function g is a transformation of the parent function f(x)=x^2 the graph of g is a translation right 5 untits and up 1 unit of the graph of f. write the equation for g in the form y=ax^2+bx+c.
To translate the graph of the parent function f(x) = x^2 to the right 5 units and up 1 unit, we can use the equation:
g(x) = a(x - h)^2 + k
where (h, k) represents the coordinates of the vertex of the graph. In this case, h represents the horizontal translation and k represents the vertical translation.
Since we want to translate the graph to the right 5 units, h = -5. And since we want to translate it up 1 unit, k = 1.
Substituting these values into the equation, we get:
g(x) = a(x - (-5))^2 + 1
Simplifying further:
g(x) = a(x + 5)^2 + 1
This is the equation for g in the form y = ax^2 + bx + c, where a = 1, b = 10, and c = 1.
To find the equation of the function g, which is a transformation of the parent function f(x) = x^2, we need to consider the translations applied to f(x).
1. Translation right 5 units:
For a translation right 5 units, we need to replace x in f(x) with (x - 5). This ensures that the graph of f is shifted 5 units to the right. Now we have:
f(x) = (x - 5)^2
2. Translation up 1 unit:
For a translation up 1 unit, we need to add 1 to the entire expression. Now we have:
g(x) = (x - 5)^2 + 1
To put the equation in the form y = ax^2 + bx + c, let's expand and simplify the equation:
g(x) = (x - 5)^2 + 1
g(x) = (x - 5)(x - 5) + 1
g(x) = (x^2 - 10x + 25) + 1
g(x) = x^2 - 10x + 26
Therefore, the equation for g(x) in the form y = ax^2 + bx + c is:
y = x^2 - 10x + 26
To find the equation for the transformed function g(x), we need to consider the translations made in the x and y directions.
For a translation right 5 units, we need to replace x with (x - h), where h is the amount of units translated. In this case, h = 5.
For a translation up 1 unit, we need to add a constant value to the function.
So, applying the translations to the parent function f(x) = x^2, we get:
g(x) = (x - 5)^2 + 1
Expanding the expression:
g(x) = (x - 5)(x - 5) + 1
= x^2 - 10x + 25 + 1
= x^2 - 10x + 26
Therefore, the equation for g in the form y = ax^2 + bx + c is:
y = x^2 - 10x + 26.