function g is a transformation of the parent function f(x)=x^2, The graph of g is a translation left 4 units down and down 2 units of the graph of f. Write the equation for g in the form y=ax^2+bx+c
To translate the graph of f left 4 units and down 2 units, we need to subtract 4 from the x-values and subtract 2 from the y-values of the points on the graph of f. So, the equation for g can be written as:
g(x) = (x - 4)^2 - 2
Expanding this equation, we get:
g(x) = x^2 - 8x + 16 - 2
Simplifying further:
g(x) = x^2 - 8x + 14
Therefore, the equation for g in the form y = ax^2 + bx + c is:
y = x^2 - 8x + 14
To find the equation for g in the form y = ax^2 + bx + c, let's analyze the given transformations.
Translation left 4 units:
We know that a left translation of a function is represented by (x + h), where h is the amount of units to shift to the left. So, for g, the translation left 4 units can be represented as (x - 4).
Translation down 2 units:
Similarly, a down translation of a function is represented by (y - k), where k is the amount of units to shift downwards. Therefore, for g, the translation down 2 units can be represented as (y - (-2)) = (y + 2).
Combining these transformations, the equation for g is:
g(x) = f(x) translated left 4 units and down 2 units
g(x) = f(x - 4) + 2
Substituting the parent function f(x) = x^2, we get:
g(x) = (x - 4)^2 + 2
Simplifying this equation, we have:
g(x) = x^2 - 8x + 14
Therefore, the equation for g in the form y = ax^2 + bx + c is y = x^2 - 8x + 14.
To find the equation for the transformed function g(x), we need to apply the given translations to the parent function f(x) = x^2.
First, let's start with the translation left 4 units. To achieve this, we replace x with (x + 4). This shifts the graph of f four units to the left.
Next, we have a translation down 2 units. To achieve this, we subtract 2 from the whole function. Therefore, we have (x + 4)^2 - 2.
To write the equation for g in the form y = ax^2 + bx + c, we need to expand and simplify the expression:
(x + 4)^2 - 2
= (x + 4)(x + 4) - 2
= (x^2 + 8x + 16) - 2
= x^2 + 8x + 14
So, the equation for g is y = x^2 + 8x + 14.