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.

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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.

Given $f(x) = x^2$ , find $g(x)$ after shifting $f(x)$ right 2 units and up 3 units.

Horizontal shift: $f(x-2) = (x-2)^2$ . 2. Vertical shift: $g(x) = (x-2)^2 + 3$ .

Given $f(x) = |x|$ , find $g(x)$ after reflecting $f(x)$ over the x-axis and stretching vertically by 2.

Reflection: $-f(x) = -|x|$ . 2. Vertical stretch: $g(x) = -2|x|$ .

How to find the equation after reflecting over the y-axis and shifting left by 1?

Reflection: $f(-x)$ . 2. Horizontal shift: $g(x) = f(-(x+1)) = f(-x-1)$ .

Describe the steps to transform $f(x)$ to $2f(x-1) + 3$ .

Shift right 1 unit: $f(x-1)$ . 2. Vertical stretch by 2: $2f(x-1)$ . 3. Shift up 3 units: $2f(x-1) + 3$ .

How do you determine the transformations from $f(x)$ to $g(x) = f(2x) - 1$ ?

Horizontal compression by a factor of 2. 2. Vertical shift down by 1 unit.

If $f(x) = sqrt{x}$ , what is the equation after a horizontal stretch by 3?

$g(x) = \sqrt{\frac{1}{3}x}$

Given $f(x) = x^3$ , find the equation after reflection over the x-axis and shift down by 4.

Reflection: $-f(x) = -x^3$ . 2. Vertical shift: $g(x) = -x^3 - 4$ .

How to transform $f(x)$ to $g(x) = -f(x + 2) - 1$ ?

Shift left 2 units: $f(x+2)$ . 2. Reflect over x-axis: $-f(x+2)$ . 3. Shift down 1 unit: $-f(x+2) - 1$ .

Describe the effect of $g(x) = f(-x) + 2$ on the graph of $f(x)$ .

Reflect over the y-axis. 2. Shift up by 2 units.

If $f(x) = |x|$ , find the equation after a vertical compression by a factor of 0.5 and a shift up by 2 units.

Vertical compression: $0.5|x|$ . 2. Vertical shift: $g(x) = 0.5|x| + 2$ .

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.