Daily Check Give the transformations for each of the following functions? f(x) = (x - 2)2 + 4 f(x) = -3x2 f(x) = ½ (x+3)2 Write the equation in vertex.

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Presentation transcript:

Daily Check Give the transformations for each of the following functions? f(x) = (x - 2)2 + 4 f(x) = -3x2 f(x) = ½ (x+3)2 Write the equation in vertex form for the following graph.

Warm-up Multiply out each of the following functions. y = (x – 1)2 + 8 This is how you convert from vertex form to standard form.

a (+ up; – down) Vertex Form Vertex: (h, k) Axis: x = h Rate: Summary of Day One Findings Parabolas Vertex Form Vertex: (h, k) Axis: x = h Rate: a (+ up; – down)

COMPLETING THE SQUARE • What's the pattern? (x + 6)2 x2 + 12x + 36 • How about these? x2 + 4x ______ (x _____ )2 + 4 + 2 x2 + 10x ______ (x _____ )2 + 25 + 5 x2 – 14x ______ (x _____ )2 + 49 – 7

COMPLETING THE SQUARE • Converting from standard form to vertex form can be easy… x2 + 6x + 9 (x + 3)2 x2 – 2x + 1 = (x – 1)2 x2 + 8x + 16 = (x + 4)2 x2 + 20x + 100 = (x + 10)2 … but we're not always so lucky

COMPLETING THE SQUARE • The following equation requires a bit of work to get it into vertex form. y = x2 + 8x + 10 y = (x2 + 8x ) + 10 + 16 – 16 16 is added to complete the square. 16 is sub-tracted to maintain the balance of the equation. y = (x + 4)2 – 6 The vertex of this parabola is located at ( –4, –6 ).

COMPLETING THE SQUARE • Lets do another. This time the x2 term is negative. y = –x2 + 12x – 5 Un-distribute a negative so that when can complete the square y = (–x2 + 12x ) – 5 y = –(x2 – 12x ) – 5 y = –(x2 – 12x ) – 5 + 36 + 36 The 36 in parentheses becomes negative so we must add 36 to keep the equation balanced. y = – (x – 6)2 + 31 The vertex of this parabola is located at ( 6, 31 ).

COMPLETING THE SQUARE Find the value to add to the trinomial to create a perfect square trinomial: (Half of “b”)2 [A] [B] [C] [D]

Example 1 Type 1: a = 1 Write in vertex form. Identify the vertex and axis of symmetry. [B] [A]

Example 2 Type 1: a≠1 Write in standard form. Identify the vertex and axis of symmetry. [B] [A]

Method #2: SHORTCUT Find the AXIS of SYMMETRY : Find VERTEX (h, k) h = x k is found by substituting “x” “a” – value for vertex form should be the same coefficient of x2 in standard form. Check by using another point (intercept)

Method #2 Example Given f(x) = x2 + 8x + 10 1) Find a, b, and c. 2) Find the line of symmetry or “h” using x = -b/2a 3) Find the y value of the vertex, or “k” by substituting “x” into the equation. So, the vertex is at (-4, 6). 4) Write the equation in vertex form using the “h” and “k” found. “a” will be the same thing as in Step 1.

Write in vertex form. Find vertex and axis of symmetry. PRACTICE METHOD #2: Write in vertex form. Find vertex and axis of symmetry. [1] [2]

Write in vertex form. Find vertex and axis of symmetry. PRACTICE METHOD #2: Write in vertex form. Find vertex and axis of symmetry. [3] [4]