The graph of the function g(x) has the same shape and direction of opening as the graph of f(x)= 3(x-2)^2+9. The graph of g(x) has a vertex that is 2 units to the right and 5 units down from the vertex of the graph of f(x).
A.) Determine an equation of the function g(x)
B.) State the domain and range
C.) Write another function h(x) with the same vertex and shape but whose graph opens in the opposite direction.
If you could show me step by step how to to do, that would be awesome. Thanks
the vertex of f(x) is (2,9)
The vertex of g(x) is to the right 2 units and 5 units down, so its vertex must be (4,4)
since the "shape" has to be the same, we need the same 3 in front
so g(x) = 3(x-4)^2 + 4
domain: the set of real numbers
range: y ≥ 4 , since the parabola opens up
http://www.wolframalpha.com/input/?i=plot+y+%3D+3(x-2)%5E2%2B9+,+y+%3D+3(x-4)%5E2+%2B+4
c) how about h(x) = -3(x-4)^2 + 4
http://www.wolframalpha.com/input/?i=plot+y+%3D+3(x-4)%5E2+%2B+4,+y+%3D+-3(x-4)%5E2+%2B+4
A) To find the equation of the function g(x), we need to determine the new vertex coordinates based on the given transformations.
1. The vertex of the original function f(x) is (2, 9). Since the vertex of g(x) is 2 units to the right and 5 units down, the new vertex of g(x) is (2 + 2, 9 - 5), which simplifies to (4, 4).
2. Since the shape and direction of opening remain the same, the equation of g(x) will be of the form g(x) = a(x - h)^2 + k, where (h, k) represents the vertex. Substituting the new vertex coordinates, we have g(x) = a(x - 4)^2 + 4.
B) To determine the domain and range, we consider the characteristics of the parabola.
1. The domain of a quadratic function is always all real numbers, so the domain of g(x) is (-∞, ∞).
2. Since the parabola opens upward (based on the shape and direction information), the vertex represents the minimum point. Thus, the range of g(x) is the set of all y-values greater than or equal to the y-coordinate of the vertex. Therefore, the range is [4, ∞).
C) To write another function h(x) with the same vertex and shape but opens in the opposite direction, we need to reflect the graph of f(x) about the x-axis.
1. The vertex of h(x) will be the same as f(x) since they share the same vertex coordinates. Thus, the vertex of h(x) is (2, 9).
2. To create the opposite direction, the equation of h(x) will be of the form h(x) = -a(x - h)^2 + k. Substituting the vertex coordinates, we have h(x) = -a(x - 2)^2 + 9.
Therefore, the step-by-step solutions are as follows:
A) The equation of g(x): g(x) = a(x - 4)^2 + 4
B) Domain of g(x): (-∞, ∞) and range of g(x): [4, ∞)
C) The equation of h(x): h(x) = -a(x - 2)^2 + 9
Sure! Let's go step by step to solve this problem:
A.) Determine an equation of the function g(x):
1. Start with the given function f(x) = 3(x - 2)^2 + 9. Rewrite it in vertex form:
f(x) = 3(x - h)^2 + k, where (h, k) is the vertex of the parabola.
Comparing this with the given function, we can see that the vertex of f(x) is (2, 9).
2. Since g(x) has the same shape and direction of opening as f(x), the vertex of g(x) should be 2 units to the right and 5 units down from the vertex of f(x). Therefore, the vertex of g(x) is (2 + 2, 9 - 5) = (4, 4).
3. Now, let's write the equation of g(x) in vertex form:
g(x) = a(x - 4)^2 + 4, where a is a constant that determines the "steepness" of the parabola.
4. We still need to find the value of a. To do that, we can use another point on the graph of f(x) or g(x). Let's use the vertex of f(x) (2, 9).
Plug in these coordinates into the equation g(x):
9 = a(2 - 4)^2 + 4
9 = a(-2)^2 + 4
9 = 4a + 4
4a = 9 - 4
4a = 5
a = 5/4
5. Substitute the value of a back into the equation for g(x):
g(x) = (5/4)(x - 4)^2 + 4
So, the equation of the function g(x) is g(x) = (5/4)(x - 4)^2 + 4.
B.) State the domain and range:
The domain of both f(x) and g(x) is all real numbers, since there are no restrictions on the possible input values x.
For the range, we can observe that the vertex of g(x) is the lowest point on the graph. Since a > 0 in the equation of g(x), the parabola opens upward, and therefore, the range of g(x) is all real numbers greater than or equal to 4.
So, the domain is (-∞, ∞) and the range is [4, ∞).
C.) Write another function h(x) with the same vertex and shape but whose graph opens in the opposite direction:
To find a function with the same vertex and shape but with a graph that opens in the opposite direction, we will simply change the sign of the coefficient a.
1. The vertex of h(x) is still (4, 4).
2. Change the sign of the coefficient a:
h(x) = (-5/4)(x - 4)^2 + 4
So, the equation of the function h(x) is h(x) = (-5/4)(x - 4)^2 + 4.