Given f(x) = x², after performing the following transformations: shift upward 76 units and shift 31 units to the right, the new function g(x)=

Question

Given f(x) = x², after performing the following transformations: shift upward 76 units and shift 31 units to the right, the new function g(x)=

Thank you,

[3.5A_5]

Answer 1

I recommend referencing the standard quadratic equation to help with these transformations.

f(x) = a(x - h)² + k

In this problem, a is equal to 1, so

f(x) = (x - h)² + k

Now with this equation, h and k represent the vertex of the parabola if you were to graph it. If both h and k are 0, you get your starting equation of f(x) = x². Now as you change h, this represents a shift along the x axis. And as you change k, this represents a shift along the y axis (similar to the y-intercept for a linear equation).

In the work problem, you need to shift the equation up (along the y-axis) by 76 units. This can be done by making k 76 in the quadratic equation. Similarly, you need to shift the equation right (along the x-axis) by 31 units, making h 31. Now we can plug these back into the quadratic equation to get g(x).

g(x) = (x - 31)² + 76.

ADDED NOTE: If you need to shift down, your k will ne negative. If you need to shift left, h will also be negative but will cancel out the subtraction sign to create a positive. For example, if you shift down 5 and left 3 you will get:

g(x) = (x + 3)² - 5.