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Quadratic function transformations involve shifting, stretching, compressing, and reflecting the graph of parabola. Vertical and horizontal shifts are made by adding constants to the function or its input, while vertical stretching or compressing is achieved by multiplying by a constant. Reflections over the x-axis occur by multiplying the function by -1, altering the graph's position and shape while retaining its parabolic form.

Algebra 2

## How Will This Worksheet on "Quadratic Functions Transformations" Benefit Your Student's Learning?

• Helps students improve their skills in drawing and understanding quadratic graphs.
• Teaches students to modify functions in specific ways, enhancing their problem-solving abilities.
• Demonstrates the application of quadratic functions to real-world situations, making abstract concepts easier to grasp.
• Deepens understanding of how algebraic transformations affect the graph.
• Helps students identify and correct errors in their graphs and calculations by observing how transformations change the function.
• Encourages creative thinking by exploring different ways to transform functions to achieve specific goals or solve problems.

## How to Quadratic Functions Transformations?

• Add or subtract constants to shift the graph vertically (up/down) or horizontally (left/right). For example, $$y = (x - h)^2 + k$$ shifts the graph right by $$h$$ units and up by $$k$$ units.
• Multiply the function by a constant to stretch or compress it vertically. For instance, $$y = a(x - h)^2$$ stretches the graph if $$|a| > 1$$ and compresses it if $$0 < |a| < 1$$.
• Multiply the function by -1 to reflect it over the x-axis. For example, $$y = -x^2$$ flips the graph upside down.
• Apply multiple transformations together, such as $$y = a(x - h)^2 + k$$, to shift, stretch/compress, and reflect the graph in a single equation.

## Solved Example

Q. Find $g(x)$, where $g(x)$ is the translation $8$ units up of $f(x) = x^2$.$\newline$Write your answer in the form $a(x – h)^2 + k$, where $a$, $h$, and $k$ are integers.$\newline$$g(x) =$ ______$\newline$
Solution:
1. Identify transformation rule: Identify the transformation rule for translating a function vertically. To translate a function $k$ units up, we add $k$ to the original function $f(x)$.
2. Apply transformation rule: Apply the transformation rule to the given function $f(x) = x^2$. Since we want to translate the function $8$ units up, we set $k$ to $8$ and add it to $f(x)$.$\newline$$g(x) = f(x) + 8$
3. Substitute given function: Substitute the given $f(x)$ into the transformation equation to find $g(x)$.$\newline$$g(x) = x^2 + 8$
4. Rewrite function in desired form: Rewrite $g(x)$ in the desired form $a(x - h)^2 + k$. Since there is no horizontal shift, $h$ is $0$. The coefficient $a$ is $1$ because the shape of the parabola does not change, only its position. The value of $k$ is $8$, representing the vertical shift.$\newline$$g(x) = 1(x - 0)^2 + 8$

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