widths of parabolas

The value of a, the coefficient of the term in a quadratic function, affects the width of the parabola as well as the direction in which it opens.

·  When then the parabola is steeper, (or ______) than y = x2

·  When then the parabola is not as steep, (or ______) than y = x2

Order each group of quadratic functions from widest to narrowest graph:

10.) 11.)


D. Applications

12.) A monkey drops an banana from a branch 64 feet above the ground. Gravity causes the banana to fall. The function gives the height of the banana h in feet after t seconds.

a) Graph this quadratic function b) When does the banana hit the ground?

t / / (t, h(t))


13) A bungee jumper dives from a platform. The function h = -16t2 + 160 describes her height, h, after t seconds in the air.

a) What will her height be after 1 second? b) what will her height be after 2 seconds?

c) How far did she fall between 1 and 2 seconds in the air?


Notes #24

9-2: Quadratic Functions

y = ax2 + bx + c

One key characteristic of a parabola is its vertex (min/max point). Yesterday we found the vertex after we graphed the function. It would help to find the vertex first.

Vertex
- find x =
- plug this x-value into the function (table)
- this point (___, ___) is the vertex of the parabola / Graphing
- put the vertex you found in the center of
your x-y chart.
- choose 2 x-values less than and 2 x-values more than your vertex.
- plug in these x values to get 4 more points.
- graph all 5 points

Find the vertex of each parabola. Graph the function and find the requested information

1.) f(x)= -x2 + 2x + 3 a = ____, b = ____, c = ____
/

Vertex: ______
Max or min? ______
Direction of opening? ______
Wider or narrower than y = x2 ?
______
Domain: ______
Range: ______
x-intercepts: ______
Axis of symmetry: ______
2.) h(x) = 2x2 + 4x + 1
/

Vertex: ______
Max or min? ______
Direction of opening? ______
Wider or narrower than y = x2 ?
______
Domain: ______
Range: ______
x-intercepts: ______
Axis of symmetry: ______
3.) k(x) = 2 – x –x2
/

Vertex: ______
Max or min? ______
Direction of opening? ______
Wider or narrower than y = x2 ?
______
Domain: ______
Range: ______
x-intercepts: ______
Axis of symmetry: ______

Without graphing the quadratic functions, complete the requested information:

4.)
What is the direction of opening? ______
Is the vertex a max or min? ______
Wider or narrower than y = x2 ? ______/ 5.)

What is the direction of opening? ______
Is the vertex a max or min? ______
Wider or narrower than y = x2 ? ______

6.)
What is the direction of opening? ______
Is the vertex a max or min? ______
Wider or narrower than y = x2 ? ______/
7.)

What is the direction of opening? ______
Is the vertex a max or min? ______
Wider or narrower than y = x2 ? ______

B. Application

8.) Suppose a particular star is projected from an aerial firework at a starting height of 520 feet with an initial upward velocity of 88 ft/s. How long will it take for the star to reach its maximum height? How far above the ground will it be? The equation gives the star’s height h in feet at time t in seconds.