Graph each function. Compare the graph with the graph of y = x 2 then label the vertex and axis of symmetry. 1.y = 5x 2 2.y = 4x y = -2x 2 – 6x 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 = ax 2 + bx + c, a ≠ 0 Today we will learn how to graph quadratic functions in two more forms: Vertex Form Intercept Form 4.2

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EXAMPLE 1 Graph a quadratic function in vertex form Graph y = – (x + 2) a = – (1/4) h = – 2 k = 5 a < 0, so the parabola opens down. 1.Identify the vertex: (h, k) = ( – 2, 5) The axis of symmetry x = – 2. Vertex Form: y = a ( x – h) 2 + k 2. Make a Table: xy Graph:

GUIDED PRACTICEGraph the function. Label the vertex and axis of symmetry. 1. y = (x + 2) 2 – 3 Vertex (h, k) = ( – 2, – 3). Axis of symmetry is x = – y = – (x + 1) Vertex (h, k) = ( – 1, 5). Axis of symmetry is x = – 1.

GUIDED PRACTICE 3. f (x) = (x – 3) 2 – Vertex (h, k) = (3, – 4). Axis of symmetry is x = 3.

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EXAMPLE 3 Graph a quadratic function in intercept form Graph y = 2(x + 3) (x – 1). 1. x - intercepts occur at the points (– 3, 0) and (1, 0). 2. Find the vertex: x = p + q 2 – = – 1= y = 2(– 1 + 3)(– 1 – 1) = – 8 a = 2 p = -3 q = 1 y = a (x - p) (x – q) Vertex is (-1, -8) 3. Graph using vertex and x- intercepts: x = -1 is also the axis of symmetry

Graph the function. Label the vertex, axis of symmetry, and x - intercepts. 4. 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). 5. 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 PRACTICE 6. 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) = x 2 + 7x + 4x + 28 = x x + 28

EXAMPLE 5 Change from intercept form to standard form. Write y = – 2 (x + 5) (x – 8) in standard form. y = – 2 (x + 5) (x – 8) Write original function. = – 2 (x 2 – 8x + 5x – 40) Multiply using FOIL. = – 2 (x 2 – 3x – 40) Combine like terms. = – 2x 2 + 6x + 80 Distributive property

EXAMPLE 6 Change from vertex form to standard form. Write f (x) = 4 (x – 1) in standard form. f (x) = 4(x – 1) Write original function. = 4(x – 1) (x – 1) + 9 = 4(x 2 – x – x + 1) + 9 Multiply using FOIL. Rewrite (x – 1) 2. = 4(x 2 – 2x + 1) + 9 Combine like terms. = 4x 2 – 8x Distributive property = 4x 2 – 8x + 13 Combine like terms.

GUIDED PRACTICE Write the quadratic function in standard form. 7. y = – (x – 2) (x – 7) y = – (x – 2) (x – 7) = – (x 2 – 7x – 2x + 14) = – (x 2 – 9x + 14) = – x 2 + 9x – f(x) = – 4(x – 1) (x + 3) = – 4(x 2 + 3x – x – 3) = – 4(x 2 + 2x – 3) = – 4x 2 – 8x + 12

GUIDED PRACTICE 9. y = – 3(x + 5) 2 – 1 y = – 3(x + 5) 2 – 1 = – 3(x + 5) (x + 5) – 1 = – 3(x 2 + 5x + 5x + 25) – 1 = – 3(x x + 25) – 1 = – 3x 2 – 30x – 75 – 1 = – 3x 2 – 30x – 76 Homework: p. 249: 3-52 (EOP)

Graphing y=x 2 StandardInterceptVertex y = ax 2 +bx+c y = a (x - p) (x - q) y= a (x - h) 2 +k axis: axis: x = h vertex ( x, y ) vertex ( h, k ) C is y-intercept x-intercept = p, q a>0 U shaped, Minimum a<0 ∩ shaped, Maximum