5.1: Graphing Quadratic Functions

Objectives:Students will be able to:• Graph a quadratic function in

standard, vertex, and intercept form. • Write a quadratic function in standard

form• Find the vertex of a quadratic function

on a graphing calculator

QUADRATIC FUNCTIONS!!!!!

STANDARD FORM: y = ax2 + bx +c

The graph is called a parabola

The lowest or highest point of the graph is called the vertex

The graph is symmetric about a vertical line through the vertex called the Axis of Symmetry

Examples of GraphsAxis of Symmetry

Vertex

Axis of Symmetry

Vertex

Standard Form: y=ax2+ bx +cCharacteristics of the graph:If a > 0, opens up, vertex is a minimum pointIf a < 0, opens down, vertex is a maximum point

If |a| < 1, parabola widensIf |a| > 1, parabola narrows

x coordinate at the vertex is found by:

Equation for Axis of Symmetry:

a

b

2

a

bx

2

How to Graph a Quadratic in Standard Form

1. Notice value of a (does it open up, down, narrow, wide??)

2. Find the Axis of Symmetry (remember, this also gives you the x coordinate at the vertex!!):

3. Find coordinates of vertex:

4. Graph vertex and Axis of Symmetry.

5. Pick x values and evaluate function. This gives you extra points. Graph its reflection on the other side of AOS, and draw a smooth curve through points!

a

bx

2

a

bf

a

b

2,

2

Graph the following:

62

1.4

1062.3

12.2

34.1

2

2

2

2

xxy

xxy

xxy

xxy

Graph the following on the same

coordinate plane. 1. y= x2 2. y = x2

y = (x+1)2 y = x2 +1

y = (x-1)2 y = x2 -1

3. y= x2 2. y = x2

y = 2x2 y = (x+2)2 -3

y = -2x2 y = (x- 2)2 -3

Vertex Form y= a (x-h)2 + k• Effect of a is the same

• Vertex: (h, k) (h is always opposite sign)

• Axis of Symmetry: x = h

• h describes horizontal translation of parent function, k describes vertical translation of parent function

To Graph in Vertex Form:

1. Identify vertex (h, k) and Axis of Symmetry. Graph.

2. Pick x values to evaluate in function. Be careful of order of operations!!

3. Graph points from step 2 and their reflections in the Axis of Symmetry.

4. Sketch a smooooooth curve

Examples: Graph.

25.3

322

1.2

312.1

2

2

2

xy

xy

xy

Intercept Form y= a (x- p)(x- q)• Effects of a are the same

• x-intercepts are p and q (opposite signs)

• Axis of Symmetry is halfway between (p, 0) and

(q, 0)

To Graph in Intercept Form1. Identify the intercepts of the graph.

2. Find the Axis of Symmetry:

3. Use the Axis of Symmetry to find y coordinate at the vertex.

2

qp

Examples: Graph

)2)(4(.2

)1)(3(2.1

xxy

xxy

Write the quadratic function in

Standard Form. • Use Algebra and Order of Operations!!!

)5)(1(3.2

322

1.1 2

xxy

xy

The Vertex of a Parabola

…..It is a powerful point!!!!!

It represents the maximum and minimum value of the function.

The y coordinate at the vertex tells you the max or min value

The x coordinate at the vertex tells you where the max or min value occurs.

Example:The function h(t)= -16t2 +48t + 96 represents the height of your calculator, in feet, as you throw it off a 96 ft. cliff at time t, in seconds.

When does it reach the maximum height?

What is the maximum height that the calculator reaches?

Let’s Graph on your CalculatorEnter function in Y1. (make sure to use an appropriate window).

To find the vertex of your graph:1. 2nd Trace2. Calculate: 3: Minimum or 4: Maximum3. Use your left and right arrows to move cursor to

the left bound of the vertex. Hit enter. 4. Use your left and right arrow to move cursor to

the right bound of vertex. Hit enter. 5. Hit enter.

Example:Suppose that a group of high school students conducted an experiment to determine the number of hours of study that leads to the highest score on a comprehensive year end exam. The exam score y for each student who studied x hours can be modeled by

y= -0.853x2 +17.48x +6.923

Which amount of studying produced the highest score on the exam? What is the highest score?

Example:The path of a ball thrown by a baseball player forms a parabola with equation

where x is the horizontal distance in feet of the ball from the player and y is the height in feet of the ball.

a.) How far does the ball travel before it again reaches the same height from which it was thrown?

b.) How high was the ball at its highest point?

5.8492401

3 2 xy

Example:The archway that forms the ceiling of a tunnel can be modeled by the equation y= -0.0355x2 +0.923x +10where x is the horizontal distance in feet from one wall of the tunnel to the other and y is the height in feet of the ceiling from the floor of the tunnel. How many feet from the walls of the tunnel does the ceiling reach its max height? What is the max height of the tunnel?