Question

how to graph transformations of quadratic functions

Answer 1

Answer:

Shift "h" units to the right, "k" units up, and reflect over the x or y axis when needed.

Step-by-step explanation:

1) I want to talk about reflections first.

• Reflections across the x-axis --> $y = ax^2$, a is the coefficient. if a is negative, then the equation should be reflected across the x-axis. This is known as a vertical reflection.
• Reflections across the y-axis --> $y=a(bx)^2$, b is the coefficient. If b is negative, then reflect the equation over the y-axis. There are cases where the reflection across the y-axis does not change anything. But, let's say its $y=(x-3)^2$... the reflection across the y-axis is different (that equation is: $y=(-x-3)^2$ )

2) Rigid transformations

• Horizontal transformations (to the left or right): $y=a(bx-h)^2$, factor out b from "bx-h" and whatever h equals is the units to the right. If h is a negative number, then you move to the left.
• Vertical transformations (up and down): $y = a(bx-h)^2+k$... k is just the units up... if k is negative then we move it down.

Example (check image for visual)

We transform $y = x^2$ to $y = -(-x-3)^2+3$ , you move right 3, then reflect across the x-axis, then reflect across y-axis, then move 3 up.

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Note: In the image, the red line is the original function, the blue one is the transformed function. See if you can follow along with the verbal instructions I gave above.