The point (1,-2) is on the graph of f(x). Describe the following transformations on f(x), and determine the resulting point.
a)
g(x)=2f(x)+3
b)
g(x)=f(x+1)−3
c)
g(x)=−f(2x)
d)
g(x)=−f(−x−1)+3
(a) stretch by 2 and shift up 3
(b) shift left 1 and down 3
(c) shrink by 2 and reflect in x-axis
(d) shift left 1, reflect in y-axis, reflect in x-axis, shift up 3.
Just plug in your numbers to find the image of the point.
a) In this transformation, the graph of f(x) is stretched vertically by a factor of 2 and shifted upward by 3 units. To find the resulting point, we apply these transformations to the given point (1, -2) on f(x):
Vertical stretch by 2: y-coordinate becomes -2 * 2 = -4
Shift upward by 3 units: y-coordinate becomes -4 + 3 = -1
So, the resulting point on g(x) is (1, -1).
b) In this transformation, the graph of f(x) is shifted left by 1 unit and downward by 3 units. To find the resulting point, we apply these transformations to the given point (1, -2) on f(x):
Shift left by 1 unit: x-coordinate becomes 1 - 1 = 0
Shift downward by 3 units: y-coordinate becomes -2 - 3 = -5
So, the resulting point on g(x) is (0, -5).
c) In this transformation, the graph of f(x) is reflected across the y-axis and horizontally compressed by a factor of 2. To find the resulting point, we apply these transformations to the given point (1, -2) on f(x):
Reflect across y-axis: x-coordinate becomes -1
Horizontal compression by 2: x-coordinate becomes -1 / 2 = -0.5
So, the resulting point on g(x) is (-0.5, -2).
d) In this transformation, the graph of f(x) is reflected across the x-axis, then the y-axis, shifted right by 1 unit, and shifted upward by 3 units. To find the resulting point, we apply these transformations to the given point (1, -2) on f(x):
Reflect across x-axis: y-coordinate becomes 2
Reflect across y-axis: x-coordinate becomes -1
Shift right by 1 unit: x-coordinate becomes -1 + 1 = 0
Shift upward by 3 units: y-coordinate becomes 2 + 3 = 5
So, the resulting point on g(x) is (0, 5).
Remember, these transformations are altering the shape and position of the graph of f(x) to create the graph of g(x).
a) Transformation: Vertical stretch by a factor of 2, followed by a vertical translation upwards by 3 units.
Resulting point: The point (1, -2) on f(x) will become the point (1, -1) on the graph of g(x).
b) Transformation: Horizontal shift to the left by 1 unit, followed by a vertical translation downwards by 3 units.
Resulting point: The point (1, -2) on f(x) will become the point (0, -5) on the graph of g(x).
c) Transformation: Horizontal compression by a factor of 2, followed by a reflection across the x-axis.
Resulting point: The point (1, -2) on f(x) will become the point (0.5, 2) on the graph of g(x).
d) Transformation: Reflection across the y-axis, followed by a horizontal shift to the right by 1 unit, and finally a vertical translation upwards by 3 units.
Resulting point: The point (1, -2) on f(x) will become the point (-2, 1) on the graph of g(x).
To describe the transformations and determine the resulting point for each function, we need to understand the effects of each transformation and apply them to the given point (1, -2).
a) Transformation: g(x) = 2f(x) + 3
Explanation: This transformation involves scaling vertically by a factor of 2 and shifting upward by 3 units.
1. Scaling vertically by a factor of 2:
Multiply the y-coordinate of the given point (-2) by 2 to get -4.
2. Shifting upward by 3 units:
Add 3 to the y-coordinate obtained in step 1 (-4 + 3 = -1).
Result: The resulting point on the graph of g(x) is (1, -1).
b) Transformation: g(x) = f(x + 1) - 3
Explanation: This transformation involves shifting the graph horizontally to the left by 1 unit and shifting downward by 3 units.
1. Shifting horizontally to the left by 1 unit:
Subtract 1 from the x-coordinate of the given point (1 - 1 = 0).
2. Shifting downward by 3 units:
Subtract 3 from the y-coordinate of the given point (-2 - 3 = -5).
Result: The resulting point on the graph of g(x) is (0, -5).
c) Transformation: g(x) = -f(2x)
Explanation: This transformation involves compressing the graph horizontally by a factor of 2 and reflecting it over the x-axis.
1. Compressing horizontally by a factor of 2:
Divide the x-coordinate of the given point (1) by 2 to get 0.5.
2. Reflecting over the x-axis:
Take the negative of the y-coordinate of the given point (-2) to get 2.
Result: The resulting point on the graph of g(x) is (0.5, 2).
d) Transformation: g(x) = -f(-x - 1) + 3
Explanation: This transformation involves reflecting the graph over the y-axis, shifting horizontally to the right by 1 unit, reflecting over the x-axis, and shifting upward by 3 units.
1. Reflecting over the y-axis:
Take the negative of the x-coordinate of the given point (1) to get -1.
2. Shifting horizontally to the right by 1 unit:
Add 1 to the x-coordinate obtained in step 1 (-1 + 1 = 0).
3. Reflecting over the x-axis:
Take the negative of the y-coordinate of the given point (-2) to get 2.
4. Shifting upward by 3 units:
Add 3 to the y-coordinate obtained in step 3 (2 + 3 = 5).
Result: The resulting point on the graph of g(x) is (0, 5).
By understanding the transformations involved in each function and applying them step-by-step, we can determine the resulting point on the graph.