Translation of parabola in cartesian plane

Question

Reading through my year 11 textbook (second last year for highschool in Australia). I have just entered the section on transformations.

I'm curious as to why these values of $h$ translate the graph. Also, why is it not (x+h)^2 but (x-h)^2. None of this is explained and so in order to understand how this formula works, and not just memorize how it works, I would like to know the why behind these things.

Also, the textbook here does not explain why when you extract the value of h from the binomial expression, you reverse the sign to get $h$? What's with that?

Answer 1

Some examples may be helpful. With $y = (x - 2)^2$ the graph, a parabola, appears to be shifted to the right by $2$. With the original equation, $y = x^2$, we obtain $y = 0$ when $x = 0$. However, with $y = (x -2)^2$, we get $y = 0$ when $x = +2$, i.e. we subtracted $2$ from $x$ in the equation, so we need add it back and look at $x = 2$ to get the same answer. In a way the sign changes as you correctly noted. This is true for any other value of $x$ or $h$.

Answer 2

Let's compare the function of $f(x)$ and $g(x)=f(x+h)$.

Notice that we have $g(x-h)=f((x-h)+h)=f(x)$.

Hence $(x, f(x))$ is effectively shifted to $(x-h, f(x))$, hence the graph is being translated.

As for your question of how to obtain the value of $h$.

We are talking about the function of the form of $(x\color{red}-\color{blue}h)^2$, the sign in front of $h$ is negative.

Hence $(x\color{red}-\color{blue}2)^2$ corresponds to value $h=2$.

$(x+3)^2=(x\color{red}-\color{blue}{(-3)})^2$ corresponds to value $h=-3$.

Answer 3

When you solve $f(x) = (x+h)^2 = 0$ you're finding all the points where the graph intersects the x-axis. Changing the value of $h$ changes the solution to $f(x) = (x+h)^2 = 0$ so the graph gets moved around - either to the left or the the right. The reason the signs are opposite is because when you solve the equation and isolate $x$ you'll get $-h$ - the opposite of what ever is in the parentheses.