Transformations can really change how a quadratic function looks on a graph! Let’s break down some important effects:
Vertical Shifts: If you add or subtract a number, like in ( f(x) = x^2 + 3 ), the graph moves up or down. So, adding 3 moves it up!
Horizontal Shifts: When you change what you input, for example in ( f(x) = (x - 2)^2 ), the graph shifts left or right. Here, subtracting 2 moves it to the right.
Stretching and Compressing: Changing the number in front of ( x^2 ) makes the graph either wider or narrower. For example, in ( f(x) = 2x^2 ), the graph gets narrower. On the other hand, ( f(x) = \frac{1}{2}x^2 ) makes it stretch wider.
Reflections: If there's a negative sign, like in ( f(x) = -x^2 ), it flips the graph upside down over the x-axis.
Think of it this way: you have a whole box of tools to change a quadratic function and make it look the way you want!
Transformations can really change how a quadratic function looks on a graph! Let’s break down some important effects:
Vertical Shifts: If you add or subtract a number, like in ( f(x) = x^2 + 3 ), the graph moves up or down. So, adding 3 moves it up!
Horizontal Shifts: When you change what you input, for example in ( f(x) = (x - 2)^2 ), the graph shifts left or right. Here, subtracting 2 moves it to the right.
Stretching and Compressing: Changing the number in front of ( x^2 ) makes the graph either wider or narrower. For example, in ( f(x) = 2x^2 ), the graph gets narrower. On the other hand, ( f(x) = \frac{1}{2}x^2 ) makes it stretch wider.
Reflections: If there's a negative sign, like in ( f(x) = -x^2 ), it flips the graph upside down over the x-axis.
Think of it this way: you have a whole box of tools to change a quadratic function and make it look the way you want!