Graph each function. Label the vertex and axis of symmetry.

1. y = 5x2

2. y = 4x2 + 1

3. y = -2x2 – 6x + 3

4.2

Vertex is (0,0).

Axis of symmetry is x = 0.

Vertex is (0,1).

Axis of symmetry is x = 0.

Vertex is (-1.5,7.5).

Axis of symmetry is x = -1.5.

In the previous lesson we graphed quadratic functions in Standard

Form: y = ax2 + bx + c, a ≠ 0

Today we will learn how to graph quadratic functions in two more forms:

Vertex Form

Intercept Form

4.2

Vertex Form

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This is another text box to take up room!1. The graph is a parabola

with vertex (h, k).

Graph Vertex Form: y = a (x – h)2 + k

Characteristics of the graph of:

y = a (x – h)2 + k

2. The axis of symmetry is x = h

3. The graph opens up if a > 0 and opens down if a < 0.

(h, k)y = x2 (h, k)

y = a (x – h)2 + k

x = h Axis of symmetry

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GUIDED PRACTICEGraph the function. Label the vertex and axis of symmetry.

2. y = (x + 2)2 – 3

Vertex (h, k) = (– 2, – 3).

Axis of symmetry is x = – 2.

3. y = – (x + 1)2 + 5

Vertex (h, k) = (– 1, 5).

Axis of symmetry is x = – 1.

Opens down a < 0. (-1 < 0)

GUIDED PRACTICE4. f (x) = (x – 3)2 – 412

Vertex (h, k) = ( 3, – 4).

Axis of symmetry is x = 3.

Opens up a > 0. (1/2 < 0)

Intercept FormGraph Intercept Form: y = a (x – p) (x – q)

Characteristics of the graph of y = a (x – p) (x – q):

1. The x – intercepts are p and q, so the points (p, 0) and (q, 0) are on the graph.

2. The axis of symmetry is halfway between (p, 0) and (q, 0) and has equation: x = (p + q) / 2

3. The graph opens up if a > 0 and opens down if a < 0.

(p, 0)

(q, 0)

x = (p + q) / 2

y = a (x – p) (x – q)

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Graph the function.

6. y = (x – 3) (x – 7)

x - intercepts occur at the points (3, 0) and (7, 0).

Axis of symmetry is x = 5

Vertex is (5, – 4).

7. f (x) = 2(x – 4) (x + 1)

x - intercepts occur at the points (4, 0) and (– 1, 0).

Axis of symmetry is x = 3/2.

Vertex is (3/2, 25/2).

GUIDED PRACTICE8. y = – (x + 1) (x – 5)

x - intercepts occur at the points (– 1, 0) and (5, 0).

Vertex is (2, 9).

Axis of symmetry is x = 2.

Changing quadratic functions from intercept form or vertex form to

standard form.

FOIL Method:To multiply two expressions that each contain two terms, add the

products of the First terms, the Outer terms, the Inner terms, and the Last terms.

Example:

F O I L

(x + 4)(x + 7) = x2 + 7x + 4x + 28

= x2 + 11x + 28

EXAMPLE 5Change from intercept form to standard form.

Write y = – 2 (x + 5) (x – 8) in standard form.

y = – 2 (x + 5) (x – 8)

= – 2 (x2 – 8x + 5x – 40)

= – 2 (x2 – 3x – 40)= – 2x2 + 6x + 80

EXAMPLE 6Change from vertex form to standard form.

Write f (x) = 4 (x – 1)2 + 9 in standard form.

f (x) = 4(x – 1)2 + 9

= 4(x – 1) (x – 1) + 9= 4(x2 – x – x + 1) + 9

= 4(x2 – 2x + 1) + 9= 4x2 – 8x + 4 + 9= 4x2 – 8x + 13

GUIDED PRACTICEWrite the quadratic function in standard form.

7. y = – (x – 2) (x – 7)

y = – (x – 2) (x – 7)

= – (x2 – 7x – 2x + 14)

= – (x2 – 9x + 14)

= – x2 + 9x – 14

8. f(x) = – 4(x – 1) (x + 3)

= – 4(x2 + 3x – x – 3)

= – 4(x2 + 2x – 3)

= – 4x2 – 8x + 12

GUIDED PRACTICE

9. y = – 3(x + 5)2 – 1

y = – 3(x + 5)2 – 1

= – 3(x + 5) (x + 5) – 1= – 3(x2 + 5x + 5x + 25) – 1

= – 3(x2 + 10x + 25) – 1= – 3x2 – 30x – 75 – 1= – 3x2 – 30x – 76

Homework: p. 249: 3-52 (EOP)