2.2.3 Period 5 by Kristen Ritter

Notation

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When a number is added to the y-value, there is a vertical shift, while when a number is added to the x-value, there is a horizontal shift.

you don't need to have the equation; you can manipulate either the x values or the y values

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Notation is where w and y values are changed to move the equation on a graph.

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Notation

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This is different because we are modifying the y-value of the function, so the y value (f(x)) is changed.

Notaition

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When A is negative it has a vertical reflection across the x-axis, when x is negative it has a Horizontal reflection across the y-axis.

f(-x) vs -f(x)

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If the negative is under the radical, then the graph flips over the y-axis. If the negative is outside of the radical, then the graph flips over the x-axis.

-x changes the x values while -f(x) changes the y values

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Notation is the different ways you manipulate x and y through a, h, and k.

These equations change the quadrant the graph will be on and sometimes the direction it faces.

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Notation

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when a number is placed with the x in the parenthesis, it changes the x values on the table. when its placed outside it shifts the y values on the table

Transformation notation is the effect on a function where a changes the width, h changes the x inputs, and k changes the y inputs.

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f(-x) vs f(x)

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Having a negative x values makes for a horizontal shift when the negative is attached to the x so when it looks like f(-x).

When given the function, you don't need the equation. Instead, you can manipulate the table by adding a number to the x value or the y value.

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Depending on what you are trying to multiple, subtract, divide, or add you put it in the correct column to give that property to the correct variable.

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Negative y is a vertical reflection over the x-axis which changes the y intercepts, and negative x is a horizontal reflection over the y axis which changes the x intercepts.

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Even functions stay in their original positions. Odd equations flip. And radical x doesn't follow either of these rules.

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Even/Odd/Neither Functions

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The even functions are called even because the y values stay positive the entire time, they are perfectly symmetrical. The odd functions are called odd because they are partially symmetrical, but not completely. The neither function is not symmetrical at all

notation

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when numbers are added or taken away the is a vertical change and then when there is a change in the x there is a horizontal change

-x is the same as x in the even functions, -f(x) and f(-x) is the same in odd functions, and f(x), f(-x), and -f(x) are all different in the radical function

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when the - is on the "outside" of the function the x becomes opposite, while when the - in on the "inside" the y's become opposite

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f(-x) is a horizontal shift because the x value is being changed, while -f(x) is a vertical shift because the y value is being changed.

f(-x) vs -f(x)

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For f(-x) it is a reflected across the a-axis whereas the -f(x) reflects off the y axis.

f(x) gets altered by an additional variable moving it up/down, left/right, flipping it, or changing the stretch factor

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Even functions of absolute value and quadratic are symmetrical between their x values, the odd functions of cubic and rational are symmetrical between -x and -y, while rational functions don't have any symmetry.

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Even/Odd/Neither

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Even functions are symmetrical about the origin, while an odd function is symmetrical about the y-axis. If it is neither of these conditions, it is classified as neither.

Even/Odd/Neither Functions

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Even functions include y=|x| and y=x^2 where f(x) is equal to f(-x). Both of these can reflect and look the same because they are symetrical.

-f(x) f(-x)

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-f(x) is a vertical change due to the y values change. f(-x) is a horizontal shift due to the change in the x.

Even functions don't matter if you do or don't add the negative with the variable cause its the same. Odd functions it doesn't matter if you have the negative in front of with the variable cause its the same as well.

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Notation

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Notation is where you manipulate the x-values and y-values using 'h,' 'k,' and 'a,' where 'h' is written with the 'x' on a table, and the others with 'y.'

when f(-x) there will be a horizontal reflection by flipping which Xs are positive and which are negative in relation to which Y. -f(x) means that the Ys corresponding to the positive Xs will become negative, causing a vertical reflection

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The even functions f(x)=f(-x), meaning when mirrored over the y they dont move. while the odd function are -f(x)=f(-x), meaning that when mirrored the function switched to opposite quadrants.

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f(-x) vs -f(x)

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A negative 'a' implies a vertical reflection, or a reflection across the x-axis, and a negative 'h' causes a horizontal reflection, across the y-axis.

even functions are when both sides of the graph go in the same direction, and are not affected by f(-x). odd functions are when the sides of the graph go in opposite directions and become reflected across the Y axis when f(-x)

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even/odd/neither functions

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even odd and neither functions are a way to group the parent functions we know based on there reflect ability. even means horizontal reflections don't change the values, odd means vertical reflections don't change the values, neither means neither

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Even functions will always be symmetrical across the x-axis through the origin, odd functions are the same across the y-axis through the origin, and neither function will never be symmetrical.

Even/Odd/Neither Functions

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Even functions, like absolute value and quadratics, are symmetrical, which causes the horizontal flip to be the same as without. Odd functions, like cubic and exponential, have identical vertical and horizontal flips, whereas radicals, or neither functions, have a different representation every time.