Curve Shifters Worksheet
Curve Shifters
Rational Function Transformations Lab
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Objective: Analyze the transformations applied to the parent function \( f(x) = \frac{1}{x} \) to create the new function \( g(x) = \frac{a}{x-h} + k \). Identify shifts, stretches, and reflections.
The function \( f(x) = \frac{1}{x} \) is transformed to \( g(x) = \frac{7}{x+3} - 1.5 \). Describe the transformations.
Vertical stretch by 7, shifted left 3 units and down 1.5 units
Vertical stretch by 7, shifted right 3 units and down 1.5 units
Vertical compression by 7, shifted left 3 units and up 1.5 units
Vertical stretch by 7, shifted left 3 units and up 1.5 units
Given \( g(x) = \frac{1}{x-4} + 2 \), identify the horizontal and vertical shifts from the parent function.
The parent function is reflected across the x-axis and shifted left 5 units. Write the new equation \( g(x) \).
Identify the vertical stretch and vertical shift for the function \( g(x) = \frac{3}{x} - 1 \).
Curve Shifters Analysis Lab Continued
Which function represents \( f(x) = \frac{1}{x} \) shifted right 2 units and up 4 units?
\( g(x) = \frac{1}{x+2} + 4 \)
\( g(x) = \frac{1}{x-2} + 4 \)
\( g(x) = \frac{1}{x-2} - 4 \)
\( g(x) = \frac{1}{x+2} - 4 \)
Describe all transformations applied to create \( g(x) = \frac{-2}{x+1} - 3 \).
A rational function is compressed vertically by a factor of 0.5 and shifted down 9 units. Write the equation.
Identify the asymptotes for the function \( g(x) = \frac{1}{x+8} + 0.5 \). (Hint: Asymptotes are determined by the shifts!)
The function \( g(x) = \frac{4}{x-6} - 9 \) is a transformation of \( f(x) = \frac{1}{x} \). Which of the following is true?
It is shifted left 6 and down 9.
It is shifted right 6 and down 9.
It is shifted right 6 and up 9.
Explain how to determine the horizontal shift of a rational function just by looking at the denominator.
Curve Shifters Answer Key
Curve Shifters
Teacher's Edition: Answer Key
MASTER KEY
The function \( f(x) = \frac{1}{x} \) is transformed to \( g(x) = \frac{7}{x+3} - 1.5 \). Describe the transformations.
Vertical stretch by 7, shifted left 3 units and down 1.5 units
Given \( g(x) = \frac{1}{x-4} + 2 \), identify the horizontal and vertical shifts from the parent function.
Horizontal shift: Right 4 units
Vertical shift: Up 2 units
The parent function is reflected across the x-axis and shifted left 5 units. Write the new equation \( g(x) \).
\( g(x) = -\frac{1}{x+5} \)
Identify the vertical stretch and vertical shift for the function \( g(x) = \frac{3}{x} - 1 \).
Vertical stretch: 3
Vertical shift: Down 1
Curve Shifters Key Continued
Which function represents \( f(x) = \frac{1}{x} \) shifted right 2 units and up 4 units?
\( g(x) = \frac{1}{x-2} + 4 \)
Describe all transformations applied to create \( g(x) = \frac{-2}{x+1} - 3 \).
1. Reflection across the x-axis (due to negative)
2. Vertical stretch by 2
3. Horizontal shift: Left 1 unit
4. Vertical shift: Down 3 units
A rational function is compressed vertically by a factor of 0.5 and shifted down 9 units. Write the equation.
\( g(x) = \frac{0.5}{x} - 9 \) or \( g(x) = \frac{1}{2x} - 9 \)
Identify the asymptotes for the function \( g(x) = \frac{1}{x+8} + 0.5 \).
Vertical Asymptote: \( x = -8 \)
Horizontal Asymptote: \( y = 0.5 \)
The function \( g(x) = \frac{4}{x-6} - 9 \) is a transformation of \( f(x) = \frac{1}{x} \). Which is true?
It is shifted right 6 and down 9.
Explain how to determine the horizontal shift of a rational function just by looking at the denominator.
"The horizontal shift is found by identifying the value of \( x \) that makes the denominator equal to zero. This value represents the vertical asymptote and indicates how far the graph has moved left or right from the origin."
