Vertical translation by k units:

$g(x) = f(x) + k$

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All Flashcards

Vertical translation by k units:

$g(x) = f(x) + k$

Horizontal translation by h units:

$g(x) = f(x + h)$

Vertical dilation by a factor of a:

$g(x) = af(x)$

Horizontal dilation by a factor of 1/b:

$g(x) = f(bx)$

Reflection over the x-axis:

$g(x) = -f(x)$

Reflection over the y-axis:

$g(x) = f(-x)$

General form of combined transformations:

$g(x) = a cdot f(bx + h) + k$

How to represent a vertical stretch by a factor of 3?

$g(x) = 3f(x)$

How to represent a horizontal compression by a factor of 2?

$g(x) = f(2x)$

Formula for shifting a function 5 units to the right?

$g(x) = f(x-5)$

What is a vertical translation?

Shifting a function's graph up or down by adding/subtracting a constant.

What is a horizontal translation?

Shifting a function's graph left or right by adding/subtracting a constant to the input.

What is a vertical dilation?

Stretching or shrinking a function's graph vertically by multiplying the function by a constant.

What is a horizontal dilation?

Stretching or shrinking a function's graph horizontally by multiplying the input by a constant.

What is a reflection over the x-axis?

Flipping a function's graph over the x-axis, achieved by multiplying the function by -1.

What is a reflection over the y-axis?

Flipping a function's graph over the y-axis, achieved by replacing x with -x.

Define transformation of a function.

Altering the graph of a function by shifting, stretching, shrinking, or reflecting it.

What does 'dilation' mean in function transformations?

Stretching or compressing the graph of a function either vertically or horizontally.

What is the general form equation for combined transformations?

$g(x) = a cdot f(bx + h) + k$

Define vertical stretch.

A transformation that multiplies all y-values of a function by a factor greater than 1, making the graph taller.

Explain the effect of k in $f(x) + k$ .

Shifts the graph vertically. $k > 0$ moves the graph up, $k < 0$ moves it down.

Explain the effect of h in $f(x + h)$ .

Shifts the graph horizontally. $h > 0$ moves the graph left, $h < 0$ moves it right.

Explain the effect of 'a' in $af(x)$ .

Scales the graph vertically. $|a| > 1$ stretches, $0 < |a| < 1$ shrinks. $a < 0$ reflects over x-axis.

Explain the effect of 'b' in $f(bx)$ .

Scales the graph horizontally. $|b| > 1$ shrinks, $0 < |b| < 1$ stretches. $b < 0$ reflects over y-axis.

Why does horizontal translation appear 'opposite'?

Because $f(x+h)$ evaluates the function at a shifted x-value, requiring a shift in the opposite direction to achieve the same output.

How do transformations affect the domain and range?

Translations shift the domain/range, dilations compress/expand them, and reflections can change the direction of the range.

What is the order of transformations?

Horizontal transformations (shifts and stretches) before vertical transformations.

Explain how a vertical stretch affects the range of a function.

A vertical stretch multiplies the range values by the stretch factor, expanding or compressing the range.

Describe the impact of a negative 'a' value in $g(x) = af(x)$ .

It reflects the graph of $f(x)$ over the x-axis, changing the sign of the y-values.

What happens to the x-intercepts after a vertical stretch?

The x-intercepts remain unchanged because the y-value at these points is zero, and multiplying zero by any factor still results in zero.

All Flashcards

Vertical translation by k units:

$g(x) = f(x) + k$

Horizontal translation by h units:

$g(x) = f(x + h)$

Vertical dilation by a factor of a:

$g(x) = af(x)$

Horizontal dilation by a factor of 1/b:

$g(x) = f(bx)$

Reflection over the x-axis:

$g(x) = -f(x)$

Reflection over the y-axis:

$g(x) = f(-x)$

General form of combined transformations:

$g(x) = a cdot f(bx + h) + k$

How to represent a vertical stretch by a factor of 3?

$g(x) = 3f(x)$

How to represent a horizontal compression by a factor of 2?

$g(x) = f(2x)$

Formula for shifting a function 5 units to the right?

$g(x) = f(x-5)$

What is a vertical translation?

Shifting a function's graph up or down by adding/subtracting a constant.

What is a horizontal translation?

Shifting a function's graph left or right by adding/subtracting a constant to the input.

What is a vertical dilation?

Stretching or shrinking a function's graph vertically by multiplying the function by a constant.

What is a horizontal dilation?

Stretching or shrinking a function's graph horizontally by multiplying the input by a constant.

What is a reflection over the x-axis?

Flipping a function's graph over the x-axis, achieved by multiplying the function by -1.

What is a reflection over the y-axis?

Flipping a function's graph over the y-axis, achieved by replacing x with -x.

Define transformation of a function.

Altering the graph of a function by shifting, stretching, shrinking, or reflecting it.

What does 'dilation' mean in function transformations?

Stretching or compressing the graph of a function either vertically or horizontally.

What is the general form equation for combined transformations?

$g(x) = a cdot f(bx + h) + k$

Define vertical stretch.

A transformation that multiplies all y-values of a function by a factor greater than 1, making the graph taller.

Explain the effect of k in $f(x) + k$ .

Shifts the graph vertically. $k > 0$ moves the graph up, $k < 0$ moves it down.

Explain the effect of h in $f(x + h)$ .

Shifts the graph horizontally. $h > 0$ moves the graph left, $h < 0$ moves it right.

Explain the effect of 'a' in $af(x)$ .

Scales the graph vertically. $|a| > 1$ stretches, $0 < |a| < 1$ shrinks. $a < 0$ reflects over x-axis.

Explain the effect of 'b' in $f(bx)$ .

Scales the graph horizontally. $|b| > 1$ shrinks, $0 < |b| < 1$ stretches. $b < 0$ reflects over y-axis.

Why does horizontal translation appear 'opposite'?

Because $f(x+h)$ evaluates the function at a shifted x-value, requiring a shift in the opposite direction to achieve the same output.

How do transformations affect the domain and range?

Translations shift the domain/range, dilations compress/expand them, and reflections can change the direction of the range.

What is the order of transformations?

Horizontal transformations (shifts and stretches) before vertical transformations.

Explain how a vertical stretch affects the range of a function.

A vertical stretch multiplies the range values by the stretch factor, expanding or compressing the range.

Describe the impact of a negative 'a' value in $g(x) = af(x)$ .

It reflects the graph of $f(x)$ over the x-axis, changing the sign of the y-values.

What happens to the x-intercepts after a vertical stretch?

The x-intercepts remain unchanged because the y-value at these points is zero, and multiplying zero by any factor still results in zero.