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Wednesday, February 18 th. Warm Up. Fill in the Blank A quadratic functions makes a ____________when you graph it. Two important features of a quadratic functions are _____________and the __________. Week 6 #3. Homework Answers. Unit 2. Quadratic Functions. Today’s Goal. Graphing

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Warm up

Wednesday, February 18th

Warm Up

Fill in the Blank

A quadratic functions makes a ____________when you graph it.

Two important features of a quadratic functions are _____________and the __________.



Homework

Answers


Unit 2

Quadratic

Functions


Today’s Goal

Graphing

With

Standard Form


Standard form equation
Standard Form Equation

y=ax2 + bx + c

  • If a is positive, u opens up

    If a is negative, u opens down

  • The x-coordinate of the vertex is at

  • To find the y-coordinate of the vertex, plug the x-coordinate into the given eqn.

  • The axis of symmetry is the vertical line x=

  • Choose 2 x-values on either side of the vertex x-coordinate. Use the eqn to find the corresponding y-values.

  • Graph and label at least 3 points and axis of symmetry on a coordinate plane. Connect the points with a smooth curve.


Standard form

y

a > 0

x

a < 0

Standard Form

The standard form of a quadratic function is

y = ax2 + bx + c

The parabola will open up when the a value is positive.

The parabola will open down when the a value is negative.


Line of symmetry

Line of Symmetry

y

x

Line of Symmetry

Parabolas have a symmetric property to them.

If we drew a line down the middle of the parabola, we could fold the parabola in half.

We call this line the line of symmetry.

Or, if we graphed one side of the parabola, we could “fold” (or REFLECT) it over, the line of symmetry to graph the other side.

The line of symmetry ALWAYS passes through the vertex.


Finding the line of symmetry
Finding the Line of Symmetry

When a quadratic function is in standard form

For example…

Find the line of symmetry of y = 3x2 – 18x + 7

y = ax2 + bx + c,

The equation of the line of symmetry is

Using the formula…

This is best read as …

the opposite of b divided by the quantity of 2 times a.

Thus, the line of symmetry is x = 3.


Finding the vertex
Finding the Vertex

y = –2x2 + 8x –3

We know the line of symmetry always goes through the vertex.

STEP 1: Find the line of symmetry

Thus, the line of symmetry gives us the x – coordinate of the vertex.

STEP 2: Plug the x – value into the original equation to find the y value.

To find the y – coordinate of the vertex, we need to plug the x – value into the original equation.

y = –2(2)2 + 8(2) –3

y = –2(4)+ 8(2) –3

y = –8+ 16 –3

y = 5

Therefore, the vertex is (2 , 5)


A quadratic function in standard form

USE the equation

A Quadratic Function in Standard Form

The standard form of a quadratic function is given by

y = ax2 + bx + c

There are 3 steps to graphing a parabola in standard form.

Plug in the line of symmetry (x – value) to obtain the y – value of the vertex.

STEP 1: Find the line of symmetry

MAKE A TABLE

using x – values close to the line of symmetry.

STEP 2: Find the vertex

STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve.


y

x

A Quadratic Function in Standard Form

Let's Graph ONE! Try …

y = 2x2 – 4x – 1

STEP 1: Find the line of symmetry

Thus the line of symmetry is x = 1


y

x

A Quadratic Function in Standard Form

Let's Graph ONE! Try …

y = 2x2 – 4x – 1

STEP 2: Find the vertex

Since the x – value of the vertex is given by the line of symmetry, we need to plug in x = 1 to find the y – value of the vertex.

Thus the vertex is (1 ,–3).


y

x

y

x

2

3

A Quadratic Function in Standard Form

Let's Graph ONE! Try …

y = 2x2 – 4x – 1

STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve.

–1

5


Example 1 graph y 2x 2 8x 6
Example 1: Graph y=2x2-8x+6

  • Axis of symmetry is the vertical line x=2

  • Table of values for other points: x y

  • 0 6

  • 1 0

  • 2 -2

  • 3 0

  • 4 6

  • * Graph!

  • a=2 Since a is positive the parabola will open up.

  • Vertex: use

    b=-8 and a=2

    Vertex is: (2,-2)



Now you try one y x 2 x 12 open up or down vertex axis of symmetry table of values with 5 points
Now you try one!y= -x2+x+12* Open up or down?* Vertex?* Axis of symmetry?* Table of values with 5 points?


(.5,12)

(-1,10)

(2,10)

(-2,6)

(3,6)

X = .5


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