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) Well, with the transformation g(x) = 2f(x) + 3, we can imagine that f(x) just won the lottery and is now feeling twice as awesome as before! And to celebrate, they decide to treat themselves to a fancy meal that costs 3 dollars more than usual. So the resulting point for g(x) would be (1, -2) -> (1, 2( -2) + 3) -> (1, -1)
b) Ah, with the transformation g(x) = f(x + 1) - 3, f(x) decides to take a little vacation and shifts one unit to the left. But don't worry, they'll keep in touch and let you know their new location. And as a parting gift, f(x) leaves you with a 3-dollar voucher to a local coffee shop. So the resulting point for g(x) would be (1, -2 + 1 - 3) -> (1, -4)
c) Oh, this transformation g(x) = -f(2x) means that f(x) turns into its evil twin, who likes to do everything backwards! Imagine f(x) wearing a villainous mustache and cackling. But that's not all, the evil twin also decides to stretch time by a factor of 2, making everything happen twice as fast. So the resulting point for g(x) would be (1, -f(2 * 1)) -> (1, -f(2)) -> (1, -f(x))
d) With the transformation g(x) = -f(-x - 1) + 3, f(x) decides to change their orientation and become the mirror image of their former self. To make things interesting, they also decide to play hide and seek, hiding one unit to the right. But don't worry, they'll add a nice touch of mystery to their location by keeping the 3 dollars they found as a secret surprise. So the resulting point for g(x) would be (1, -f(-1 - 1) + 3) -> (1, -f(-2) + 3) -> (1, -f(x) + 3)
a) The transformation in function g(x) = 2f(x) + 3 is a vertical stretch by a factor of 2, followed by a vertical shift upwards by 3 units. To find the resulting point, we substitute the x-coordinate of the given point into the function f(x), multiply by 2, and then add 3 to the result:
g(1) = 2f(1) + 3
Since the point (1, -2) is on the graph of f(x), we have -2 = f(1). Substituting this into the equation, we get:
g(1) = 2(-2) + 3 = -4 + 3 = -1
So the resulting point on the graph of g(x) is (1, -1).
b) The transformation in function g(x) = f(x+1) - 3 is a horizontal shift to the left by 1 unit, followed by a vertical shift downwards by 3 units. To find the resulting point, we subtract 1 from the x-coordinate of the given point, substitute it into the function f(x), and then subtract 3 from the result:
g(1-1) = f(1-1) - 3
Simplifying, we have:
g(0) = f(0) - 3
Since the point (1, -2) is on the graph of f(x), we have -2 = f(1). Substituting this into the equation, we get:
g(0) = -2 - 3 = -5
So the resulting point on the graph of g(x) is (0, -5).
c) The transformation in function g(x) = -f(2x) is a horizontal compression by a factor of 2, followed by a reflection across the x-axis. To find the resulting point, we substitute half of the x-coordinate of the given point into the function f(x), and then negate the result:
g(1/2) = -f(2(1/2))
Simplifying, we have:
g(1/2) = -f(1)
Since the point (1, -2) is on the graph of f(x), we have -2 = f(1). Substituting this into the equation, we get:
g(1/2) = -(-2) = 2
So the resulting point on the graph of g(x) is (1/2, 2).
d) The transformation in function g(x) = -f(-x-1) + 3 is a horizontal reflection across the y-axis, followed by a horizontal shift to the right by 1 unit, a vertical stretch by a factor of -1, and finally a vertical shift upwards by 3 units. To find the resulting point, we substitute the opposite of the x-coordinate of the given point into the function f(x), add 1, multiply by -1, and then add 3 to the result:
g(-1) = -f(-(-1) - 1) + 3
Simplifying, we have:
g(-1) = -f(0) + 3
Since the point (1, -2) is on the graph of f(x), we have -2 = f(1). Substituting this into the equation, we get:
g(-1) = -(-2) + 3 = 2 + 3 = 5
So the resulting point on the graph of g(x) is (-1, 5).
To describe the transformations and determine the resulting point for each of the given functions, we need to understand the effect of each transformation on the original graph of f(x). Let's go through each transformation one by one:
a) g(x) = 2f(x) + 3
In this transformation, the original function f(x) is scaled vertically by a factor of 2, and then shifted upward by 3 units. To find the resulting point, we need to apply both transformations to the given point (1, -2).
1. Scaling vertically by 2: Multiply the y-coordinate of the original point by 2.
New y-coordinate = 2 * (-2) = -4
2. Shifting upward by 3 units: Add 3 to the new y-coordinate.
Resulting y-coordinate = -4 + 3 = -1
Therefore, the resulting point for function g(x) = 2f(x) + 3 is (1, -1).
b) g(x) = f(x + 1) - 3
In this transformation, the original function f(x) is shifted horizontally to the left by 1 unit, and then shifted downward by 3 units. Again, we will use the given point (1, -2) to find the resulting point.
1. Shifting horizontally to the left by 1 unit: Subtract 1 from the x-coordinate of the original point.
New x-coordinate = 1 - 1 = 0
2. Shifting downward by 3 units: Subtract 3 from the y-coordinate of the new point.
Resulting y-coordinate = -2 - 3 = -5
Therefore, the resulting point for function g(x) = f(x + 1) - 3 is (0, -5).
c) g(x) = -f(2x)
In this transformation, the original function f(x) is compressed horizontally by a factor of 2, and then reflected across the x-axis. Let's use the given point (1, -2) to find the resulting point.
1. Compressing horizontally by a factor of 2: Divide the x-coordinate of the original point by 2.
New x-coordinate = 1 / 2 = 0.5
2. Reflection across the x-axis: Change the sign of the y-coordinate of the new point.
Resulting y-coordinate = -(-2) = 2
Therefore, the resulting point for function g(x) = -f(2x) is (0.5, 2).
d) g(x) = -f(-x - 1) + 3
In this transformation, the original function f(x) is reflected across the y-axis, shifted horizontally to the right by 1 unit, reflected across the x-axis, and then shifted upward by 3 units. Let's use the given point (1, -2) to find the resulting point.
1. Reflecting across the y-axis: Change the sign of the x-coordinate of the original point.
New x-coordinate = -(1) = -1
2. Shifting horizontally to the right by 1 unit: Add 1 to the new x-coordinate.
Resulting x-coordinate = -1 + 1 = 0
3. Reflecting across the x-axis: Change the sign of the y-coordinate of the new point.
New y-coordinate = -(-2) = 2
4. Shifting upward by 3 units: Add 3 to the new y-coordinate.
Resulting y-coordinate = 2 + 3 = 5
Therefore, the resulting point for function g(x) = -f(-x - 1) + 3 is (0, 5).
By applying the appropriate transformations to the given point (1, -2), we have determined the resulting points for each of the defined functions.