5 – 1: Graphing Quadratic Functions (Day 1 ) Objective: CA 10: Students graph quadratic functions and determine the maxima, minima, and zeros of the function.

5 – 1: Graphing Quadratic 5 – 1: Graphing Quadratic FunctionsFunctions(Day 1(Day 1 ) )

Objective:

CA 10: Students graph quadratic functions and determine the maxima, minima, and

zeros of the function.

The graph of a quadratic function is U – shaped and is called a parabola.

The graph of y = x2 and y = - x2 are

shown.

2

-2g x = -x2

f x = x2

2

-2g x = -x2

f x = x2

The origin is at the bottom of the graph y = x2 and the highest point of the graph y

= - x2.

The lowest or highest point on the graph of a quadratic function is called the vertex.

2

-2g x = -x2

f x = x2

The graphs of y = x2 and y = - x2 are symmetric about the y – axis, called the axis of

symmetry.

In general, the axis of symmetry for the graph is the vertical line through the

vertex.

#1. A quadratic function has the form (standard form)

2y ax bx c

where a ≠ 0.

Quadratic functions have 3 forms:

The graph of a quadratic function

2y ax bx c is a parabola with these characteristics:

• The parabola opens up if a > 0 and opens down if a < 0.

• The parabola is wider than the graph of y = x2 if |a| < 1 and narrower than the graph of y = x2 if |a| > 1

2ba

The x-coordinate of the vertex is

The axis of symmetry is the vertical line

2b

xa

(characteristics continued)

2Vertex form: y a x h k

Characteristics of graph:

• The vertex is (h, k)

• The axis of symmetry is x = h

#2.

Intercept form: y a x p x q

Characteristics of graph:

• The x – intercepts are p and q.

• The axis of symmetry is halfway between (p, 0) and (q, 0).

#3.

Example 1: Graphing a Quadratic Function.

22 8 6y x x

1. Coefficients for this function are:

Graph

a = 2 b = -8 c = 6

2. Since a > 0 the parabola opens upward.

3. Find and plot the vertex.

8 8

22 2 2 4

bx

a

The y - coordinate is:

2

2

2 8 6

2 2 8 2 6

8 16 6

2

y x x

y

Vertex = (2, -2)

4. Draw the axis of symmetry x = 2

2

-2

5

Vertex (2, -2)

X=2

Plot two points on one side of the axis of symmetry, such as (1, 0) and (0, 6).

Use symmetry to plot two more points such as (3, 0) and (4, 6).

6

4

2

-2

5

(0, 6) (4, 6)

(3,0)(1, 0)

Vertex (2, -2)

X=2

Draw the parabola through the points.

6

4

2

-2

5

(3, 1)

(4, 6)(0, 6)

(0, 1)

X=2

(2, -2)

f x = 2x2-8x +6

Example 2: Graphing a Quadratic function in Vertex from.

Graph 213 4

2y x

2y a x h k

1

2a graph opens downward

because a < 0.

What we know:

The vertex is (-3, 4)

The A.o.S is x = - 3

(-3,4)

Graphing a Quadratic function in Intercept form.

Graph 2 4y x x y a x p x q

From observation we know the following

1a The parabola opens downward

Intercept Form:

The x – intercepts occur at:

2 0 4 0

2 4

x x

x x

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

The axis of symmetry lies half way between –2 and 4 which is x = 1

Example 4: Write the quadratic function in standard form.

2

2

2

4 9 Given

9 4 36 Multiply using FOIL

5 36 Combine like terms

5 36 Use the distributive property

y x x

x x x

x x

x x

4 9y x x

23 1 8y x

Write the quadratic function in standard form.

2

2

2

2

2

2

3 1 8 Given

3 1 1 8 Expand 1

3 1 8 Multiply using FOIL

3 2 1 8 Combine like terms

3 6 3 8 Use the distributive property

3 6 11 Combine like te

y x

x x x

x x x

x x

x x

x x

rms.

Homework:

Page 253 #17 – 19, #21 – 43 odd

Investigating Parabolas page 249

1. Use a graphing calculator to graph each of these functions in the same viewing

windows:

2 2 2 21, , 2 ,and 3

2y x y x y x y x

2. Repeat Step 1 for these functions:

2 2 2 21, , 2 ,and 3

2y x y x y x y x

3. What are the vertex and the axis of symmetry of the graph of y = ax2?

(0, 0); x = 0

4. Describe the effect of a on the graph of y = ax2?

The graph opens up if a > 0, the graph opens down if a < 0.

213 4

2y x

2y a x h k

By observation we know the following about this function.

1

2a this means that the graph opens

downward because a < 0.

The vertex is (-3, 4).

The axis of symmetry is x = - 3

To graph the function: plot the vertex (-3, 4).

Draw the axis of symmetry x = -3

6

4

2

x=-3

Vertex(-3, 4)

Plot two points to the right such as (-1, 2) and (1, -4).Use the axis of symmetry to plot two points to the left (-5, 2) and (-7, -4 )

4

2

-2

-4

-6

-5

(-7, -4) (1, -4)

(-1, 2)(-5, 2)

Vertex(-3, 4)