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  2. AP Pre Calculus
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What is a vertical translation?

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

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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)=acdotf(bx+h)+kg(x) = a cdot f(bx + h) + kg(x)=acdotf(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.

Given f(x)=x2f(x) = x^2f(x)=x2, find g(x)g(x)g(x) after shifting f(x)f(x)f(x) right 2 units and up 3 units.

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

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

  1. Reflection: −f(x)=−∣x∣-f(x) = -|x|−f(x)=−∣x∣. 2. Vertical stretch: g(x)=−2∣x∣g(x) = -2|x|g(x)=−2∣x∣.

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

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

Describe the steps to transform f(x)f(x)f(x) to 2f(x−1)+32f(x-1) + 32f(x−1)+3.

  1. Shift right 1 unit: f(x−1)f(x-1)f(x−1). 2. Vertical stretch by 2: 2f(x−1)2f(x-1)2f(x−1). 3. Shift up 3 units: 2f(x−1)+32f(x-1) + 32f(x−1)+3.

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

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

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

g(x)=13xg(x) = \sqrt{\frac{1}{3}x}g(x)=31​x​

Given f(x)=x3f(x) = x^3f(x)=x3, find the equation after reflection over the x-axis and shift down by 4.

  1. Reflection: −f(x)=−x3-f(x) = -x^3−f(x)=−x3. 2. Vertical shift: g(x)=−x3−4g(x) = -x^3 - 4g(x)=−x3−4.

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

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

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

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

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

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

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)y = f(-x)y=f(−x) look like compared to y=f(x)y = f(x)y=f(x)?

It's a reflection of y=f(x)y = f(x)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.