5.1 – GRAPHING QUADRATIC FUNCTIONS (DAY 1)

Algebra 2

Reward for sitting through 5.1 notes rehearsal

Objectives

Graph quadratic functions in standard form

Relate quadratic functions to real-life problems

Quadratic function: A two-variable equation that has its greatest exponent on a variable raised to the 2nd power

Form:2y ax bx c

Note: the variable “a” cannot equal 0. Why?

What is a quadratic function?

Vocabulary

Parabola: a U-shaped graph of a quadratic function

Vertex: the highest or lowest point of a parabola

Axis of symmetry: the vertical line through the vertex of a parabola

Vocabulary

axis of symmetry

parabola

vertex

If an equation is in standard form…

1. Label a, b, and c2. Find the vertex

a) Use for the x-coordinateb) Plug x in to find the y-coordinate

3. Pick the next two integers greater than the x value of your vertex and find the corresponding y-values

4. Plot the ordered pairs5. Use symmetry to plot two “mirror image”

ordered pairs

2b

xa

Example 1

22 8 6y x x

Vertex?

Axis of symmetry?

(2, -2)

x = 2

Example 2

2 6 11y x x

What is different about this graph compared to the others we’ve seen?

Vertex?

Axis of symmetry?

(-3, -2)

x = -3

Example 3

215 4

2y x x

Vertex?

Axis of symmetry?

(-4, -3)

x = -4

Exit Slip

22 4 2y x x

1. Graph the following quadratic function. Identify the vertex and the axis of symmetry.

2. Rate your understanding of today’s lesson on a scale 1-5.

(1 = I’m lost! 3 = I’m okay. 5 = This is easy!)

Vertex?

Axis of symmetry?

(1, -4)

x = 1

Homework/Reminders

Due on Monday:pg. 253 #20-23 (make nice, accurate graphs)

Quiz 5.1-5.3 on Tuesday, November 20th

5.1 – GRAPHING QUADRATIC FUNCTIONS (DAY 2)

Algebra 2

Quadratic function

Objectives

Graph quadratic functions in vertex form

Graph quadratic functions in intercept form

Use quadratic functions to solve real-life problems

Vertex Form

An equation in vertex form is written as:

2( )y a x h k

Does this form look similar to anything we’ve done in the past?

Discovery Activity

Using your graphing calculator, find the vertex of the following functions. Try to identify a pattern.

1.y = 2(x – 1)2 + 6

2.y = -0.5(x – 2)2 – 7

3.y = 3(x + 3)2 – 1

4.y = 0.25(x – 1)2

5.y = 2x2 + 6

(1, 6)

(2, -7)

(-3, -1)

(1, 0)

(0, 6)

Vertex Form

Equations in vertex form highlight the vertex of a quadratic equation.

2( )y a x h k

Vertex: (h, k)

If an equation is in vertex form…

1. Identify the vertex, (h, k), and plot the ordered pair

2. Pick the next two integers greater than the x value of your vertex and find the corresponding y-values

3. Plot the ordered pairs4. Use symmetry to plot two “mirror image”

ordered pairs

Example 1

22( 2) 6y x

Vertex?

Axis of symmetry?

(2, 6)

x = 2

Example 2

21( 3) 4

2y x

Vertex?

Axis of symmetry?

(-3, 4)

x = -3

Intercept Form

An equation in intercept form is written as:

( )( )y a x p x q

Examples:• y = 2(x – 2)(x – 6)

• y = 4(x + 1)(x – 1)

• y = -(x + 8)(x + 7)

How can these be quadratic functions if there is no exponent of 2 on a variable?

Intercept Form Standard Form

Use FOIL to convert the quadratic equation in intercept form into a quadratic equation in standard form.

• y = (x – 2)(x – 6)

• y = 4(x + 2)(x – 1)

• y = -(x + 4)(x + 3)

y = x2 – 8x + 12

y = 4x2 + 4x – 8

y = -x2 – 7x – 12

Discovery Activity

Using your graphing calculator, find the intercepts of the following functions. Try to identify a pattern.

1.y = 2(x – 1)(x + 2)

2.y = -0.5(x – 2)(x + 4)

3.y = .25(x + 3)(x – 8)

(1, 0) and (-2, 0)

(2, 0) and (-4, 0)

(-3, 0) and (8, 0)

Intercept Form

Equations in vertex form highlight the x-intercepts of a quadratic equation.

( )( )y a x p x q

x-intercepts: (p, 0) and (q, 0)

If an equation is in intercept form…

1. Find the x-intercepts (set each factor to zero and solve for x) and plot

2. Take the average of the x-intercepts to find the h of the vertex, and then find k, and plot.

3. Pick the next integer greater than the x value of your x-intercept and find the corresponding y-value

4. Plot the ordered pair5. Use symmetry to plot the “mirror image”

ordered pair

Example 3

( 2)( 4)y x x

x-intercepts?

vertex?

axis of symmetry?

(-2, 0) and (4, 0)

(1, 9)

x = 1

Example 4

2(2 4)( 1)y x x

x-intercepts?

vertex?

axis of symmetry?

(2, 0) and (-1, 0)

(0.5, -9)

x = 0.5

Homework/Reminders

Due tomorrow:pgs. 253-254 #28, 30, 32, 36, 40, 42 (make nice, accurate graphs)

Quiz 5.1-5.3 on Tuesday, November 20th

Mechanical pencil