Transformations are really important when we want to understand how different types of functions look on a graph. These changes can move, stretch, shrink, or flip the graphs. Let’s see how this works for three kinds of functions: linear, quadratic, and cubic.
Linear functions, like (f(x) = mx + c), create straight lines. Here’s how transformations change these lines:
Vertical Shifts: If we add or subtract a number, it moves the graph up or down. For example, in (f(x) = x + 2), the line moves up by 2 units.
Horizontal Shifts: When we change the input, like in (f(x) = x - 3), the graph slides to the right by 3 units.
Reflections: If we multiply the function by -1, like in (f(x) = -x), the line flips over the x-axis.
Quadratic functions, shown as (f(x) = ax^2 + bx + c), create U-shaped graphs called parabolas. Here’s how transformations affect them:
Vertical Shifts: Adding a number to the function moves the parabola up or down. For example, (f(x) = x^2 + 3) makes it go up by 3 units.
Horizontal Shifts: For a function like (f(x) = (x - 2)^2), the graph moves to the right by 2 units.
Stretching or Compressing: The number (a) changes how wide or narrow the parabola is. If (a > 1), like in (f(x) = 2x^2), the shape becomes steeper. If (0 < a < 1), like (f(x) = \frac{1}{2}x^2), it becomes wider.
Reflections: If we change the sign of (a), like in (f(x) = -x^2), the parabola flips upside down.
Cubic functions are written as (f(x) = ax^3 + bx^2 + cx + d) and can be a little more complex:
Vertical Shifts: Just like quadratics, in (f(x) = x^3 + 1), the graph moves up by 1 unit.
Horizontal Shifts: For (f(x) = (x - 1)^3), the graph shifts to the right by 1 unit.
Stretching or Compressing: Changing (a) also affects cubic functions. For example, (f(x) = 3x^3) makes it stretch tall, while (f(x) = \frac{1}{4}x^3) makes it shorter.
Reflections: If we flip it around with (f(x) = -x^3), the graph mirrors itself over the x-axis.
Understanding these transformations helps us see and predict how different functions behave, making graphing much easier!
Transformations are really important when we want to understand how different types of functions look on a graph. These changes can move, stretch, shrink, or flip the graphs. Let’s see how this works for three kinds of functions: linear, quadratic, and cubic.
Linear functions, like (f(x) = mx + c), create straight lines. Here’s how transformations change these lines:
Vertical Shifts: If we add or subtract a number, it moves the graph up or down. For example, in (f(x) = x + 2), the line moves up by 2 units.
Horizontal Shifts: When we change the input, like in (f(x) = x - 3), the graph slides to the right by 3 units.
Reflections: If we multiply the function by -1, like in (f(x) = -x), the line flips over the x-axis.
Quadratic functions, shown as (f(x) = ax^2 + bx + c), create U-shaped graphs called parabolas. Here’s how transformations affect them:
Vertical Shifts: Adding a number to the function moves the parabola up or down. For example, (f(x) = x^2 + 3) makes it go up by 3 units.
Horizontal Shifts: For a function like (f(x) = (x - 2)^2), the graph moves to the right by 2 units.
Stretching or Compressing: The number (a) changes how wide or narrow the parabola is. If (a > 1), like in (f(x) = 2x^2), the shape becomes steeper. If (0 < a < 1), like (f(x) = \frac{1}{2}x^2), it becomes wider.
Reflections: If we change the sign of (a), like in (f(x) = -x^2), the parabola flips upside down.
Cubic functions are written as (f(x) = ax^3 + bx^2 + cx + d) and can be a little more complex:
Vertical Shifts: Just like quadratics, in (f(x) = x^3 + 1), the graph moves up by 1 unit.
Horizontal Shifts: For (f(x) = (x - 1)^3), the graph shifts to the right by 1 unit.
Stretching or Compressing: Changing (a) also affects cubic functions. For example, (f(x) = 3x^3) makes it stretch tall, while (f(x) = \frac{1}{4}x^3) makes it shorter.
Reflections: If we flip it around with (f(x) = -x^3), the graph mirrors itself over the x-axis.
Understanding these transformations helps us see and predict how different functions behave, making graphing much easier!