To compare the graphs of $f(x) = 3x + 5$ and $g(x) = 3(x + 10) + 5$, let us analyze the transformation step by step:

1. Start with $f(x) = 3x + 5$:
   This is a linear function with a slope of 3 and a y-intercept of 5.

2. Transform to $g(x) = 3(x + 10) + 5$:
   Simplify $g(x)$: $g(x) = 3(x + 10) + 5 = 3x + 30 + 5 = 3x + 35$

   The slope remains the same ($3$), but the y-intercept changes from 5 to 35.

3. Interpret the transformation:
   The graph of $g(x) = 3x + 35$ is parallel to $f(x)$ because they have the same slope ($3$).
   The change in the y-intercept indicates a vertical shift: $g(x)$ is shifted upward by 30 units compared to $f(x)$.

Final Comparison:

• The graph of $g(x)$ is a vertical shift of $f(x)$ upward by 30 units.
• Both graphs have the same slope, so their steepness and orientation remain identical.

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Questions:

1. What is the effect of changing the constant in a linear function's equation on its graph?
2. How can you tell if two linear functions are parallel?
3. What happens to the graph of $f(x) = mx + b$ if $m$ changes while $b$ remains constant?
4. What are other types of transformations besides vertical shifts in linear functions?
5. How would the graph look if $g(x) = 3(x - 10) + 5$ instead?

Tip:

For linear functions, transformations like shifting, stretching, or reflecting depend on changes to the slope ($m$) and intercept ($b$). Keep these changes in mind to quickly identify transformations.