Essential Question: What is the shape of a quadratic graph?

10-1: Exploring Quadratic GraphsQuadratic Function: a function that can be

written in the form y = ax2 + bx + c. This is called the standard form of a quadratic function.Examples: y = 5x2

Examples: y = x2 + 7Examples: y = x2 – x – 3

The simplest quadratic function is y = x2. This is called the quadratic parent function.

The graph of a quadratic function is aU-shaped curve called a parabola.

10-1: Exploring Quadratic GraphsYou can fold a parabola so that the two sides match

exactly. This property is called symmetry. The fold/line that divides the parabola into two equal parts is called the axis of symmetry.

The highest or lowest point on a parabola is called the vertex, which is always on the axis of symmetry.If a > 0 (is positive) in ax2 + bx + c, the parabola

opens up, and the vertex represents the minimum of the graph.

If a < 0 (is negative) in ax2 + bx + c, the parabola opens down, and the vertex represents the maximum of the graph.

10-1: Exploring Quadratic GraphsExample 1: Identifying a Vertex

Identify the vertex of each graph. Tell whether it is a minimum or maximum

The vertex is at (1, -2) The vertex is at (-2,4)It is a minimum It is a maximum

10-1: Exploring Quadratic GraphsYOUR TURN

Identify the vertex of each graph. Tell whether it is a minimum or maximum

The vertex is at (4, 3)It is a maximum

The vertex is at (-3, -3)It is a minimum

10-1: Exploring Quadratic GraphsNo need to copy

You can use the fact that a parabola is symmetric to graph it quickly. First, find the coordinates of the vertex, then a few points on either side of the vertex. Then reflect the points across the axis of symmetry.

For functions in the form y = ax2, the vertex is at the origin

10-1: Exploring Quadratic GraphsExample 2: Graphing y = ax2

Make a table of values and graph the quadratic functiony = ½ x2

x y = ½ x2 (x, y)

0 ½ (0)2 = 0

(0, 0)

2 ½ (2)2 = 2

(2, 2)

4 ½ (4)2 = 8

(4, 8)

10-1: Exploring Quadratic GraphsYOUR TURN

Make a table of values and graph the quadratic functiony = -2x2

x y = -2x2 (x, y)

0 -2(0)2 = 0

(0, 0)

1 -2(1)2 = -2

(1, -2)

2 -2(2)2 = -8

(2, -8)

1 2 3 4 5–1–2–3–4–5 x

1

–1

–2

–3

–4

–5

–6

–7

–8

–9

y

10-1: Exploring Quadratic GraphsAssignment

Worksheet #10-1Problems 1, 3, 5, 13, 15

Graph each function (don’t follow directions for #1 – 5)

Essential Question: What is the shape of a quadratic graph?

10-1: Exploring Quadratic GraphsThe y-axis is also the axis of symmetry for

functions in the form y = ax2 + c, so like yesterday, start your table of values with x = 0.

Fundamentally, the value of c translates (shifts) the graph up or down.

Example 4: Graph y = 2x2 + 3x y = 2x2 + 3

(x, y)

0 2(0)2 + 3 = 3

(0, 3)

1 2(1)2 + 3 = 5

(1, 5)

2 2(2)2 + 3 = 11

(2, 11)

10-1: Exploring Quadratic GraphsYOUR TURN

Make a table of values and graph the quadratic functiony = x2 – 4x y = x2 – 4 (x, y)

0 (0)2 – 4 = -4

(0, -4)

1 (1)2 – 4 = -3

(1, -3)

2 (2)2 – 4 = 0

(2, 0)

1 2 3 4 5–1–2–3–4–5 x

1

–1

–2

–3

–4

–5

–6

–7

–8

–9

y

10-1: Exploring Quadratic GraphsYou can model the height of an object moving due

to gravity using a quadratic function.As an object falls, its speed continues to increase.

Ignoring air resistance, you can find the approximate height of a falling object using the function

h = -16t2 + cExample 5

Suppose you see an eagle flying over a canyon. The eagle is 30 ft above sea level when it drops a rock. The function y = -16t2 + 30 gives the height of the rock after t seconds.

Graph this quadratic function.

10-1: Exploring Quadratic GraphsExample 5

Suppose you see an eagle flying over a canyon. The eagle is 30 ft above sea level when it drops a rock. The function h = -16t2 + 30 gives the height of the rock after t seconds.

Graph this quadratic function.t h = -16t2 + 30

(t, h)

0 -16(0)2 + 30 = 30

(0, 30)

1 -16(1)2 + 30 = 14

(1, 14)

2 -16(2)2 + 30 = -34

(2, -34)

10-1: Exploring Quadratic GraphsYOUR TURN

Suppose a squirrel is in a tree 24 ft above the ground. She drops an acorn. The function h = -16t2 + 24 gives the height of the acorn after t seconds. Graph this function.

Graph on board

t h = -16t2 + 24

(t, h)

0 -16(0)2 + 24 = 24

(0, 24)

1 -16(1)2 + 24 = 8

(1, 8)

2 -16(2)2 + 24 = -40

(2, -40)

10-1: Exploring Quadratic GraphsAssignment

Worksheet 10-1 (You got it yesterday)Problems 17 – 27, odds