91Ó°ÊÓ

Understanding graph translations is crucial when modifying the position of a quadratic function like \( y = x^2 \). Translations involve shifting the graph in any direction without altering its shape. For instance, if you translate it up, you simply add a constant to the function. In our exercise, translating up by 3 units means adding 3 to the function: \( y = x^2 + 3 \). This shift affects only the graph's vertical position.

Horizontal translations involve adjustments to the \( x \)-value inside the function. When translating right by 2 units, subtract 2 from \( x \) resulting in \( (x-2)^2 \). So the combined equation for both translations is \( y = (x-2)^2 + 3 \).

• Translate up: Add to the entire function.
• Translate right: Subtract from \( x \).

These simple shifts ensure that the same parabolic shape moves to a new position on the graph.

Vertical stretches modify the graph's shape by making it taller or shorter. When stretching a quadratic function \( y = x^2 \), multiply the entire function by a constant greater than 1 to make it taller.

In our example, the function is stretched vertically by a factor of 3. This changes \( y = x^2 \) to \( y = 3x^2 \). The value 3 indicates the stretch factor, making each point on the graph three times further from the \( x \)-axis.

• Stretch factor \( >1 \): Graph becomes taller.
• Stretch factor \( <1 \): Graph becomes shorter.

Remember, this transformation affects the width of the parabola, making it either narrower or wider depending on the coefficient.

Reflections across the \( x \)-axis flip the graph over this line, presenting a mirror image of the original graph. For the quadratic function \( y = x^2 \), reflection is achieved by multiplying the entire function by -1.

This transformation changes \( y = x^2 \) to \( y = -x^2 \), altering the graph's direction. Consequently, every point flips vertically, plotting downward instead.

• Multiply by -1 for reflection over the \( x \)-axis.
• Inverts the graph.

This reflection can be combined with other transformations. For example, reflecting first and then translating up by 4 units results in \( y = -x^2 + 4 \), which lifts the entire flipped graph upward.