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Questions and Answers
The graph of $f(x) = |x|$ is vertically stretched by a factor of 2, then shifted 3 units downward. Write the equation of the transformed function, $g(x)$.
The graph of $f(x) = |x|$ is vertically stretched by a factor of 2, then shifted 3 units downward. Write the equation of the transformed function, $g(x)$.
$g(x) = 2|x| - 3$
Describe the transformations applied to the graph of $f(x) = \sqrt{x}$ to obtain the graph of $g(x) = \sqrt{x+1} - 2$.
Describe the transformations applied to the graph of $f(x) = \sqrt{x}$ to obtain the graph of $g(x) = \sqrt{x+1} - 2$.
The graph of $f(x)$ is shifted 1 unit to the left and 2 units downward.
The function $f(x) = x^3$ is transformed to $g(x)$ by a horizontal shift of 2 units to the right, followed by a reflection across the x-axis. Write the equation for $g(x)$.
The function $f(x) = x^3$ is transformed to $g(x)$ by a horizontal shift of 2 units to the right, followed by a reflection across the x-axis. Write the equation for $g(x)$.
$g(x) = -(x-2)^3$
Given the function $f(x) = x^2$, describe the single transformation that results in the function $g(x) = (2x)^2$. What effect does the transformation have on the graph?
Given the function $f(x) = x^2$, describe the single transformation that results in the function $g(x) = (2x)^2$. What effect does the transformation have on the graph?
Suppose the point (1, 4) lies on the graph of $y = f(x)$. After a series of transformations, including a horizontal shift 3 units to the left and a vertical shift 2 units upward, what are the new coordinates of the transformed point on the graph of the new function $g(x)$?
Suppose the point (1, 4) lies on the graph of $y = f(x)$. After a series of transformations, including a horizontal shift 3 units to the left and a vertical shift 2 units upward, what are the new coordinates of the transformed point on the graph of the new function $g(x)$?
Flashcards
Vertical Shift
Vertical Shift
Shifts the graph of f(x) vertically. If 'b' is positive, the shift is upward; if 'b' is negative, the shift is downward.
Horizontal Shift
Horizontal Shift
Shifts the graph of f(x) horizontally. If 'a' is positive, the shift is to the left; if 'a' is negative, the shift is to the right. It's the opposite of what you might expect!
f(x - a) Transformation
f(x - a) Transformation
Replacing 'x' with '(x - a)' in a function shifts the graph horizontally. Positive 'a' shifts right, negative 'a' left.
Describing Transformations
Describing Transformations
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Combining Shifts
Combining Shifts
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Study Notes
- The topic is application of functions involving transformations such as shifting, reflection, stretching, compressing and rotation.
Vertical Shift
- Given b > 0, the graph of f(x) + b shifts along the y-axis.
- For y = f(x) + b, the graph shifts upward by b units.
- For y = f(x) - b, the graph shifts downward by b units.
- Vertically shifting is:
- Upward when b>0
- Downward when b<0
Horizontal Shift
- For y = f(x − a), the graph shifts horizontally by a units.
- If a > 0, the graph moves to the right.
- If a < 0, the graph moves to the left.
- Remember, for horizontal shifts, it is opposite of what you see in the brackets.
Reflections
- Graphs of y = f(x) and y = −f(x) are reflectionsm about the x-axis.
- Graphs of y = f(x) and y = f(-x) are reflections about the y-axis.
Vertical Stretch and Shrink
- Given y = cf(x) is a function, and c is a positive real number.
- If c > 1, the graph of y = cf(x) is the graph of y = f(x) vertically stretched by multiplying each of its y-coordinates by c.
- If 0 < c < 1, the graph of y = cf(x) is the graph of y = f(x) vertically shrunk by multiplying each of its y-coordinates by c.
Horizontal Stretch and Shrink
- Given y = f(x) is a function, and c is a positive real number.
- If c > 1, the graph of y = f(cx) is the graph of y = f(x) horizontally shrunk by multiplying each of its x-coordinates.
- If 0 < c < 1, the graph of y = f(cx) is the graph of y = f(x) horizontally stretched by multiplying each of its x-coordinates.
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Description
Explore transformations of functions including vertical and horizontal shifts. Learn about reflections across the x and y-axis. Stretching or compressing a function can be done vertically or horizontally.
