Question

1. $g(x)=x^{2}-2$
2. $g(x)=\frac{1}{2}|x|$
3. $g(x)=4x$
4. $g(x)=|5x|$
5. which types of transformations in exercises 4 - 11 do not change the shape of a graph? which types of transformations change the shape of a graph? explain.

Explanation:

Step1: Recall transformation types

Transformations include translations (shifts), reflections, dilations (stretches and compressions). Translations and reflections are rigid - motions and do not change the shape of the graph. Dilations change the shape of the graph.

Step2: Analyze given functions

For linear functions like $g(x)=4x$, it is a non - vertical line. For absolute - value functions like $g(x)=\frac{1}{2}|x|$ and $g(x)=|5x|$, and quadratic functions like $g(x)=x^{2}-2$.

• Translations (horizontal or vertical shifts) just move the graph around without changing its shape. For example, if we have $y = f(x)$ and $y=f(x)+k$ (vertical shift) or $y = f(x - h)$ (horizontal shift), the shape remains the same. Reflections (over $x$ - axis: $y=-f(x)$; over $y$ - axis: $y = f(-x)$) also do not change the shape.
• Dilations change the shape. For a function $y = af(x)$ (vertical dilation) or $y=f(bx)$ (horizontal dilation), the graph is stretched or compressed. For $g(x)=4x$, compared to $y = x$, it is a vertical dilation. For $g(x)=\frac{1}{2}|x|$ compared to $y = |x|$, it is a vertical compression, and for $g(x)=|5x|$ compared to $y = |x|$, it is a horizontal compression.

Answer:

Translations and reflections do not change the shape of a graph because they are rigid motions that only move or flip the graph. Dilations change the shape of a graph because they stretch or compress the graph either vertically or horizontally.