Quadratic Functions Algebra III, Sec. 2.1

Objective

You will learn how to sketch and analyze graph of functions.

Important Vocabulary

Axis – the line of symmetry for a parabola

Vertex – the point where the the axis intersects the parabola

The Graph of a Quadratic Fn

A polynomial function of x with degree n is

A quadratic function is

The Graph of a Quadratic Fn (cont.)

A quadratic function is a polynomial function of _____________ degree. The graph of a quadratic function is a special “U” shaped curve called a _____________ .

second

parabola

The Graph of a Quadratic Fn (cont.)

If the leading coefficient of a quadratic function is positive, the graph of the function opens ________ and the vertex of the parabola is the ____________ y-value on the graph.

up

minimum

The Graph of a Quadratic Fn (cont.)

If the leading coefficient of a quadratic function is negative, the graph of the function opens ________ and the vertex of the parabola is the ____________ y-value on the graph.

down

maximum

The Graph of a Quadratic Fn (cont.)

The absolute value of the leading coefficient determines ____________________________________.

If |a| is small, _________________________________________________________________________________.

how wide the parabola opens

the parabola opens wider than when |a| is large

Example 1Sketch and compare the graphs of the quadratic functions.

a)

Reflects over the x-axisVertical stretch by 3/2

Example 1Sketch and compare the graphs of the quadratic functions.

b)

Vertical shrink by 5/6

Standard Form of a Quadratic Fn

The standard form of a quadratic function is ________________________________.

The axis of the associated parabola is ___________ and the vertex is ____________.

a ≠ 0

x = h (h, k)

Standard Form of a Quadratic Fn

To write a quadratic function in standard form…

Use completing the square

add and subtract the square of half the coefficient of x

Example 2Sketch the graph of f(x). Identify the vertex and axis.

Write original function.

Add & subtract (b/2)2 within parentheses.

Regroup terms.

Simplify.

Write in standard form.

Example 2Sketch the graph of f(x). Identify the vertex and axis.

a = 1 h = 5 k = 0

Vertex: (5, 0) Axis: x = 5

Standard Form of a Quadratic Fn

To find the x-intercepts of the graph…

You must solve the quadratic equation.

Example 3 Sketch the graph of f(x). Identify the vertex and x-intercepts.

Write original function.

Factor -1 out of x terms.

Add & subtract (b/2)2 within parentheses.

Regroup terms.

Simplify.

Write in standard form.

Example 3 Sketch the graph of f(x). Identify the vertex and x-intercepts.

a = -1 h = -2 k = 25

Vertex: (-2, 25) Axis: x = -2

Example 3 Sketch the graph of f(x). Identify the vertex and x-intercepts.

Set original function =0.

Factor out -1.

Factor.

Set factors =0.

Solve.

Example (on your handout)

Sketch the graph of f(x). Identify the vertex and x-intercepts.

Write original function.

Add & subtract (b/2)2 within parentheses.

Regroup terms.

Simplify.

Write in standard form.

Example (on your handout)

Sketch the graph of f(x). Identify the vertex and x-intercepts.

a = 1 h = -1 k = -9

Vertex: (-1, -9) Axis: x = -1

Example (on your handout)

Sketch the graph of f(x). Identify the vertex and x-intercepts.

Set original function =0.

Factor.

Set factors =0.

Solve.

Applications of Quadratic Fns

For a quadratic function in the form ,

the x-coordinate of the vertex is given as ___________

& the y-coordinate of the vertex is given as ________.

Example 5

The height y (in feet) of a ball thrown by a child is

given by , where x is the horizontal

distance (in feet) from where the ball is thrown.

How high is the ball when it is at its maximum

height? a = - 1/8 b = 1 c = 4

Maximum height = vertex

Example (on your handout)

Find the vertex of the parabola defined by