Graphing Graphing Quadratic FunctionsQuadratic Functions

IntroductionIntroduction LessonLesson AssessmentAssessment

IntroductionIntroduction

• Subject: Pre-CalculusSubject: Pre-Calculus

• Grade Level: 12Grade Level: 12thth grade grade

ObjectivesObjectives• The students will be able to calculate the The students will be able to calculate the

vertex of any quadratic functionvertex of any quadratic function

• The students will be able to label the axis of The students will be able to label the axis of symmetry of any given quadratic functionsymmetry of any given quadratic function

• The students will be able to sketch the The students will be able to sketch the graphs of simple quadratic functions given graphs of simple quadratic functions given in the form of f(x)=axin the form of f(x)=ax22+bx+c+bx+c

User GuideUser Guide• Click the home button to go to the title slideClick the home button to go to the title slide

• Click the forward or backward arrow button at Click the forward or backward arrow button at the bottom of each slide to respectively move to the bottom of each slide to respectively move to the next slide or go back to the previous slidethe next slide or go back to the previous slide

• After going through the lesson, try the After going through the lesson, try the assessment to see how well you understand the assessment to see how well you understand the materialmaterial

LessonLessonA quadratic function is written in the form

f (x) = ax2 + bx + cwhere a, b, and c are constants (number coefficients)

Examples:

f (x) = 3x2 + 4 f (x) = ¼x – 2x2 – 7We can say that We can say that

a= 3, b= 0, and c= 4 a= -2, b= ¼, and c= -7

The Basic GraphThe Basic GraphThe graph of any quadratic equation has the shape of a tall arch. We call this shape a PARABOLA.

A parabola has one

“hump” and its two sides

will extend outward

forever – it never stops

going.

Where does it go?Where does it go?- A quadratic function opens UPWARD if the “a” value is a positive number.

- A quadratic function opens DOWNWARD if the “a” value is a negative number.

+ a - a

The Vertex of a ParabolaThe Vertex of a ParabolaThe vertex of a parabola is the point

(x,y) on the graph located at the top (maximum) or the bottom (minimum) of the arch:

Vertex

(maximum)

Vertex

(minimum)

Finding the VertexFinding the VertexThe x-coordinate of the vertex

(x,y) can be calculated as:

X = -b/(2a)

The y-coordinate of the vertex can be found by solving:

y = f (-b/2a)

For example, f (x) = x2 – 4x + 1

Since a=1, b= -4, and c=1:

X = - (- 4) / 2(1) = 4/2 = 22

To find y = f (2), plug in (2) everywhere there is an x in f (x) and simplify:

f(2) = (2)2 – 4(2) + 1

= 4 – 8 + 1 = - 3- 3

Therefore, the vertex of f(x) is the point (2, -3)(2, -3)

Are you ready?Are you ready?You have completed the lesson on graphing

quadratic functions…

Think you can handle the quiz??

Um, I think I need to review some moreAbsolutely!

AssessmentAssessment1.1. List the a, b, and c values in order List the a, b, and c values in order

for the given quadratic function:for the given quadratic function:

f(x) = ½x -3x²f(x) = ½x -3x²

½, -3, 0

-3, ½, 1

3, ½, 2

-3, ½, 0

Good Job!Good Job! You remembered the general formula for quadratic functions:

f (x) = ax2 + bx + c

Next question…

Hmm… not quite!Hmm… not quite!

Hint: Think of the general formula for all quadratic functions

Try again…

Here’s another question…Here’s another question…2. In which direction does the graph of 2. In which direction does the graph of

f(x) = 3 – 2x² – 5x keep going on forever?f(x) = 3 – 2x² – 5x keep going on forever?

UP

DOWNRIGHT

LEFT

Good Job!Good Job! You remembered to look at

the sign of the “a” value

And you knew that a negative “a” means the graph opens, or

goes on forever, in the downward direction

Let’s try a harder one…

Nope, sorry!Nope, sorry!

Try again…

Hint: Do you remember what the sign of the “a”

value means?

Ok, last one…Ok, last one…3. What are the coordinates of the vertex of 3. What are the coordinates of the vertex of

this quadratic function: f(x) = x² + 4x + 5this quadratic function: f(x) = x² + 4x + 5

(2, 17)

(-2, 1)

(-4, 5)

(-8, 37)

Good Good Job!Job!

References

You remembered that the vertex (x,y) is given by

X = -b/2a and y = f (-b/2a)

You really seem to understand the basic graph of a quadratic function. I’m impressed!

Think hard, you can do it!Hint: Think of the formula for finding the point (x,y) that gives you the vertex of a parabola

Try again…

ReferencesReferences

• Google images

• Math GV graphing program

• Graphing quadratic functions website