Write the function f(x)= 2x^2-4x+7 in standard form, then identify the vertex and the equation for the axis of symmetry.
To write the function f(x) = 2x^2 - 4x + 7 in standard form, we need to expand and simplify the equation.
So, let's start by multiplying each term individually:
f(x) = 2x^2 - 4x + 7
Now, we square each term:
f(x) = 2x * 2x - 4 * 4x + 7
This simplifies to:
f(x) = 4x^2 - 16x + 7
Now, we have the function f(x) in standard form, which is f(x) = 4x^2 - 16x + 7.
To identify the vertex and the equation for the axis of symmetry, we need to convert the equation to vertex form, which is of the form f(x) = a(x - h)^2 + k, where (h, k) represents the vertex.
Let's complete the square to convert the equation to vertex form:
f(x) = 4x^2 - 16x + 7
First, we take out the common factor of 4 from the first two terms:
f(x) = 4(x^2 - 4x) + 7
Next, we need to complete the square inside the parentheses. To do this, we take half the coefficient of the x-term (-4) and square it to get 4.
Adding and subtracting 4 within the parentheses, we have:
f(x) = 4(x^2 - 4x + 4 - 4) + 7
Now, we can rewrite this as:
f(x) = 4((x - 2)^2 - 4) + 7
Simplifying further, we have:
f(x) = 4(x - 2)^2 - 16 + 7
f(x) = 4(x - 2)^2 - 9
Now, we have the equation f(x) = 4(x - 2)^2 - 9 in vertex form.
From the vertex form, we can identify the vertex, which is (h, k). In this case, h = 2 and k = -9. Therefore, the vertex is (2, -9).
To find the equation for the axis of symmetry, we look at the x-value of the vertex. So, the equation for the axis of symmetry is x = 2.