If f(x)=2(x-3)^2-5 find the vertex, the x intercepts the y intercepts sketch the graph, choose two points on the graph and find the rate of change
F(x) = Y = 2(x-3)^22(-5
Y = 2(x^2--6x+9)-5
Y = 2x^2-12x+18-5
Y = 2x^2-12x+13
h = -B/2A = 12/4 = 3
k = 2(3-3)^2-5 = -5
V(h,k) = (3,-5)
Use Quad. Formula and get:
X = 4.5,and 1.5. = X- Intercepts
To find Y-int.,replace x in given Eq
with 0 and solve for y.
Y-int = 4.
Correction: F(x) = Y = 2(x-3)^2-5
Use the following points for graphing:
(X,Y)
(1,3)
(2,-7)
V(3,-5)
(4,-3)
(5,3)
Correction: (2,-7) should be (2,-3).
To find the vertex of the quadratic function f(x) = 2(x-3)^2 - 5, you can use the vertex form of a quadratic equation, which is given by f(x) = a(x-h)^2 + k. In this equation, (h, k) represents the coordinates of the vertex.
Comparing the given function f(x) = 2(x-3)^2 - 5 to the vertex form, we see that a = 2, h = 3, and k = -5. Therefore, the vertex of the function is (3, -5).
To find the x-intercepts, set f(x) = 0 and solve for x. In this case, we have:
2(x-3)^2 - 5 = 0
Adding 5 on both sides:
2(x-3)^2 = 5
Dividing by 2:
(x-3)^2 = 2.5
Taking the square root of both sides:
x - 3 = ±√2.5
Adding 3 on both sides:
x = 3 ± √2.5
Therefore, the x-intercepts are approximately x = 3 + √2.5 and x = 3 - √2.5.
To find the y-intercept, set x = 0 in the function and solve for f(x):
f(0) = 2(0-3)^2 - 5
f(0) = 2(-3)^2 - 5
f(0) = 2(9) - 5
f(0) = 18 - 5
f(0) = 13
So, the y-intercept is (0, 13).
To sketch the graph, plot the vertex at (3, -5). Draw a smooth curve that opens upward since the coefficient of x^2 (which is 2) is positive. Also, plot the x-intercepts at approximately x = 3 + √2.5 and x = 3 - √2.5. Finally, plot the y-intercept at (0, 13).
To find the rate of change, choose two points on the graph and determine the slope between those points. Let's choose the points (1, -1) and (5, 13) on the graph.
The formula to find the rate of change (slope) between two points (x1, y1) and (x2, y2) is given by:
rate of change = (y2 - y1) / (x2 - x1)
Using the points (1, -1) and (5, 13), we have:
rate of change = (13 - (-1)) / (5 - 1)
rate of change = 14 / 4
rate of change = 3.5
So, the rate of change between these two points on the graph is 3.5.