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

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

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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.

How does a vertical stretch affect the steepness of a graph?

It increases the steepness if the stretch factor is greater than 1 and decreases it if the factor is between 0 and 1.

How does a horizontal shift affect the x-intercepts of a graph?

It shifts the x-intercepts by the same amount as the horizontal shift. A shift to the right increases the x-intercept values, and a shift to the left decreases them.

What does a reflection over the x-axis do to the y-values on a graph?

It changes the sign of all y-values, flipping the graph over the x-axis.

How does a horizontal compression affect the period of a periodic function?

It decreases the period by the compression factor, making the function oscillate more rapidly.

What does the graph of $y = f(-x)$ look like compared to $y = f(x)$ ?

It's a reflection of $y = f(x)$ over the y-axis.

How can you identify a vertical translation from a graph?

The entire graph is shifted up or down without changing its shape.

What happens to the vertex of a parabola after a horizontal shift?

The x-coordinate of the vertex shifts by the same amount as the horizontal shift.

How does a vertical compression affect the maximum and minimum values of a function?

It reduces the maximum and minimum values by the compression factor.

What does a reflection over the y-axis do to a function's symmetry?

If the original function was even (symmetric about the y-axis), the reflection doesn't change the graph. If it was odd (symmetric about the origin), the reflection changes the sign of the function.

How does a horizontal stretch affect the domain of a function?

It expands the domain by the stretch factor.

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?

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