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Understanding Transformations of Functions: A Guide to Graphing and Vertex Movements

This guide explores the concept of function transformations in algebra, focusing on how the vertex of various parent functions can move on the graph. The original vertex starts at (0, 0), and transformations can shift it left, right, up, or down. It covers key types of functions including quadratic, constant, absolute value, radical, and linear. Examples illustrate how vertex movements are determined by changes inside parentheses or outside value expressions. Learn to identify graph shapes and their transformations for effective algebra practice.

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Understanding Transformations of Functions: A Guide to Graphing and Vertex Movements

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  1. Function Transformations Notes from Adv Algebra Week of Sept 24, 2012

  2. Transformation(s) This means the graphed line can move on the graph. The vertex can move up or down or left or right. The original (or parent function) vertex was at 0,0 (the blue dot). The next function was transformed (moved) and now the vertex is changed (the red dot). It was moved to the left 3 and up 3.

  3. Transformations You will need to know the names and shapes of the transformations. 1. Quadratic - it is U shaped and the equation is y = x2 (parabola)

  4. Transformations 2. Constant: It is a horizontal line. (think horizon = constant) y = Here y = 3

  5. Transformations Absolute Value - these graphs are V shaped. Y = | x | Think absolute value = v –shaped.

  6. Transformations 4. Radical - shaped like a curve or half of a U y = √x

  7. Transformations 5. Linear - line y = mx + b

  8. OK, now the graphed vertices move….. Anytime you have an x and a number INSIDE parentheses, under a radicand or in-between absolute value marks, your vertex has moved the OPPOSITE way. Ex. y = ( x + 3)2 means your X has moved to the LEFT 3 spaces Yes, adding means go to the left or……………………… left = + (the little T looks like a plus sign) Ex. Y = (x -- 5)2 means your X has moved to the RIGHT 5 spaces right = minus (they both have that long i sound) By the way, what kind of graph is this? Hint: squared……

  9. OK, now the graphed vertices move….. It is a quadratic graph and the line will be shaped like a U Once you figure out which directions your X goes in (PLUS = LEFT, MINUS = RIGHT) Look at what is outside of the parentheses, radicand, absolute value marks Ex y = |x + 2 | + 5 There is a plus 5. On the OUTSIDE PLUS = UP MINUS = DOWN (like an elevator) So this last example means our vertex moved to the left 2 spaces and up 5 spaces.

  10. It may look something like this The original vertex is the blue one, the transformed one is the red one

  11. More examples Y = √x + 1 - 2 Bear with me, the radicand is supposed to be over the x + 1 Your original vertex was at 0, 0 how far to the left or right did it move? ( + 1, plus moves LEFT 1 space) How far up or down did it move? (--2, minus means DOWN 2) What kind of graph is this? Radical so your line is shaped like a half a U or a curve Where is your second vertex? (--2, 1)

  12. More examples Y = | x + 7 | -- 2 Your original vertex was at 0,0 How far left or right did it move? (+ 7….PLUS means LEFT…it moved left 7 spaces How far up or down did it move (--2, MINUS means DOWN, it moved down 2 spaces What shape is your line? Absolute value = v-shaped

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