Chapter 6, Section 6.1, Question 036
The function g(x) is obtained by shifting the graph of y=x^2
If g(3) = 25
(a) g is the result of applying only a horizontal shift to y=x^2
g(x) =
(b) g is the result of applying only a vertical shift to y=x^2
g(x) =
g left-parenthesis x right-parenthesis equals
(c) g is the result of applying a horizontal shift right 2 units and an appropriate vertical shift of y=x^2
g(x) =
(a) g is the result of applying only a horizontal shift to y=x^2
g(x) = (x - h)^2, where h represents the amount of horizontal shift applied.
(b) g is the result of applying only a vertical shift to y=x^2
g(x) = x^2 + k, where k represents the amount of vertical shift applied.
(c) g is the result of applying a horizontal shift right 2 units and an appropriate vertical shift of y=x^2
g(x) = (x - 2)^2 + k
To determine the function g(x) that is obtained by shifting the graph of y=x^2, we need to know how the shifts are applied.
(a) If g(3) = 25 is the result of only a horizontal shift, it means that moving the graph to the right or left changes the x-value but does not affect the y-value. In this case, the function g(x) would be:
g(x) = x^2
(b) If g(3) = 25 is the result of only a vertical shift, it means that moving the graph up or down changes the y-value but does not affect the x-value. In this case, the function g(x) would be:
g(x) = x^2 + k, where k is the vertical shift.
(c) If g(3) = 25 is the result of a horizontal shift right 2 units and an appropriate vertical shift, it means that the graph is shifted 2 units to the right and the y-values are also adjusted. In this case, the function g(x) would be:
g(x) = (x - 2)^2 + k, where k is the vertical shift.
Without additional information about the value of k, we cannot determine the exact form of g(x) for case (b) or (c).
To answer these questions, we need to understand how shifting affects the graph of a function.
(a) If g(x) is obtained by applying only a horizontal shift to y = x^2, it means the graph of g(x) is the same as the graph of y = x^2, but shifted horizontally. To find the equation of g(x), we can determine how much the graph of y = x^2 has shifted horizontally.
Since g(3) = 25, it means that when x equals 3, g(x) equals 25. In the original graph y = x^2, when x equals 3, y equals 9. So, the shift in this case is 25 - 9 = 16 units to the right.
Therefore, g(x) = (x - 3 + 16)^2. Simplifying this expression, we get g(x) = (x + 13)^2.
(b) If g(x) is the result of applying only a vertical shift to y = x^2, it means the graph of g(x) is the same as the graph of y = x^2, but shifted vertically. To find the equation of g(x), we can determine how much the graph of y = x^2 has shifted vertically.
Since g(3) = 25, it means that when x equals 3, g(x) equals 25. In the original graph y = x^2, when x equals 3, y equals 9. So, the shift in this case is 25 - 9 = 16 units up.
Therefore, g(x) = x^2 + 16.
(c) If g(x) is the result of applying a horizontal shift right 2 units and an appropriate vertical shift to y = x^2, it means the graph of g(x) is a horizontally shifted version of y = x^2, and it is also vertically shifted.
To find the equation of g(x), we need to determine the amount of horizontal and vertical shifting. Since g(3) = 25, when x equals 3, g(x) equals 25. In the original graph y = x^2, when x equals 5, y equals 25. So, the horizontal shift in this case is 5 - 3 = 2 units to the right. And since the vertical shift is not explicitly given, we can assume it is 0.
Therefore, g(x) = (x - 2)^2.
(a) 5^2 = 25, and we want
(3-h)^2 = 25
so, 3-h = ±5
g(x) = (x+2)^2 or (x-8)^2
(b) g(x) = x^2+16
(c) shift right 2 is (x-2)^2
Since (3-2)^2 = 1,
g(x) (x-2)^2 + 24