How do I write the quadratic function
f(x) = 1/2 (6x^2 - 24x + 22) in standard form, and go about graphing the function?
You would complete the square, but let's first clean it up a bit
f(x) = 3x^2 - 12x + 11
now ...
= 3(x^2 - 4x + ..... ) + 11
= 3(x^2 - 4x + 4 - 4) + 11
= 3((x-2)^2 - 4) + 11
= 3(x-2)^2 - 12 + 11
= 3(x-2)^2 - 1
up to you ....
To write the quadratic function f(x) = 1/2 (6x^2 - 24x + 22) in standard form and graph it, follow these steps:
Step 1: Expand the equation
f(x) = 1/2 (6x^2 - 24x + 22)
Distributing the 1/2, we get:
f(x) = 3x^2 - 12x + 11
Step 2: Rearrange in standard form
The standard form for a quadratic function is:
f(x) = ax^2 + bx + c
Comparing this to our equation, we have:
a = 3
b = -12
c = 11
Rearranging, we get:
f(x) = 3x^2 - 12x + 11
Therefore, the quadratic function f(x) in standard form is:
f(x) = 3x^2 - 12x + 11
Step 3: Graphing the function
To graph the quadratic function, follow these steps:
a) Identify the vertex:
The vertex of a quadratic function in the form f(x) = ax^2 + bx + c can be calculated using the formulas:
h = -b / (2a)
k = f(h)
For our function f(x) = 3x^2 - 12x + 11, let's calculate the vertex:
h = -(-12) / (2 * 3) = 2
k = f(2) = 3(2)^2 - 12(2) + 11 = -7
So, the vertex is (2, -7).
b) Determine the axis of symmetry:
The axis of symmetry is a vertical line passing through the vertex. Its equation is:
x = h
In our case, the axis of symmetry is x = 2.
c) Find the y-intercept:
To find the y-intercept, substitute x = 0 into the function and calculate f(0):
f(0) = 3(0)^2 - 12(0) + 11 = 11
So, the y-intercept is (0, 11).
d) Find two additional points:
Choose two other values for x and calculate the corresponding y-values. For example, let's use x = -2 and x = 4:
For x = -2:
f(-2) = 3(-2)^2 - 12(-2) + 11 = 29
So, the point is (-2, 29).
For x = 4:
f(4) = 3(4)^2 - 12(4) + 11 = 11
So, the point is (4, 11).
e) Plot the points and draw the graph:
Start by plotting the vertex (2, -7), the y-intercept (0, 11), and the two additional points (-2, 29) and (4, 11). Then, draw a smooth curve that passes through these points. Remember to consider the shape of the quadratic function (in this case, a positive leading coefficient indicates an upward-facing parabola).
This should provide you with a visual representation of the graph of the quadratic function f(x) = 3x^2 - 12x + 11.